Grundlehren der
mathematischen Wissenschaften 293
A Series of Comprehensive Studies in Mathematics
Editors
S. S. Chern B. Eckmann P. de la Harpe
H. Hironaka F. Hirzebruch N. Hitchin
L. Hormander M.-A. Knus A. Kupiainen
J. Lannes G. Lebeau M. Ratner D. Serre
Ya. G.Sinai N. J.A. Sloane J.Tits
M.Waldschmidt S.Watanabe
Managing Editors
M. Berger J. Coates S. R. S. Varadhan

Springer
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Daniel Revuz
Marc Yor
Continuous Martingales
and Brownian Motion
Third Edition
With 8 Figures
Springer

Daniel Revuz
Universite de Paris VII
Departement de Mathematiques
2, place Jussieu
F-75251 Paris Cedex 05, France
Marc Yor
Universite Pierre et Marie Curie
Laboratoire de Probabilites
4, place Jussieu, Tour 56
F-75252 Paris Cedex 05, France
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Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Revuz, Daniel: Continuous martingales and Brownian motion / Daniel Revuz;
Marc Yor. - 3. ed. - Berlin; Heidelberg; New York; Barcelona; Hong Kong;
London; Milan; Paris; Singapore; Tokyo: Springer, 1999
(Grundlehren der mathematischen Wissenschaften; 293)
ISBN 3-540-64325-7
Mathematics Subject Classification A991): 6oGo7, 60H05
ISSN 0072-7830
ISBN 3-540-64325-7 Springer-Verlag Berlin Heidelberg New York
ISBN 3-540-57622-3 2nd edition Springer-Verlag Berlin Heidelberg New York
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Preface to
As we did to preface the second edition, we would like here again to single out a
few topics closely related with the matter of this book, that have been the subject
of some intensive study since 1994.
- In a number of applications, processes with long-range dependence seem to
fit the random phenomena under study better than do semi-martingales, and
particularly diffusion processes; hence, the development of stochastic integration
with respect to fractional Brownian motions;
- the domain of validity of Ito's formula, and its interpretations, are constantly
being extended (Follmer-Protter-Shyriaev, Lyons-Zheng, Russo-Vallois, ...);
- anticipative stochastic calculus (Nualart-Pardoux, ...);
- the study of processes with independent increments in the light of previously
found results for Brownian motion, and, more generally, diffusions; the publi-
publication of J. Bertoin's excellent book: Levy processes (Cambridge Univ. Press,
1996) gives great impetus to these developments;
- asymptotics of diffusions in random environments (Kawazu-Tanaka, ...).
Paris, July 1998 Daniel Revuz
Marc Yor

Preface to the Second Edition
Since the beginning of the nineties, the main advances made in relation to the
material found in this book seem to go in the following directions:
i) More and more classes of discontinuous processes are discovered for which
some of the main "structural" properties of diffusions or martingales hold, for
instance:
- Pitman's type theorems (representation of the BES3 process by means of the
one-dimensional BM) for Levy processes (Bertoin) or random walks (Tanaka);
- Knight's theorem on orthogonal martingales as time changes of independent
ones for a large class of martingales (Kallenberg);
- the chaos representation property not only for BM and the compensated Poisson
process, but also for a large family of discontinuous martingales, including
Azema's martingales, as shown in a series of papers by Emery.
ii) The branching properties of the Brownian paths and/or Brownian excursions
have been studied extensively (Aldous, Le Gall, Neveu, Pitman ...).
iii) The increased familiarity with the local times of one-dimensional BM
and/or diffusions has led to the construction and study of intersection local times
in dimensions 2 or 3. A beautiful account of this topic and its many applications
to multiple points of rf-dimensional BM is now found in Le Gall's S'-Flour lecture
course (LNM 1527 Springer, 1992).
iv) (closely related to ii)) The importance of random measures is perhaps one of
the reasons why superprocesses, i.e. measure-valued processes, became so popular
(Dawson, Dynkin ...). More generally, many studies now focus on defining and
studying Brownian motion with more and more general state spaces such as: sets
of trajectories, graphs, fractals, ... (Barlow, Bass, Evans, Le Gall, Perkins, ...).
v) Central-limit theorems for Brownian functionals extend to various classes
of Levy processes (Rosen) or Gaussian processes (Berman).
This short list of topics shows obviously how much alive the studies of, and
around, Brownian motion are. Researchers' interests in this field are going back
and forth between the "simple" Brownian motion model and much more elaborate
random processes. Despite, or may be because of, these various developments,
we decided not to overhaul our initial exposition. We have contented ourselves
with the correction of a few mistakes and the streamlining of a few proofs. We
have also added some remarks and exercises which either answer questions raised

VIII Preface to the Second Edition
by readers about the material in the original text, or follow from some recent
developments. In particular, we have included, mainly in the form of exercises,
a more concise and, at the same time, "richer" proof of the arcsine law; some
applications of excursion theory to the computation of laws related to the Brownian
bridge; some important relations between exponentials of Brownian motion and
Bessel processes; further examples of, and remarks about, local martingales that
are not martingales (a seemingly hazardous topic ...); a beautiful identity due to
F. Knight. The bibliography has been increased accordingly, but the more one
digs, the more there is to fill in afterwards ...
Talking about "digging", some readers, including ourselves sometimes, have
experienced difficulties in finding where a particular result was hidden ... As a
partial solution to this problem we have enlarged the index and supplemented it
with a "catalogue" of statements that cannot find their way into the index.
Finally, it is a pleasure to thank a number of dedicated readers, among these
A. Adhikari, J. Azema, M. Emery, H. Follmer, as well as graduate students who
sent us remarks and suggestions, thus contributing to a possible improvement on
the first edition.
Paris, April 1994 Daniel Revuz
Marc Yor

Preface to the First Edition
This book focuses on the probabilistic theory of Brownian motion. This is a good
topic to center a discussion around because Brownian motion is in the intersection
of many fundamental classes of processes. It is a continuous martingale, a Gaus-
Gaussian process, a Markov process or more specifically a process with independent
increments; it can actually be defined, up to simple transformations, as the real-
valued, centered process with independent increments and continuous paths. It is
therefore no surprise that a vast array of techniques may be successfully applied
to its study and we, consequently, chose to organize the book in the following
way.
After a first chapter where Brownian motion is introduced, each of the follow-
following ones is devoted to a new technique or notion and to some of its applications
to Brownian motion. Among these techniques, two are of paramount importance:
stochastic calculus, the use of which pervades the whole book and the powerful
excursion theory, both of which are introduced in a self-contained fashion and
with a minimum of apparatus. They have made much easier the proofs of many
results found in the epoch-making book of Ito and McKean: Diffusion Processes
and their Sample Paths, Springer A965).
Furthermore, rather than working towards abstract generality, we have tried to
study precisely some important examples and to carry through the computations of
the laws of various functionals or random variables. Thus we hope to facilitate the
task of the beginner in an area of probability theory which is rapidly evolving. The
later chapters of the book however, will hopefully be of interest to the advanced
reader.
We strove to offer, at the end of each section, a large selection of exercises,
the more challenging being marked with the sign * or even **. On one hand, they
should enable the reader to improve his understanding of the notions introduced
in the text. On the other hand, they deal with many results without which the
text might seem a bit "dry" or incomplete; their inclusion in the text however
would have increased forbiddingly the size of the book and deprived the reader
of the pleasure of working things out himself. As it is, the text is written with the
assumption that the reader will try a good proportion of them, especially those
marked with the sign #, and in a few proofs we even indulged in using the results
of foregoing exercises.

X Preface to the First Edition
The text is practically self-contained but for a few results of measure theory.
Beside classical calculus, we only ask the reader to have a good knowledge of
basic notions of integration and probability theory such as almost-sure and in the
mean convergences, conditional expectations, independence and the like. Chapter 0
contains a few complements on these topics. Moreover the early chapters include
some classical material on which the beginner can hone his skills.
Each chapter ends up with notes and comments where, in particular, references
and credits are given. In view of the enormous literature which has been devoted to
Brownian motion and related topics, we have in no way tried to draw a historical
picture of the subject and apologize in advance to those who may feel slighted.
Likewise our bibliography is not even remotely complete and leaves out the many
papers which deal with the relationships of Brownian motion with other fields
of Mathematics such as Potential Theory, Harmonic Analysis, Partial Differential
Equations and Geometry. A number of excellent books have been written on these
subjects some of which we discuss in the notes and comments.
Finally, it is a pleasure to thank those who have offered useful comments on
the first drafts in particular J. Jacod, P.A. Meyer, B. Maisonneuve and J. Pit-
Pitman. Our special thanks go to J.F. Le Gall who put us straight on an inordinate
number of points and Shi Zhan who has helped us with the exercises. Each chap-
chapter of this book has been taught a number of times by the authors in the last
decade, either in a "Cours de 3° Cycle" in Paris or in "crash courses" on Brown-
Brownian motion; we would like to seize this opportunity of thanking our audiences for
their warm response. Last but not least, Josette Saman a pris une part essentielle
dans la preparation materielle du manuscrit et nous Ten remercions bien vivement.
Paris, October 1990 Daniel Revuz
Marc Yor

Table of Contents
Chapter 0. Preliminaries 1
§ 1. Basic Notation 1
§2. Monotone Class Theorem 2
§3. Completion 3
§4. Functions of Finite Variation and Stieltjes Integrals 4
§5. Weak Convergence in Metric Spaces 9
§6. Gaussian and Other Random Variables 11
Chapter I. Introduction 15
§ 1. Examples of Stochastic Processes. Brownian Motion 15
§2. Local Properties of Brownian Paths 26
§3. Canonical Processes and Gaussian Processes 33
§4. Filtrations and Stopping Times 41
Notes and Comments 48
Chapter II. Martingales 51
§ 1. Definitions, Maximal Inequalities and Applications 51
§2. Convergence and Regularization Theorems 60
§3. Optional Stopping Theorem 68
Notes and Comments 77
Chapter III. Markov Processes 79
§ 1. Basic Definitions 79
§2. Feller Processes 88
§3. Strong Markov Property 102
§4. Summary of Results on Levy Processes 114
Notes and Comments 117
Chapter IV. Stochastic Integration 119
§1. Quadratic Variations 119
§2. Stochastic Integrals 137

XII Table of Contents
§3. Ito's Formula and First Applications 146
§4. Burkholder-Davis-Gundy Inequalities 160
§5. Predictable Processes 171
Notes and Comments 176
Chapter V. Representation of Martingales 179
§ 1. Continuous Martingales as Time-changed Brownian Motions 179
§2. Conformal Martingales and Planar Brownian Motion 189
§3. Brownian Martingales 198
§4. Integral Representations 209
Notes and Comments 216
Chapter VI. Local Times 221
§ 1. Definition and First Properties 221
§2. The Local Time of Brownian Motion 238
§3. The Three-Dimensional Bessel Process 251
§4. First Order Calculus 260
§5. The Skorokhod Stopping Problem 269
Notes and Comments 277
Chapter VII. Generators and Time Reversal 281
§ 1. Infinitesimal Generators 281
§2. Diffusions and Ito Processes 293
§3. Linear Continuous Markov Processes 300
§4. Time Reversal and Applications 312
Notes and Comments 322
Chapter VIII. Girsanov's Theorem and First Applications 325
§ 1. Girsanov's Theorem 325
§2. Application of Girsanov's Theorem to the Study of Wiener's Space .... 338
§3. Functionals and Transformations of Diffusion Processes 349
Notes and Comments 362
Chapter IX. Stochastic Differential Equations 365
§ 1. Formal Definitions and Uniqueness 365
§2. Existence and Uniqueness in the Case of Lipschitz Coefficients 375
§3. The Case of Holder Coefficients in Dimension One 388
Notes and Comments 399
Chapter X. Additive Functionals of Brownian Motion 401
81. General Definitions " 401

Table of Contents XIII
§2. Representation Theorem for Additive Functionals
of Linear Brownian Motion 409
§3. Ergodic Theorems for Additive Functionals 422
§4. Asymptotic Results for the Planar Brownian Motion 430
Notes and Comments 436
Chapter XI. Bessel Processes and Ray-Knight Theorems 439
§ 1. Bessel Processes 439
§2. Ray-Knight Theorems 454
§3. Bessel Bridges 463
Notes and Comments 469
Chapter XII. Excursions 471
§1. Prerequisites on Poisson Point Processes 471
§2. The Excursion Process of Brownian Motion 480
§3. Excursions Straddling a Given Time 488
§4. Descriptions of Ito's Measure and Applications 493
Notes and Comments 511
Chapter XIII. Limit Theorems in Distribution 515
§ 1. Convergence in Distribution 515
§2. Asymptotic Behavior of Additive Functionals of Brownian Motion .... 522
§3. Asymptotic Properties of Planar Brownian Motion 531
Notes and Comments 541
Appendix 543
§ 1. Gronwall's Lemma 543
§2. Distributions 543
§3. Convex Functions 544
§4. Hausdorff Measures and Dimension 547
§5. Ergodic Theory 548
§6. Probabilities on Function Spaces 548
§7. Bessel Functions 549
§8. Sturm-Liouville Equation 550
Bibliography 553
Index of Notation 591
Index of Terms 595
Catalogue 601

Chapter 0. Preliminaries
In this chapter, we review a few basic facts, mainly from integration and classical
probability theories, which will be used throughout the book without further ado.
Some other prerequisites, usually from calculus, which will be used in some special
parts are collected in the Appendix at the end of the book.
§1. Basic Notation
Throughout the sequel, N will denote the set of integers, namely, N = {0, 1, • • •},
К the set of real numbers, Q the set of rational numbers, С the set of complex
numbers. Moreover M+ — [0, oo[ and Q+ = Q П М+. By positive we will always
mean > 0 and say strictly positive for > 0.
Likewise a real-valued function / defined on an interval of К is increasing
(resp. strictly increasing) if x < у entails f(x) < /(>¦) (resp. f(x) < f(y)).
If a, b are real numbers, we write:
а л b — min(a, b), a V b — max(a, b).
If ? is a set and / a real-valued function on E, we use the notation
/+ = /v0, /- = -(/л 0), |/| = /+ + /-,
Il/H =sup|/(*)|.
xeE
We will write an \ a (a,, f a) if the sequence (a,,) of real numbers decreases
(increases) to a.
If (E, ft) and (F, .W) are measurable spaces, we write / 6 КI.W to say that
the function / : E —> F is measurable with respect to % and ,W. If (F,.W) is
the real line endowed with the a -field of Borel sets, we write simply / e # and
if, in addition, / is positive, we write / e '6+. The characteristic function of a set
A is written 1д; thus, the statements A e % and 1д е ^T have the same meaning.
If ?2 is a set and /,, i e I, is a collection of maps from Q to measurable
spaces (Ej, #,), the smallest cr-field on Q for which the /'s are measurable is
denoted by a{fi, i e /). If ИГ is a collection of subsets of Q, then a('6) is the
smallest cr-field containing V?; we say that a('6') is generated by '6. The cr-field
cr(/,, i € I) is generated by the family Yj = [f~{ (A,), A, e <5), j e /}. Finally if

2 Chapter 0. Preliminaries
(Ci, i g I, is a family of cr-fields on Q, we denote by \Д. #,¦ the a -field generated
by (J( #,-. It is the union of the cr-fields generated by the countable sub-families
of %i',i g /.
A measurable space (E, K) is separable if <C is generated by a countable
collection of sets. In particular, if ? is a LCCB space i.e. a locally compact space
with countable basis, the a -field of its Borel sets is separable; it will often be
denoted by .Л(Е). For instance, ./}(W') is the a -field of Borel subsets of the
uf-dimensional euclidean space.
For a measure m on (E, <C) and / e К, the integral of / with respect to m,
if it makes sense, will be denoted by any of the symbols
\ S dm, J f(x)dm(x), j f(x)m(dx), m(f), (m. f),
and in case ? is a subset of a euclidean space and m is the Lebesgue measure,
If (?2, .У , P) is a probability space, we will as usual use the words random
variable and expectation in lieu of measurable function and integral and write
E[X] = X dP.
We will often write r.v. as shorthand for random variable. The law of the r.v. X,
namely the image of P by X will be denoted by Px or X(P). Two r.v.'s defined
on the same space are equivalent if they are equal P-a.s.
If & is a sub-cr-field of .Y, the conditional expectation of X with respect to '</ ,
if it exists, is written E[X \ '.?]. If X — \A, A e .W, we may write P(A \ '.?). If
Ъ = a (Xi.i e I) we also write E [X \ Xt, i e I] or P (A | X{, i e I). As is well-
known conditional expectations are defined up to P -equivalence, but we will often
omit the qualifying P-a.s. When we apply conditional expectation successively,
we shall abbreviate E [E [X \ .?{\ \ .Щ\ \oE\X\.T\\ Щ.
We recall that if Q is a Polish space (i.e. a metrizable complete topological
space with a countable dense subset), Ж the cr-field of its Borel subsets and if #
is separable, then there is a regular conditional probability distribution given '.<s .
If д and v are two a -finite measures on (E, <C), we write i±-Lv to mean that
they are mutually singular, д <С v to mean that д is absolutely continuous with
respect to v and д ~ v if they are equivalent, namely if д ^С v and v«/i. The
Radon-Nikodym derivative of the absolutely continuous part of д with respect to
v is written ^
and .У is dropped when there is no risk of confusion.
§2. Monotone Class Theorem
We will use several variants of this theorem which we state here without proof.
B.1) Theorem. Let .? be a collection of subsets of Q such that

§3. Completion 3
t) О e .У,
ii) if А, В е У and А с В, then B\A e .?,
Hi) if{An] is an increasing sequence of elements of.? then U An e .У\
//".Vе D *W where JF is closed under finite intersections then ?f э о (.F").
The above version deals with sets. We turn to the functional version.
B.2) Theorem. Let Ж be a vector space of bounded real-valued functions on Q
such that
i) the constant functions are in .Ж,
ii) if [hn} is an increasing sequence of positive elements of 3<S such that h —
supn hn is bounded, then h e .Ж.
If ^6 is a subset of.J@ which is stable under pointwise multiplication, then <Ж
contains all the bounded a {tt)-measurable functions.
The above theorems will be used, especially in Chap. Ill, in the following
set-up. We have a family fhi e /, of mappings of a set Q into measurable spaces
(?,-, %i). We assume that for each i e / there is a subclass . /f of <5/, closed
under finite intersections and such that cr{. lj') = #j. We then have the following
results.
B.3) Theorem. Let ..-V be the family of sets of the form f)i€j ffx{A,) where A,-
ranges through . \'{and J ranges through the finite subsets of I; then a{. /¦'') —
ff(/j,i€/).
B.4) Theorem. Let 3*6 be a vector space of real-valued functions on Q, contain-
containing If?, satisfying property ii) of Theorem B.2) and containing all the functions
l г for Г e . -/¦'". Then, ,Ж contains all the bounded, real-valued, о (ft, i e /)-
measurable functions.
§3. Completion
If (E, %) is a measurable space and fi a probability measure on ft, the completion
<?д of К with respect to fi is the a -field of subsets В of E such that there exist
Bx and B2 in % with B\ С В С B2 and ц(В2\В\) - 0. If у is a family of
probability measures on #, the cr -field
is called the completion of ?T with respect to у. If у is the family of all probability
measures on %, then 8Y is denoted by #* and is called the a -field of universally
measurable sets.
\f .W is a sub-ст-algebra of %Y we define the completion of\F" in ^ wft/z
respect to у as the family of sets A with the following property: for each ц & y,

4 Chapter 0. Preliminaries
there is a set В such that AAB is in <5Y and [i{AAB) — 0. This family will be
denoted .Wy\ the reader will show that it is a a -field which is larger than &"*'.
Moreover, it has the following characterization.
C.1) Proposition. A set A is in .^y if and only if for every \i e у there is a set
Вц in .W and two ^-negligible sets N^ and Мц in % such that
ВЦ\МЦ С А сВдиМ„.
Proof. Left to the reader as an exercise. ?
The following result gives a means of checking the measurability of functions
with respect to a -algebras of the .^
C.2) Proposition. For i = 1, 2, let (?,-, #j) be a measurable space, y,- a family of
probability measures on lSt and ,Щa sub-a-algebra of' %". If f is a map which
is both in K\/%2 and .&Ц-Щ and if f(fi) e yi for every \i e y\ then f is in
Proof. Let A be in ,^Y2. For \i e y\, since v = /(д) is in yi, there is a set
Bv e &~2 and two v-negligible sets Nv and Mv in ^ such that
BV\NV CACBVUMV.
The set Вц = f~\Bv) belongs to.?[, the sets N,, = f~l(Nv) and Мц = f~\Mv)
are д-negligible sets of %\ and
This entails that f~l(A) e .^y\ which completes the proof.
§4. Functions of Finite Variation and Stieltjes Integrals
This section is devoted to a set of properties which will be used constantly through-
throughout the book.
We deal with real-valued, right-continuous functions A with domain [0, oof.
The results may be easily extended to the case of R. The value of A in t is denoted
A, or A(t). Let A be a subdivision of the interval [0, t] with 0 = to < h < • ¦ • <
tn = t; the number \A\ — sup,- |f,+i — tt\ is called the modulus or mesh of A. We
consider the sum
If A' is another subdivision which is a refinement of A, that is, every point и of
A is a point of A', then plainly Sf > 5,л.

§4. Functions of Finite Variation and Stieltjes Integrals 5
D.1) Definition. The Junction A is of finite variation if for every t
S, = sup 5,л < +oo.
л
The function t -*¦ S, is called the total variation of A and S, is the variation of A on
[0, t]. The function S is obviously positive and increasing and if lim,^.,*, S, < +oo,
the function A is said to be o/bounded variation.
The same notions could be defined on any interval [a,b]. We shall say that a
function A on the whole line is of finite variation if it is of finite variation on any
compact interval but not necessarily of bounded variation on the whole of R.
Let us observe that С'-functions are of finite variation. Monotone finite func-
functions are of finite variation and conversely we have the
D.2) Proposition. Any function of finite variation is the difference of two increas-
increasing functions.
Proof. The functions E + A)/2 and E - A)/2 are increasing as the reader can
easily show, and A is equal to their difference. ?
This decomposition is moreover minimal in the sense that if A = F — G where
F and G are positive and increasing, then E + A)/2 < F and E - A)/2 < G.
As a result, the function A has left limits in any t e]0, oof. We write A,- or
A(t—) for lim.^, A, and we set Aq- = 0. We moreover set Л A, = A, — A,-; this
is the jump of A in t.
The importance of these functions lies in the following
D.3) Theorem. There is a one-to-one correspondence between Radon measures \i
on [0, oof and right-continuous functions A of finite variation given by
A, = д([0, t]).
Consequently A,- = ?i([0, t[) and ЛА, = /j,({t}). Moreover, if м({0}) = О, the
variation 5 of A corresponds to the total variation \/х\ of д and the decomposition
in the proof of Proposition D.2) corresponds to the minimal decomposition of \i
into positive and negative parts.
If / is a locally bounded Borel function on R+, its Stieltjes integral with
respect to A, denoted
/ f,dA,, I f(s)dA(s) or / f(s)dAs
JO JO J]O.t]
is the integral of / with respect to д on the interval ]0, t]. The reader will observe
that the jump of A at zero does not come into play and that /„' dAs = A, — Ao. If
we want to consider the integral on [0, t], we will write /[01] f(s)dAs. The integral
on ]0, t] is also denoted by (/ • A),. We point out that the map ?->(/• A), is
itself a right-continuous function of finite variation.
A consequence of the Radon-Nikodym theorem applied to fi and to the
Lebesgue measure Я is the

./о
6 Chapter 0. Preliminaries
D.4) Theorem. A function A of finite variation is k-a.e. differentiable and there
exists a function В of finite variation such that B' — 0 k-a.e. and
r>
Aids.
Jo
The function A is said to be absolutely continuous if В = 0. The corresponding
measure д is then absolutely continuous with respect to k.
We now turn to a series of notions and properties which are very useful in
handling Stieltjes integrals.
D.5) Proposition (Integration by parts formula). If A and В are two functions
of finite variation, then for any t,
Г V
AtB, =A0B0+ AsdBs+ I Bs_dAs.
Jo Jo
Proof. If д (resp. v) is associated with A (resp. B) both sides of the equality
are equal to д ® v([0, t]2); indeed J'Q AsdBs is the measure of the upper triangle
including the diagonal, J'Q Bs-dAs the measure of the lower triangle excluding the
diagonal and A0BQ = д ® v({0, 0}). ?
To reestablish the symmetry, the above formula can also be written
A,B,=
f' f'
,B,= / As_dBs+ / Bs_dAs + ^
Jo Jo ,<t
The sum on the right is meaningful as A and В have only countably many dis-
discontinuities. In fact, A can be written uniquely A, — Act + ?],<, AAS where Ac
is continuous and of finite variation.
The next result is a "chain rule" formula.
D.6) Proposition. If F is a C' -function and A is of finite variation, then F(A) is
of finite variation and
F(At) = F(A0) + / F'(As-)dAs + V* (F(AS) - F(AS_) - F'(AS.)AAS).
Jo jt!
Proof. The result is true for F(x) — x, and if it is true for F, it is true for xF(x)
as one can deduce from the integration by parts formula; consequently the result
is true for polynomials. The proof is completed by approximating a C1-function
by a sequence of polynomials. ?
As an application of the notions introduced thus far, let us prove the useful
D.7) Proposition. If A is a right continuous function of finite variation, then
is the only locally bounded solution of the equation
Y, = YQ+ [ Ys_dAs.
Jo

§4. Functions of Finite Variation and Stieltjes Integrals 7
Proof. By applying the integration by parts formula to Yo ]~L<r 0 + AAS) and
exp (/„' dAcA which are both of finite variation, it is easily seen that У is a solution
of the above equation.
Let Z be the difference of two locally bounded solutions and M, — sup9<r \ZS\.
It follows from the equality Z, = /„' Zs_dAs that \Z,\ < M,S, where 5 is the
variation of A; then, thanks to the integration by parts formula
\Z,\<M, f Ss.dSs < M,Sf/2,
Jo
and inductively,
M f
\Z,\ < -f / S:_dSs < MtS^/(n + 1)!
n\ Jo
which proves that Z = 0. ?
We close this section by a study of the fundamental technique of time changes,
which allows the explicit computation of some Stieltjes integrals. We consider
now an increasing, possibly infinite, right-continuous function A and for s > 0,
we define
Cs = inf {t : A, > s}
where, here and below, it is understood that inf{0} = +oo.
To understand what follows, it is useful to draw Figure 1 (see below) showing
the graph of A and the way to find C. The function С is obviously increasing so
that
C,_ =limCM
up
is well-defined for every s. It is easily seen that
C,_ = inf {t : A, > s].
In particular if A has a constant stretch at level s, then Cs will be at the right end
and Cj_ at the left end of the stretch; moreover Cv_ ф Cs only if A has a constant
stretch at level s. By convention Co- = 0.
D.8) Lemma. The function С is right-continuous. Moreover A(CS) > s and
A, = inf {s : C, > t].
Proof. That A(Cj) > s is obvious. Moreover, the set {A, > s) is the union of the
sets {A, > s + s] for e > 0, which proves the right continuity of C.
If furthermore, C, > t, then / $. {u : AM > s] and A, < s. Consequently,
A, < mf[s : Cs > t]. On the other hand, C(At) > t for every t, hence С (At+e) >
t + s > t which forces
A,+E > \vS{s : Cs > t)
and because of the right continuity of A '
A, > inf{s : C, > t]. D

Chapter 0. Preliminaries
w
Fig. 1.
Remarks. Thus A and С play symmetric roles. But if A is continuous, С is still
only right-continuous in general; in that case, however, A (Cs) = s but С (As) > s
if s is in an interval of constancy of A. As already observed, the jumps of С
correspond to the level stretches of A and vice-versa; thus С is continuous iff
A is strictly increasing. The right continuity of С does not stem from the right-
continuity of A but from its definition with a strict inequality; likewise, C,_ is left
continuous.
We now state a "change of variables" formula.
D.9) Proposition. Iff is a positive Borel function on [0, oo[,
f(u)dAu = f f(Cs)l(Cs<Oo)ds.
0,oo[ JO
Proof If / = l[o,«]> the formula reads
/•00
Av = I l(cs<v)ds
Jo
and is then a consequence of the definition of С By taking differences, the equality
holds for the indicators of sets ]u, v], and by the monotone class theorem for any /
with compact support. Taking increasing limits yields the result in full generality.
?
In the same way, we also have
f f(u)dAu = f f(Cs)li0<c,<oo)ds.
Jo Jo
[0,

§5. Weak Convergence in Metric Spaces
The right member in the proposition may also be written
f{Cs)ds,
/
Jo
because Cs < oo if and only if A^ > s.
The last result is closely related to time changes.
D.10) Proposition. If и is a continuous, поп decreasing function on the interval
[a, b], then for a non negative Borel function f on [u(a), u(b)]
f f(u(s))dAu(s) = f f(t)dA,.
J[a,b] J[u(a),u(b)]
The integral on the left is with respect to the measure associated with the right
continuous increasing function s ->¦ A(u(s)).
Proof We define v, = mf[s : u(s) > t}, then u(vt) = t and v is a measurable
mapping from [u(a), u(b)] into [a, b]. Let dA be the measure on [«(a), u(b)]
associated with A and v the image of dA by v. Then dA is the image of v by и
and therefore
/ f(u(s))dv(s) = f f(t)dA,.
[a,b] J[u(a).u(b)]
In particular, A(u(b)) — A(u{a) — ) = v([a, b]) which proves that v is associated
with the increasing function s -*¦ k(u{s)). The proposition is established. ?
§5. Weak Convergence in Metric Spaces
Let ? be a metric space with metric d and call .>? the Borel a -algebra on E. We
want to recall a few facts about the weak convergence of probability measures on
(?, .>?). If P is such a measure, we say that a subset A of ? is a P-continuity
set if P(dA) = 0 where ЗА is the boundary of A.
E.1) Proposition. For probability measures Pn, n e N, and P, the following con-
conditions are equivalent:
(i) For every bounded continuous function f on E,
lim j fdPn= j fdP;
(ii) For every bounded uniformly continuous function f on E,
lim f fdPn= I fdP;
(Hi) For every closed subset F on E, Hm Pn(F) < P(F);

10 Chapter 0. Preliminaries
(iv) For every open subset G of E, lim P,,(G) > P(G);
n
(v) for every P-continuity set A, lim P,,(A) = P(A).
E.2) Definition. If Pn and P satisfy the equivalent conditions of the preceding
proposition, we say that (Pn) converges weakly to P.
If л is a family of probability measures on (E,./?), we will say that it is
weakly relatively compact if every sequence of elements of ж contains a weakly
convergent subsequence. To prove weak convergence, one needs a criterion for
weak compactness which is the raison d 'etre of the following
E.3) Definition. A family n is tight if for every s e]0, 1[, there exists a compact
set Ke such that
P[Ke] > 1 - ?, for every P e л.
With this definition we have the
E.4) Theorem (Prokhorov's criterion). If a family n is tight, then it is weakly
relatively compact. If E is a Polish space, then a weakly relatively compact family
is tight.
E.5) Definition. If (X,,)nsN and X are random variables taking their values in a
metric space E, we say that (Xn) converges in distribution or in law to X if their
laws Рх„ converge weakly to the law Px of X. We will then write X,, —> X. We
also write X — Y to mean that X and Y have the same distribution or the same
law.
We stress the fact that the Xn's and X need not be defined on the same
probability space. If they are defined on the same probability space {Q,.^, P),
we may set the
E.6) Definition. The sequence (Х„) converges in probability to X if for every
e > 0,
lim P[d(X,,,X) > e] = 0.
We will then write P- lim Х„ = X.
In a Polish space, if P-limX,, — X, then Xn —> X. The converse is not true
in general nor even meaningful since, as already observed the Xn's need not be
defined on the same probability space. However if the Xn's are defined on the
same space and converge weakly to a constant с then they converge in probability
to the constant r.v. с as the reader can easily check.
The following remarks will be important in Chap. XIII.
E.7) Lemma. If (Х„, Yn) is a sequence of r.v. 's with values in separable metric
spaces E and F and such that
(i) (Х„, Yn) converges in distribution to (X, Y),

§6. Gaussian and Other Random Variables 11
(ii) the law ofYn does not depend on n,
then, for every Вorel function ip : F -» G where G is a separable metric space,
the sequence (X,,, <p{Yn)) converges in distribution to (X, <f{Y)).
Proof. It is enough to prove that if h ,k are bounded continuous functions,
\im E[h(Xn)k (ip(Yn))} = E [h(X)k(cp(Y))].
Set /? = к о ip; if v is the common law of the Yn's, there is a bounded continuous
function p in Ll(v) such that j \p — p\dv < e for any preassigned e. Then,
\E[h(Xn)p(Yn)]-E[h(X)p(Y)]\
< \E [h(Xn) (p(Yn) - p(Yn))]\ + \E[h(Xn)p(Yn)} - E [h(X)p(Y)}\
+ \E[h(X)(p(Y)-p{Y))]\
< 2\\h\\xe + \E [h{X,,)p{Yn)} - E [h(X)p(Y)]\ .
By taking n large, the last term can be made arbitrarily small since h and p are
continuous. The proof is complete. П
E.8) Corollary. Let (Xxn, ..., X*) be a sequence ofk-tuples ofr.v. 's with values
in separable metric spaces Sj, j = 1, . .. , k, which converges in distribution to
(Xх, .. ., Xky If for each j, the law of Xi does not depend on n, then for any
В orel functions <p} : 5} -> Uj where Uj is a separable metric space, the sequence
{<px (X\) ,...,<pk (X*)) converges in distribution to {(px (X1) , .. ., ipk (X*)).
Proof. The above lemma applied to Xn = (X,', ..., X*~'), Yn = X* permits to
replace (X*) by tpk (X*); one then takes Х„ = (X\, ..., X^2; <pk (X*)), Yn =
Xk~x and so on and so forth.
§6. Gaussian and Other Random Variables
We will write X ~ . / \m, a2) to mean that the r.v. X is Gaussian with mean m
and variance a2. In particular, X ~ . I @, 1) means that X is a Gaussian centered
r.v. with unit variance or in other words a reduced Gaussian r.v. In what follows
the constant r.v.'s are considered to be a particular case of Gaussian r.v.'s, namely
those with a2 = 0. If X ~ . / '{m, ст2), а > 0, we recall that X has the density
(y/lna\ exp {-\(x - mJ/a2) and that its characteristic function (abbreviated
c.f. in the sequel) is given by
E [eitX] = exp
We recall the

12 Chapter 0. Preliminaries
F.1) Proposition. If (Xn) is a sequence of Gaussian r.v.'s which converges in
probability to a r.v. X, then X is a Gaussian r.v., the family [\Xn\p] is uniformly
integrable and the convergence holds in Lp for every p>\.
Thus the set of Gaussian r.v.'s defined on a given probability space {?2, .'W, P)
is a closed subset of L2(i2, &", P). More specifically we set the
F.2) Definition. A Gaussian space is a closed linear subspace of a space
L2(f2, &~, P) consisting only of centered Gaussian r.v. 's.
If G is a Gaussian space and Xlt..., Xj are in G, then the d-dimensional
r.v. X = (Xi,..., Xd) is a Gaussian r.v., in other words a(X) is a real Gaussian
r.v. for every linear form a on M.d. Let us also recall that if К is a symmetric
semi-definite positive d x d-matrix (i.e. (jc, Kx) > 0 for every x e M.d), it is the
covariance matrix of a d-dimensional centered Gaussian r.v..
We recall the
F.3) Proposition. Let G,, / e /, be a family of closed subspaces of a given Gaus-
Gaussian space; then, the a-fields <r(G() are independent if and only if the spaces G,-
are pairwise orthogonal.
In particular, the components of an R^-valued centered Gaussian variable are
independent if and only if they are uncorrelated.
A few probability distributions on the line will occur throughout the book and
we recall some of their relationships.
A random variable Y follows the arcsine law if it has the density
{лу/х(\ — x)) on [0, 1]; then log Y has a characteristic function equal to
- ik).
IfN ~ . i'@, 1) then log EЛГ2) has c.f. Г (| + ik) /^/n and if e is an exponential
r.v. with parameter 1 the c.f. of log(e) is Г{\ + ik). It follows that
tf2 {U 2eY
with e and Y independent. This can also be seen directly by writing
where N' ~ . I "@, 1) is independent of N and showing that the two factors are
independent and have respectively the exponential (mean: 2) and arcsine laws.
The above identity is in fact but a particular case of an identity on Gamma
and Beta r.v.'s. Let us call ya, a > 0, the Gamma r.v. with density х"~1е~х/Г(а)
on R+, and pa<b, a and b > 0, the Beta r.v. with density x"'l(l - x)b-x/B(a, b)
on [0, 1]. Classical computations show that if ya and уь are independent, then

§6. Gaussian and Other Random Variables 13
i) Ya + Yb and ya/ (ya + yh) are independent,
ii) Ya + Yb = Ya+b, iii) Ya/ (Ya + Yb) - Pa.b-
From these properties follows the bi-dimensional equality in law
(Ya, Yb) - Ya+b {Pa,h, 1 ~ Pa.b) ,
where, on the right-hand side, уа+ь and j5aj, are assumed independent.
Further if N and N' are independent reduced Gaussian random variables then
N/N' and N/\N'\ are Cauchy r.v.'s i.e. have the density (яA 4- x2)) on the real
line. If С is a Cauchy r.v. then Y = A +C2)~i. Next if (e1, Y') is an independent
copy of (e, Y), then the c.f.'s of \og(e/e'), \og(Y/Y') and logC2 are respectively
equal to
_, /1 \ /1 \
лк/ sinh(nk), tanh(^^)/^^. and (coshGrA.)) = Г I - + ik J Г I ik\ /л.
Thus, logC2 = \og(e/e') + \og(Y/Yr). Finally, the density of logC2 is
)
The above hyperbolic functions occur in many computations. If Ф is such a
function, we give below series representations for Ф, as well as for the probability
densities ф and / defined by
Ф(к)= exp(ikxL>(x)dx = exp(-k2y/2)f(y)dy,
J-oo Jo
and the distribution function
F(x) = f f(y)dy.
Jo
a) If Ф(к) = tanhnk/nk, then also
oo
Ф(к) = Bл~2) J2 (^2 + (n - A/2)J),
and ф(х) = -^-2logtanh(^/4), f(y) = я^2 Х;и°°=1 exp (-(и - A/2)J >'/2),
from which F is obtained by term by term integration.
b) Likewise if Ф(к) = лк/ $т\\лк, then also
ос
Ф(к)= 1 +2k2^2(-\)n(k2+n2yi,
л = 1
and ф(х) = Bcosh(Jc/2)), F(x) = E^-coC)" exp(-«2jc/2).
c) Furthermore if Ф(к) = (cosh^^.), then also
Ф(к) = л~1 ^(-l)"Bn + 1) (к2 + (п

14 Chapter 0. Preliminaries
and ф(х) = Bл cosh(jc/2)r' .
00
/00 = *"' J^i-l)" (« + A/2)) exp (- (n + A/2)J у/2)).
о
d) Finally, if Ф(Л) = GrA./sinh7rA.J, then also
и=1
and
ф(х) = ((V2)cosh(A:/2)-l)/2sinh(xJ,
F(x) = Y A-и2х)ехр(-и2х/2).
n=—oo

Chapter I. Introduction
§1. Examples of Stochastic Processes. Brownian Motion
A stochastic process is a phenomenon which evolves in time in a random way.
Nature, everyday life, science offer us a huge variety of such phenomena or at
least of phenomena which can be thought of as a function both of time and of a
random factor. Such are for instance the price of certain commodities, the size of
some populations, or the number of particles registered by a Geiger counter.
A basic example is the Brownian motion of pollen particles in a liquid. This
phenomenon, which owes its name to its discovery by the English botanist R.
Brown in 1827, is due to the incessant hitting of pollen by the much smaller
molecules of the liquid. The hits occur a large number of times in any small
interval of time, independently of each other and the effect of a particular hit is
small compared to the total effect. The physical theory of this motion was set up
by Einstein in 1905. It suggests that this motion is random, and has the following
properties:
i) it has independent increments;
ii) the increments are gaussian random variables;
iii) the motion is continuous.
Property i) means that the displacements of a pollen particle over disjoint time
intervals are independent random variables. Property ii) is not surprising in view
of the central-limit theorem.
Much of this book will be devoted to the study of a mathematical model of
this phenomenon.
The goal of the theory of stochastic processes is to construct and study math-
mathematical models of physical systems which evolve in time according to a random
mechanism, as in the above example. Thus, a stochastic process will be a family
of random variables indexed by time.
A.1) Definition. Let T be a set, (E, ft) a measurable space. A stochastic process
indexed by T, taking its values in (E, ft), is a family of measurable mappings
X,, t € T,from a probability space (Q, ,W, P) into (E, ft). The space (E, ft) is
called the state space.
The set T may be thought of as "time". The most usual cases are T = N and
T = R+, but they are by no means the only interesting ones. In this book, we

16 Chapter I. Introduction
deal mainly with the case T = R+ and E will usually be M.d or a Borel subset of
M.d and <6 the Borel a -field on E.
For every со е ?2, the mapping t -*¦ X,(co) is a "curve" in E which is referred
to as a trajectory or a pa?A of X. We may think of a path as a point chosen
randomly in the space .^{T, E) of all functions from T into E, or, as we shall
see later, in a reasonable subset of this space.
To set up our basic example of a stochastic process, namely the mathematical
model of Brownian motion, we will use the following well-known existence result.
A.2) Theorem. Given a probability measure д on R, there exist a probability
space (?2, .W, P) and a sequence of independent random variables Xn, defined on
П, such that Xn(P) = д for every n.
As a consequence, we get the
A.3) Proposition. Let H be a separable real Hilbert space. There exist a proba-
probability space (?2,.W, P) and a family X(h), h e H, of random variables on this
space, such that
i) the map h —*¦ X(h) is linear;
ii) for each h, the r.v. X(h) is gaussian centered and
E[X(hJ] = \\hfH.
Proof Pick an orthonormal basis [е„] in H. By Theorem A.2), there is a probabil-
probability space (J2,3F, P) on which one can define a sequence of independent reduced
real Gaussian variables gn. The series ?)jj°(A, en)Hgn converges in L2(Q, &", P)
to a r.v. which we call X(K). The proof is then easily completed. ?
We may observe that the above series converges also almost-surely, and X(h)
is actually an equivalence class of random variables rather than a random variable.
Moreover, the space [X(h),h e H] is a Gaussian subspace of L2(i2,.>r, P)
which is isomorphic to H. In particular, E \X(h)X(h')\ = (h, A')//. Because in a
Gaussian space independence is equivalent to orthogonality, this shows that X{h)
and X(h') are independent if and only if h and h! are orthogonal in H.
A.4) Definition. Let (A,,ii,fj,) be a separable a-finite measure space. If in
Proposition A.3), we choose H = L2(A,. -it, (i), the mapping X is called a Gaus-
Gaussian measure with intensity fi on (A,. V?). When F e i H and n(F) < oo, we shall
write X(F) instead of XAF).
The term "measure" is warranted by the fact that if F e . •&, /*(F) < oo
and F = ?">„, then X(F) = ?~ X(Fn) a.s. and in L2(A,. -&, ц); however,
the exceptional set depends on F and on the sequence (Fn) and, consequently,
there is usually no true measure m(co, •) depending on w such that almost-surely
X(F)(co) = m(co, F) for every FeJ.
Let us also observe that for any two sets F and G, such that (i(F) < oo,
fi(G) < oo,

§1. Examples of Stochastic Processes. Brownian Motion 17
E[X(F)X(G)] = ii(F nG);
if F and G are disjoint sets, X(F) and X(G) are uncorrelated, hence independent.
We now take a first step towards the construction of Brownian motion. The
method we use may be extended to other examples of processes as is shown
in Exercises C.9) and C.11). We take the space (A,, H, д) to be R+ = [0, oo[,
endowed with the a-field of Borel sets and the Lebesgue measure. For each t e M.+,
we pick a random variable Bt within the equivalence class X([0, t]). We now study
the properties of the process В thus defined.
1°) The process В has independent increments i.e. for any sequence 0 = tо <
t\ < ... < h the random variables Bti — #,,_,, i = 1,2,... ,k are independent.
Indeed, Bn — Bti_t is in the class X (]?г-ь ?.]) and these classes are independent
because the corresponding intervals are pairwise disjoint.
2°) The process В is a Gaussian process, that is: for any sequence 0 = to <
t\ < ... < tn, the vector r.v. (#,0,..., Btn) is a vector Gaussian r.v. This follows
from the independence of the increments and the fact that the individual variables
are Gaussian.
3°) For each /, we have obviously E[Bf] — t; in particular, P[B0 = 0] = 1.
This implies that for a Borel subset A of the real line and t > 0
P[Bt e A] = / g,(x)dx,
Ja
where g,(x) = Bnt)~*/2exp(-x2/2t). Likewise, the increment B, - Bs has vari-
variance t — s. Furthermore, the covariance E[BsBt] is equal to inf(j, t); indeed using
the independence of increments and the fact that all the B,'s are centered, we have
for s < t,
E[B,B,] = E[BAB, + B,-B,)]
= E[B2] + E[Bs(Bt-Bs)] = s.
If we refer to our idea of what a model of the physical Brownian motion
ought to be, we see that we have got everything but the continuity of paths. There
is no reason why an arbitrary choice of B, within the class X([0, t]) will yield
continuous maps t -*¦ В,{ш). On the other hand, since we can pick any function
within the class, we may wonder whether we can do it so as to get a continuous
function for almost all w's. We now address ourselves to this question.
We first need to make a few general observations and give some definitions.
From now on, unless otherwise stated, all the processes we consider are indexed
byR+.
A.5) Definition. Let E be a topological space and ft the a-algebra of its Borel
subsets. A process X with values in (E, <C) is said to be a.s. continuous if, for
almost alibi's, the function t -*¦ Xt(oj) is continuous.

18 Chapter I. Introduction
We would like our process В above to have this property; we could then, by
discarding a negligible set, get a process with continuous paths. However, whether
in discarding the negligible set or in checking that a process is a.s. continuous, we
encounter the following problem: there is no reason why the set
{со : t -*¦ Xt (ш) is continuous }
should be measurable. Since we want to construct a process with state space K,
it is tempting, as we hinted at before Theorem A.2), to use as probability space
the set .F"(R+, K) = KR+ of all possible paths, and as r.v. X, the coordinate
mapping over t, namely Xt((o) = (o(t). The smallest a -algebra for which the Xt's
are measurable is the product ст-algebra, say .W. Each set in ,W depends only on
a countable set of coordinates and therefore the set of continuous w's is not in &".
This problem of continuity is only one of many similar problems. We will, for
instance, want to consider, for a K-valued process X, expressions such as
Г (a») = inf{? : X,(fo) > 0}, limX,, f l[-i i](Xs)ds, supjXJ,
sl' Jo v</
and there is no reason why these expressions should be measurable or even mean-
meaningful if the only thing we know about X is that it satisfies Definition A.1). This
difficulty will be overcome by using the following notions.
A.6) Definition. Two processes X and X' defined respectively on the probability
spaces (Q, <W, P) and (Q\ &"', P'), having the same state space (E, %), are said
to be equivalent if for any finite sequence t\ ,...,?„ and sets А, е У5,
P [Xu eAi,Xt2eA2 X,,, € An] = P'[X'tl eAu X',2eA2,..., X'tn еА„].
We also say that each one is a version of the other or that they are versions of the
same process.
The image of P by (Xti,..., Х,л) is a probability measure on (?", #") which
we denote by Ph ,„. The family obtained by taking all the possible finite se-
sequences (?i,..., tn) is the family of finite-dimensional distributions (abbreviated
f.d.d.) of X. The processes X and X' are equivalent if they have the same f.d.d.'s.
We observe that the f.d.d.'s of X form a projective family, that is, if {s\,..., s^)
is a subset of (t\,..., tn) and if n is the corresponding canonical projection from
E" onto ?*, then
ft, * = я (Л, ,„) ¦
This condition appears in the Kolmogorov extension Theorem C.2).
We shall denote by is/6x the indexed family of f.d.d.'s of the process X. With
this notation, X and Y are equivalent if and only
It is usually admitted that, most often, when faced with a physical phenomenon,
statistical experiments or physical considerations can only give information about
the f.d.d.'s of the process. Therefore, when coristmcting a mathematical model,

§1. Examples of Stochastic Processes. Brownian Motion 19
we may if we can, choose, within the class of equivalent processes, a version for
which expressions as those above Definition A.6) are meaningful. We now work
toward this goal in the case of Brownian motion.
A.7) Definition. Two processes X and X' defined on the same probability space
are said to be modifications of each other if for each t
X, = X't a.s.
They are called indistinguishable if for almost all ш
X,(eo) = X't(fi)) for every t.
Clearly, if X and X' are modifications of each other, they are versions of each
other. We may also observe that if X and X' are modifications of each other and
are a.s. continuous, they are indistinguishable.
In the next section, we will prove the following
A.8) Theorem. (Kolmogorov's continuity criterion). A real-valued process X
for which there exist three constants a, fi, С > 0 such that
E[\Xt+h-X,\a]<Ch'+ts
for every t and h, has a modification which is almost-surely continuous.
In the case of the process В above, the r.v. B,+h — B, is Gaussian centered
and has variance h, so that
E [(Br+h - В,L] = ЗА2.
The Kolmogorov criterion applies and we get
A.9) Theorem. There exists an almost-surely continuous process В with indepen-
independent increments such that for each t, the random variable B, is centered, Gaussian,
and has variance t.
Such a process is called a standard linear Brownian motion or simply Brownian
motion (which we will often abbreviate to BM) and will be our main interest
throughout the sequel.
The properties stated in Theorem A.9) imply those we already know. For
instance, for s < t, the increments B, — Bs are Gaussian centered with variance
t — s; indeed, we can write
B, = Bs + {B, - Bs)
and using the independence of Bs and B, — Bs, we get, taking characteristic
functions,
(tu2\ / sm2
" t)=exp v ~2~

20 Chapter I. Introduction
whence E [exp (iu(B, — Bs))] — exp (— {±=p-u2J follows. It is then easy to see
that В is a Gaussian process (see Definition C.5)) with covariance inf(.s, t). We
leave as an exercise to the reader the task of showing that conversely we could
have stated Theorem A.9) as: there exists an a.s. continuous centered Gaussian
process with covariance inf(.s, t).
By discarding a negligible set, we may, and often will, consider that all the
paths of В are continuous.
As soon as we have constructed the standard linear BM of Theorem A.9), we
can construct a host of other interesting processes. We begin here with a few.
1°) For any x e №., the process X* — x + B, is called the Brownian motion
started at x, or in abbreviated form a BM(x). Obviously, for any A e .-
€ A] = -±= I e-{*-xIl2tdy = f g,(y- x)dy.
J y/baJA Ja
2°) If fi,1, B?,..., Bf, are d independent copies of Bt, we define a process
X with state space M.d by stipulating that the i-th component of X, is B\. This
process is called the d-dimensional Brownian motion. It is a continuous Gaussian
process vanishing at time zero. Again as above by adding x, we can make it start
from x e M.d and we will use the abbreviation BMd(x).
3°) The process X, = (t, Bt) with state space M.+ x R is a continuous process
known as the heat process. We can replace B, by the d-dimensional BM to get
the heat process in l+xtf.
4°) Because of the continuity of paths
sup {Bs, 0 < s < t} — sup {B,, 0 < s < t, s e <Q>}.
Therefore, we can define another process S by setting S, = sups5, Bs. In similar
fashion, we can consider the processes \B,\, B* — supv<, \BS\ or ?+ = sup@, fi,).
5°) Finally, because of the continuity of paths, for any Borel set A, the map
(co,s)^ \A(B,(a>))
is measurable on the product [Q x K+, .7" <8> ¦*?(№.+)) and therefore
Jo
U(Bs)ds
is meaningful and defines yet another process, the occupation time of A by the
Brownian motion B.
We finally close this section by describing a few geometrical invariance prop-
properties of BM, which are of paramount importance in the sequel, especially property
(iii).
A.10) Proposition. Let В be a standard linear BM. Then, the following properties
hold:

§1. Examples of Stochastic Processes. Brownian Motion 21
(i) (time-homogeneity). For any s > 0, the process Bl+S — Bs, t > 0, is a Brownian
motion independent ofa(Bu, и < s);
(ii) (symmetry). The process —B,,t>Q,isa Brownian motion;
(Hi) (scaling). For every с > 0, the process cB,jci ,t>0,isa Brownian motion;
(iv) (time-inversion). The process X defined by Xo = 0, X, — t By, for t > 0, is
a Brownian motion.
Proof, (i) It is easily seen that X, = Bl+S — Bs is a centered Gaussian process,
with continuous paths, independent increments and variance t, hence a Brownian
motion. Property (ii) is obvious and (iii) is obtained just as (i).
To prove (iv), one checks that X is a centered Gaussian process with covari-
ance inf(.s, t)\ thus, it will be a BM if its paths are continuous and, since they
are clearly continuous on JO, oo[, it is enough to prove that lim,_>o-^ = 0. But
X,, t e]0, oo[ is equivalent to B,,t e]0, oo[ and since lim,_>o.feQ B, = 0, it fol-
follows that lim^o.reQ X, — 0 a.s. Because X is continuous on ]0, oof, we have
r X, — 0 a.s. о
Remarks. 1°) Once we have constructed a BM on a space (?2, ¦ -A, P), this propo-
proposition gives us a host of other versions of the BM on this same space.
2°) A consequence of (iv) is the law of large numbers for the BM, namely
P[liml^oor'Bl=0] = l.
3°) These properties are translations in terms of BM of invariance properties
of the Lebesgue measure as is hinted at in Exercise A.14) 2°).
A.11) Exercise. Let В be the standard linear BM on [0, 1], i.e. we consider only
t e [0, 1]. Prove that the process Bt, 0 < t < 1, defined by
B, = #i_, — B\
is another version of B, in other words, a standard BM on [0, 1].
A.12) Exercise. We denote by H the subspace of C([0, 1]) of functions h such
that h@) = 0, h is absolutely continuous and its derivative h' (which exists a.e.)
satisfies
/•l
/ h'(sJds < +oo.
Jo
Г) Prove that H is a Hilbert space for the scalar product
, h)= f g'(s)h'(s)ds.
Jo
2°) For any bounded measure /x on [0, 1], show that there exists an element h
in H such that for every / e H
and that h\s) = /j.(]s, 1]).

22 Chapter I. Introduction
[Hint: The canonical injection of И into C([0, 1]) is continuous; use Riesz's
theorem.]
3=) Let В be a standard linear BM, /i and v two bounded measures associated
as in 2°) with h and g. Prove that
X<x{w)=\ Bs(to)dn(s) and Xv(co)= Bs(o))dv(s)
Jo Jo
are random variables, that the pair (X1*, Xv) is Gaussian and that
E [X^XV] = / / infE, t)dfi(s)dv(t) = (h, g).
Jo Jo
4°) Prove also that with the notation of Exercise A.14) below
X" = f n(]u, \})dBu.
Jo
This will be taken up in Sect. 2 Chap. VIII.
* A.13) Exercise. Г) Let В be the standard linear BM. Prove that lim,^^ (B,/VF)
is a.s. > 0 (it is in fact equal to +oo as will be seen in Chap. II).
2°) Prove that В is recurrent, namely: for any real x, the set [t : Bt — x] is
unbounded.
3°) Prove that the Brownian paths are a.s. nowhere locally Holder continuous
of order a if a > | (see Sect. 2).
[Hint: Use the invariance properties of Proposition A.10).]
# A.14) Exercise. 1°) With the notation of Proposition A.3) and its sequel, we set
for / bounded or more generally in L^e(R+) for e > 0,
Y,= f f(s)dBs=X(f\[Oj]).
Jo
Prove that the process Y has a continuous version. This is a particular case of the
stochastic integral to be defined in Chapter IV and we will see that the result is
true for more general integrands.
2°) For с > 0 and / e L2(R+), set Xе(f) = cX(fc) where fc(t) = f{c2t).
Prove that Xе is also a Gaussian measure with intensity the Lebesgue measure
on (R+, -J$(R+)). Derive therefrom another proof of Proposition (l.lO)iii). Give
similar proofs for properties i) and ii) of Proposition A.10) as well as for Exercise
A.11).
A.15) Exercise. Let Б be a standard linear BM. Prove that
X(co)= / B2(co)ds
Jo
is a random variable and compute its first two moments.

§ 1. Examples of Stochastic Processes. Brownian Motion 23
# A.16) Exercise. Let В be a standard linear BM.
1°) Prove that the f.d.d.'s of В are given, for 0 < f, < t2 < ... < tn, by
Р[В„ e А,,й,2 e A2 Btn e А„]
= / gtl(x,)dx\ I gl2-lt (x2 -xi)dx2... ?,„-,„_, (х„ -xn-i)dxn.
JAX JA2 JAn
2°) Prove that for ?, < t2 < ... < tn < t,
P [B, € A|Bfl,..., fl,.] = / g,-,, {у - В,„) dy.
JA
More generally, for s < t,
P [B, eA\a (Bu, и < s)] = f g,., (y - Bs)dy.
J A
A.17) Exercise. Let В be the standard BM1 and let X be the Lebesgue measure
onR2.
1°) Prove that the sets
{fl,(a»),0<r < 1}, Y2(v) = {Bt(a)),0<t < 2},
Bi(o)),0 < t < 1}, у4(ш) = {В1+,(ш)-в1(ш),0<Г < 1},
are a.s. Borel subsets of M2 and that the maps со -*¦ X (/,(«)) are random variables.
[Hint: To prove the second point, use the fact that for instance
= jz:inf|z-B,,(a»)|=0|.]
2°) Prove that ?[A(y2)] = 2?[Я(у,)], and ?[A(y,)] - E[X(y3)] = E[X(y4)].
3°) Deduce from the equality
E[X{y3 U y4)] = ?[Я(у3) + X{yA) - Цу3 Л у4)]
that ?[А(у,)] = 0, hence that the Brownian curve has a.s. zero Lebesgue measure.
One may have to use the fact (Proposition C.7) Chap. Ill) that for a BM1 /3 and
every t, the r.v. S, = sup,<, Д is integrable.
* A.18) Exercise. Let В be the standard linear BM. Using the scaling invariance
property, prove that
converges in law, as t tends to infinity, to Si = sups<1 Bs.
[Hint: Use the Laplace method, namely ||/||p converges to ||/||oo as p tends to
+oo where || ||p is the Lp-norm with respect to the Lebesgue measure on [0, 1].]
The law of Si is found in Proposition C.7) Chap. III.

24 Chapter I. Introduction
# A.19) Exercise. P) If X is a BMrf, prove that for every x e M.d with ||x|| = 1,
the process {x, Xt) is a linear BM.
2°) Prove that the converse is false. One may use the following example: if
B = (B\ B2) is aBM2, set
XI pi p2 \r2 p2 I pi
t - Й2г/3 ~ йг/3. Xi = Й2г/3 + йг/3-
A.20) Exercise (Polar functions and points). A continuous function / from M+
into Ш2 is said to be polar for BM2 if for any x e M2, P [Гх] = 0 where
Гх — {3t > 0 : B, +x — /(?)} (the measurability of the set Г, will follow from
results in Sect. 4). The set of polar functions is denoted by л.
1°) Prove that / is polar if and only if
P [3r > 0 : Bt = f(t)] = 0.
[Hint: See P) in Exercise C.14).]
2°) Prove that л is left invariant by the transformations:
a) / -» -/;
b) /->Го/ where Г is a rigid motion of M2;
c) f^(t^
3°) Prove that if / ? л, then E [Л ({B, + f(t), t > 0})] > 0 where Л is the
Lebesgue measure in M2. Use the result in Exercise A.17) to show that one-point
sets are polar i.e. for any x e M2,
P [3f > 0 : B, = x] = 0.
Extend the result to BMrf with d > 3. Another proof of this important result
will be given in Chap. V.
4°) Prove that almost all paths of BM2 are in л.
[Hint: Use the independent copies B1 and B2 of BM2, consider B1 — B2 and
apply the result in 3°).]
** A.21) Exercise. Let X = B+ or |B| where В is the standard linear BM, p be a
real number > 1 and q its conjugate number (/?~' + q~l — l).
1°) Prove that the r.v. Jp — supr>0 (X, — tp/2) is a.s. strictly positive and finite
and has the same law as sup,>0 (X,/ (l + tpl2)L.
2°) Using time-inversion, show that
and conclude that E[JP] < oo.
[Hint: Use Theorem B.1).]
3°) Prove that there exists a constant CP(X) such that for any positive r.v. L
E[XL] < Cp{X)\\Lx'2\\p.

§1. Examples of Stochastic Processes. Brownian Motion 25
[Hint: For fi > 0, write E[XL] = E[XL - /j.Lp/2] + /j.E[Lp/2] and using
scaling properties, show that the first term on the right is less than (j.~{q/p)E[Jp].]
4°) Let LM be a random time such that
XL,-nLf = sup (X, -nt"'2).
Prove that LM is a.s. unique and that the constant CP(X) — pxlp (#?[•//,]I/<? is
the best possible.
5°) Prove that
E[XLl\jp]=qJp,
** A.22) Exercise. A continuous process X is said to be self-similar (of order 1) if
for every X > 0
№u, t > 0) (=' (XX,, t > 0)
i.e. the two processes have the same law (see Sect. 3).
1°) Prove that if В is the standard BM then X, — B,i is self-similar.
2°) Let henceforth X be self-similar and positive and, for p > 1, set
Sp = sup (X, -sp) and X* = sup Xs.
Prove that there is a constant cp depending only on p such that for any a > 0
P [cp (X\)p > a] < P [Sp > a] where ^-1 + p = 1.
[Hint: Sp = sup^Q (X* - tp); apply the self-similarity property to X*.]
3°) Let к > 1; prove that for any a > 0
P [Sp > a] < IP [{kx;)" > a] + f^ P [(***)* > k"pa].
[Hint: Observe that P [Sp > a] < sup {P [X*L - Lp > a], L positive random
variable } and write Q = {L < al'p} U |Jn>o {k"al'p <L< kn+xaVp).]
4°) Prove that if g is a positive convex function on M+ and g@) = 0, then
E [g (cP №)')] < E [g (Sp)] < (l + щ^) E [g ((kX*)<)].
[Hint: If X and Y are two positive r.v.'s such that P[X > a] < ?„ anP[Y >
Pna], then
E[g(X)] = Г Р[Х > a]dg{a) < ? ~
J0 V Pn

26 Chapter 1. Introduction



2. Local Properties of Brownian Paths


Our first task is to figure out what the Brownian paths look like. It is helpful, when
reasoning on Brownian paths, to draw a picture of the paths of the heat process
or, in other words, of the graph of the mapping t -+ B, (U) (Fig. 2 below). This
graph should be very "wiggly". How wiggly is the content of this section which
deals with the local behavior of Brownian paths.


B,


o


Fig.2.


We begin with a more general version of the Kolmogorov criterion which was
stated in the preceding section. We consider a Banach-valued process X indexed
by a d-dimensional parameter. The nOnTI we use on ]Rd is Itl = SUPi Itil and we
also denote by 1 1 the nOnTI of the state space of X. We recal1 that a Banach-valued
function 1 on ]Rd is 10cal1y H6lder of order a if, for every L > 0,


sup {I/U) - l(s)I/lt - sla; It/, Isi :s L, t i= s} < 00.


(2.1) Theorem. Let X" t E [0, 1 [d, be a Banach-va/ued process for which there
exist three strictly positive constants y, C, E such that


E [IX t - XsIY] :S clt - sld+e;


then, there is a modification X of X such that


E [(s.

 (IX t - x"" 1/ It - sla) rJ < +00


for every a E [0, E/Y[. ln particu/ar, the paths of X are Ho/der continuous of
order a.


Proo! For mEN, let Dm be the set of d-uples


(2 - m . 2 -m' )
s = 11,..., Id




§2. Local Properties of Brownian Paths 27
where each ^ is an integer in the interval [0, 2m[ and set D = UmDm. Let further
Am be the set of pairs (s, t) in Dm such that \s — t\ — 2~'n; there are fewer than
2(m+\)d suc^ pajrs Finally for s and t in D, we say that s < t if each component
of .5 is less than or equal to the corresponding component of t.
Let us now set Kt = sup(s ,)ezl \XS — X,\. The hypothesis entails that for a
constant J,
(s.t)eA,
For a point s (resp.: t) in D, there is an increasing sequence (sn) (resp.: (/„))
of points in D such that sn (resp.: tn) is in Dn, sn < s(tn < t) and sn = s(tn = t)
from some n on.
Let now s and t be in D and \s — t\ < 2~m; either sm = tm or (sm, tm) e Am,
and in any case
xs -x, = J2 (xSi+l - xSl) + х,„ - xtm + J
i=m i—
where the series are actually finite sums. It follows that
\X,-X,\ <Km
As a result, setting Ma = sup{|X, - Xs\/\t - s\a; s, t € D. s ф г}, we have
Ma < supJ2('"+1)a sup \X, ~Xs\:s,t e D. ^ r 1
m+\
< sup 2.2
""+"«
For у > 1 and a < e/y, we get, with ./' = 2a+] J[/y,
\\Ma\\y <2a+lJ22""WKiWy ^ J'^2i{a-{tlY)) < oo.
(=0 /=0
For у < 1, the same reasoning applies to E[(Ma)r] instead of ||Ma||y.
It follows in particular that for almost every со, X is uniformly continuous on
D and it makes sense to set
it, (со) = \imXs(co).
By Fatou's lemma and the hypothesis, X, = X, a.s. and X is clearly the desired
modification.

28 Chapter I. Introduction
Remark. Instead of using the unit cube, we could have used any cube whatsoever.
In the case of Brownian motion, we have
E[(B, -B.,J] = \t-s\
and because the increments are Gaussian, for every p > 0,
E[\Br-Bs\2p] = Cp\t-s\"
for some constant Cp. From this result we deduce the
B.2) Theorem. The linear Brownian motion is locally Holder continuous of order
a for every a < 1/2.
Proof. As we have already observed, a process has at most one continuous mod-
modification (up to indistinguishability). Theorem B.1) tells us that BM has a mod-
modification which is locally Holder continuous of order a for a < (p — \)/2p —
1/2 — l/2p. Since p can be taken arbitrarily large, the result follows. D
From now on, we may, and will, suppose that all the paths of linear BM are
locally Holder continuous of order a for every a < 1/2. We shall prove that the
Brownian paths cannot have a Holder continuity of order a, for a > 1/2 (see also
Exercise A.13)). We first need a few definitions for which we retain the notation
of Sect. 4 Chap. 0.
For a real-valued function X defined on Ш+, we set
At variance with the Sf of Chap. 0, it is no longer true that T,A' > T,A if A' is a
refinement of A and we set
B.3) Definition. A real-valued process X is of finite quadratic variation if there
exists a finite process (X, X) such that for every t and every sequence {А„} of
subdivisions of[0, t] such that \А„\ goes to zero,
P-lim7;4' = (X, X),.
The process {X, X) is called the quadratic variation of X.
Of course, we may consider intervals [s, t] and, with obvious notation, we will
then have
P-lim Tsu," = (X, X), - (X, X),;
thus, {X, X) is an increasing process.
Remark. We stress that a process may be of finite quadratic variation in the sense
of Definition B.3) and its paths be nonetheless a.s. of infinite quadratic variation
in the classical sense, i.e. sup4 TtA = oo for every t > 0; this is in particular the
case for BM. In this book the words "quadratic variation" will be used only in the
sense of Definition B.3).

§2. Local Properties of Brownian Paths 29
B.4) Theorem. Brownian motion is of finite quadratic variation and {В, В), = t
a.s. More generally, if X is a Gaussian measure with intensity fi, and F is a set
such that fi(F) < oo, for every sequence [F?] ,n = 1,2,... of finite partitions of
F such that sup4 /x (F4") —* 0,
in the L2-sense.
Proof. Because of the independence of the X (FtB)'s, and the fact that E \x(F?J~\
I ? X{F»kf - (
and since for a centered Gaussian r.v. Y, E[Y4] = 3E[Y2]2, this is equal to
which completes the proof. D
Remarks. 1°) This result will be generalized to semimartingales in Chap. IV.
2°) By extraction of a subsequence, one can always choose a sequence (А„)
such that the above convergence holds almost-surely; in the case of BM, one can
actually show that the a.s. convergence holds for any refining (i.e. А„ с A,+i)
sequence (see Proposition B.12) in Chap. II and Exercise B.8) in this section).
B.5) Corollary. The Brownian paths are a.s. of infinite variation on any interval.
Proof. By the foregoing result, there is a set J?oCl2 such that P(f2o) — 1 and
for any pair of rationale p < q there exists a sequence (A") of subdivisions of
[p, q] such that \A"\ -+ 0 and
lim ]T (BlM (со) - В„ (со)J = q-p
tie A"
for every to e Qo.
Let V(to) < +oo be the variation of t -*¦ B,(to) on [p, q]. We have
? (Bti+I(co) - Btl(io)J < isup\Bli+l(co) - B,,(co)\\ V(co).
By the continuity of the Brownian path, the right-hand side would converge to 0
as и —»• oo if V(co) were finite. Hence, V(co) = +oo a.s. D

In the following, we will say that a function is nowhere locally Holder contin-
continuous of order a if there is no interval on which it is Holder continuous of order
a.
B.6) Corollary. The Brownian paths are a.s. nowhere locally Holder continuous
of order a for a > 1/2.
Proof. It is almost the same as that of Corollary B.5). If \В,(ш) — Bs(a))\ <
k\t - s\a for p < s, t <q and a > 1/2 then
BllM (w) - Btl (w)J <k2(q- p) sup |r,+, - U
2a-1
and we conclude as in the previous proof. D
Theorem B.2) and Corollary B.6) leave open the case a = 1/2. The next
result shows in particular that the Brownian paths are not Holder continuous of
order 1/2 (see also Exercise B.31) Chap. III).
B.7) Theorem (Levy's modulus of continuity). Ifh(t) = Bf log(l/f)I/2,
= 1.
liml sup \Btl-Bu\ Д(?) -
?->0 \ 0<Г]<г2<1 ' |
/T -I] <f
Proof. Pick a number S in ]0, 1 [ and consider the quantity
Ln = P Inua^ \Вк2-„ - %_,J-"| < A -
By the independence of the increments, Ln is less than
е л
dx
1-.5)^2 log 2" V27T
By integrating by parts the left side of the inequality
the reader can check that
f-H)
Й2\„ а
db > -5—- exp
H)-
Using this and the inequality 1 — s < e ¦*, we see that there exists a constant
С > 0 such that

It follows from the Borel-Cantelli lemma that the lim in the statement is a.s.
> 1 — S, and it is a.s. > 1 since S is arbitrary. We shall now prove the reverse
inequality.
Again, we pick S e]0, 1[ and e > 0 such that A + eJ(l - S) > 1 + S. Let К
be the set of pairs (i, j) of integers such that 0 < i < j < 2" and 0 < j — i < 2"s
and for such a pair set к — j — i. Using the inequality
Г exp (-b2/2) db < f exp (~b2/2) (b/a)db = a exp (-a2/2) ,
Ja Ja
and setting L = P [max4eAr (|B/2- - Ba-n\ /h{k2~n)) > 1 + e], we have
/•00
/ exp(-x2/2)dx
(•00
L <
k V1Ж Jd+sJIogti-'I»)
where D is a constant which may vary from line to line. Since k~x is always
larger than 2~"s, we further have
к
Moreover, there are at most 2"(I+5) points in К and for each of them
so that finally
By the choice of e and S, this is the general term of a convergent series; by the
Borel-Cantelli lemma, for almost every со, there is an integer n(co) such that for
и > n(co),
\Bj2- -B,-2-»| < (\+s)h(k2-")
where к е К and к = j — i. Moreover, the integer и may be chosen so that for
и > n(co),
?/7B""') < D ¦ h{2~") ? (P2-p)l/2 < 8h B-("+1»1-«).
Let со be a path for which these properties hold; pick 0 < t\ < ti < 1 such that
h ~ tx < 2-я(<")A-*>. Next pick an integer n > n(co) such that
may find integers i, j, pr, qs such that
Г, = i2~" - 2"'" - 2"''2 - ..., t2 = j2~" + 2~41 + 2~42 +...,

with n < p\ < p2 < ..., n < q\ < цг < ... and 0 < j — i < (t2 — U) 2 " < 2"s.
Since В is continuous, we get
|В„ (со) - B,2(co)\ < |B.-2-(ш) - Bh (co)\ + \Bj2-(со) - Bi2-(со)I
+ \B,2(co)-Bj2-(co)\
< 2A + e) ?*B-') + A + e)A ((; - iJ"")
< 2A + s)Sh B-("+"(|-г») + A + fi)A ((y - 02"").
Since h is increasing in a neighborhood of 0, for ti — ti sufficiently small, we get
\Bu(co) - Bh(co)\ < B(l+e)a+(l+e))*(fc-fi).
But s and S can be chosen arbitrarily close to 0, which ends the proof.
# B.8) Exercise. If В is the BM and A,, is the subdivision of [0, t] given by the
points tj = j2~"t, j — 0, 1,..., 2", prove the following sharpening of Theorem
B.4):
lim T. " = t almost-surely.
и
[Hint: Compute the exact variance of T,A" — t and apply the Borel-Cantelli
lemma.]
This result is proved in greater generality in Proposition B.12) of Chap. II.
* B.9) Exercise (Non-differentiability of Brownian paths). 1°) If g is a real-
valued function on R+ which is differentiable at t, there exists an integer / such
that if / = [nt] + 1 then
UU/n)-g{(j-l)/n)\<71/n
for / < j < i + 3 and и sufficiently large.
2°) Let D, be the set of Brownian paths which are differentiable at t > 0.
Prove that (J, D, is contained in the event
и+l i+3
г = U lim U П №и/п) - B((j - i)/n)| < //«}
and finally that Р(Г) = 0.
#* B.10) Exercise. Let (Xf) be a family of Шк -valued continuous processes where
t ranges through some interval of WL and a is a parameter which lies in W1. Prove
that if there exist three constants у, с, s > 0 such that
("sup |X," - Xf|rl <c\a-
then, there is a modification of (X") which is jointly continuous in a and t and is
moreover Holder continuous in a of order a for a < s/y, uniformly in t.

B.11) Exercise (p-variation of BM). 1°) Let В be the standard linear BM and,
for every n, let t,¦. = i/n, i = 0 n. Prove that for every p > 0
и-1
1=0
converges in probability to a constant vp as и tends to +oo.
[Hint: Use the scaling invariance properties of BM and the weak law of large
numbers.]
2°) Prove moreover that
V=o /
converges in law to a Gaussian r.v.
B.12) Exercise. For p > 0 given, find an example of a right-continuous but
discontinuous process X such that
E[\Xt-Xl\e]<C\t-s\
for some constant С > 0, and all s, t > 0.
[Hint: Consider X, = l(y<o for a suitable r.v. Y.]
§3. Canonical Processes and Gaussian Processes
We now come back more systematically to the study of some of the notions
introduced in Sect. 1. There, we pointed out that the choice of a path of a process
X amounts to the choice of an element of .F(T, E) for appropriate T and E.
It is well-known that the set .F~(T, E) is the same as the product space ET. If
w e ,F~(T, E) it corresponds in ET to the product of the points w(t) of E.
From now on, we will not distinguish between .F~(T, E) and ET. The functions
Y,, t e T, taking their values in E, defined on ET by Y,(w) = w(t) are called the
coordinate mappings. They are random variables, hence form a process indexed
by T, if ET is endowed with the product a -algebra K1'. This a -algebra is the
smallest for which all the functions Y, are measurable and it is the union of the
ff-algebras generated by the countable sub-families of functions F,, t e T. It is
also the smallest ст-algebra containing the measurable rectangles ГЬег ^' wnere
A, e fC for each t and A, = E but for a finite sub-family (t\, ...,tn) of T.
Let now X,J e 7\ be a process defined on (i2,.F~, P) with state space
(?, %). The mapping ф from Q into ET defined by
ф(ш)Ц) = X,(co)
is measurable with respect to .7" and <C ' because Y, оф is measurable for each t.
Let us call Px the image of P by </>. Plainly, for any finite subset (fj f,,) of
T and sets Л, е X,

Р [Х„ еА Х,,б А„] = px[Y,,gAi,..., У,, е А„],
that is, the processes X and Y are versions of each other.
C.1) Definition. The process Y is called the canonical version of the process X.
The probability measure P\ is called the law ofX.
In particular, if X and X' are two processes, possibly defined on different
probability spaces, they are versions of each other if and only if they have the
same law, i.e. Px — /V on ET and this we will often write for short
X(=>X' or (X,,t>0) = (X',,t>0).
For instance, property iii) in Proposition A.10) may be stated
(B,,t >0) = (cBc-2,,t >0).
Suppose now that we want to construct a process which models some physical
phenomenon. Admittedly, nature (i.e. statistical experiments, physical considera-
considerations, ...) gives us a set of f.d.d.'s, and the goal is to construct a process with
these given distributions. If we can do that, then by the foregoing we also have
a canonical version; actually, we are going to construct this version in a fairly
general setting.
We recall from Sei 1 that the family ,. /6% of the finite-dimensional distri-
distributions of a process X forms a projective family. The existence of the canonical
version of a process of given finite-dimensional distributions is ensured by the
Kolmogorov extension theorem which we recall here without proof.
C.2) Theorem. IfE is a Polish space and % the a-algebra of its Borel subsets,
for any set T of indices and any projective family of probability measures on finite
products, there exists a unique probability measure on (ET, <5T) whose projections
on finite products are the given family.
For1 our present purposes, this theorem might have been stated: if (?, ?>)
is Polish, given a family ../? of finite-dimensional distributions, there exists a
unique probability measure on (E7', <CT) such that, /6Y = .-/6 for the coordinate
process Y.
The above result permits to construct canonical versions of processes. How-
However, for the same reason as those invoked in the first section, one usually does not
work with the canonical version. It is only an intermediate step in the construction
of some more tractable versions. These versions will usually have some continuity
properties. Many processes have a version with paths which are right continuous
and have left-hand limits at every point, in other words: lims;, Xv(w) = X,(w)
and Нщф Xs(w) exists; the latter limit is then denoted by X,-(w). We denote by
D(K+, E) or simply D the space of such functions which are called cadlag. The
space D is a subset of ET; to say that X has a version which is a.s. cadlag is

equivalent to saying that for the canonical version of X the probability measure
px on ?>T gives the probability 1 to every measurable set which contains D.
In that case, we may use D itself as probability space. Indeed, still calling
Y, the mapping w -*¦ w(t) on D, let #"J be the ex-algebra o(Y,, t > 0); plainly,
%т = <CT П D. The reader will easily check that although D is not measurable
in %T, one defines unambiguously a probability measure Q on K? by setting
0(Г) = Px(f)
where Г is any set in К T such that Г — Г П D. Obviously, the process Y defined
on (D, &?, Q) is another version of X. This version again is defined on a space
of functions and is made up of coordinate mappings and will also be referred to as
canonical; we will also write Px instead of Q; this causes no confusion so long
as one knows the space we work with.
Finally, if X is a process defined on (i2,.V, P) with a.s. continuous paths,
we can proceed as above with C(R+, E) instead of D and take the image of P
by the map ф defined on a set of probability 1 by
We can do that in particular in the case of Brownian motion and we state
C.3) Proposition. There is a unique probability measure W on C(R+,R) for
which the coordinate process is a Brownian motion. It is called the Wiener measure
and the space C(ffi+, ffi) is called the Wiener space.
Proof. One gets W as the image by ф of the probability measure of the version
of BM already constructed. D
It is actually possible to construct directly the Wiener measure, hence a con-
continuous version of BM, without the knowledge of the results of Sect. 1. Let . /.
be the union of the ex-fields on С = C(R+, R) generated by all the finite sets of
coordinate mappings;. ¦/¦ is an algebra which generates the cx-field on C. On. -/,,
we can define W by the formula in Exercise A.16) Iе) and then it remains to
prove that the set function thus defined extends to a probability measure on the
whole cx-field.
For the Brownian motion in Rd, we define similarly the Wiener space
i+, WLd) and the Wiener measure W as the only probability measure on the
Wiener space for which the coordinate process is a standard BMrf. The Wiener
space will often be denoted by W or W' if we want to stress the dimension. It
is interesting to restate the results of Section 2 in terms of Wiener measure, for
instance: the Wiener measure is carried by the set of functions which are Holder
continuous of order a for every a. < 1 /2 and nowhere Holder continuous of order
a fora > 1/2.
We now introduce an important notion. On the three canonical spaces we have
defined, namely ER\ ?>(R+. E), C(R+. E), we can define a family of transfor-
transformations в,, t 6 R+, by

where as usual Y is the coordinate process. Plainly, 6t о 6S = 6t+s and в, is
measurable with respect to a (X.s. s > t) hence, to a (Xs, s > 0). The effect of в,
on a path w is to cut off the part of the path before t and to shift the remaining
part in time. The operators 6,,t >0, are called the shift operators.
C.4) Definition. A process X is stationary if for every t\ t,, and t and every
A, eX
P [X,+h 6 Ax,..., X,+tn e An] = P [Х„ бЛ, Х,„ б A,,].
Another way of stating this is to say, using the canonical version, that for
every t, 6,(PX) = Px', in other words, the law of X is invariant under the shift
operators.
We now illustrate the above notions by studying the case of Gaussian processes.
C.5) Definition. A real-valued process Xt,t e T, is a Gaussian process if for any
finite sub-family (fi,..., tn) of T, the vector r.v. (X, X,J is Gaussian. The
process X is centered ifE[X,] = 0 W 6 T.
In other words, X is a Gaussian process if the smallest closed subspace of
l?(Q,.V, P) containing the r.v.'s X,,f 6 T is a Gaussian subspace. As was
already observed, the standard linear BM is a Gaussian process.
C.6) Definition. IfX,,teT,isa Gaussian process, its covariance Г is the func-
function defined on T x T by
r(s, t) = cov (X,, X,) = E [(X, - ?[X.S])(X, - ?[X,])].
Let us recall that a semi-positive function Г on 7" is a function from T x T
into R such that for any rf-uple (J\ tj) of points in T and any integer d, the
d x cf-matrix (r(f;, tj)) is semi-definite positive (Sect. 6, Chap. 0).
C.7) Proposition. The covariance of a Gaussian process is a semi-definite positive
function. Conversely, any symmetric semi-definite positive function is the covari-
covariance of a centered Gaussian process.
Proof. Let (t,-) be a finite subset of T and a,- some complex numbers; then
? ПГ,-. tjteaj = e\\ J> (X,, - E [X,,]) И > 0
i.j L j J
which proves the first statement.
Conversely, given a semi-definite positive function Г, for every finite subset
t\ t,, of 7\ let Pn tn be the centered Gaussian probability measure on R"
with covariance matrix (r(f,, f,)) (see Sect. 6 Chap. 0). Plainly, this defines a
projective family and under the probability measure given by the Kolmogorov
extension theorem, the coordinate process is a Gaussian process with covariance
Г. We stress the fact that the preceding discussion holds for a general set T and
not merely for subsets of M. D

Remark. The Gaussian measures of Definition A.4) may be constructed by apply-
applying Proposition C.7) with T = . / and Г(А, В) = ц(А П В).
We have already seen that the covariance of the standard linear BM is given
by inf(^, /). Another large set of examples is obtained in the following way. If fi
is a symmetric probability measure on Ш, its Fourier transform
<P(t)
= / e"
J—oc
is real and we get a covariance on Г = К by setting F(t, t') — <j>(t — t') as
the reader will easily show. By Proposition C.7), the process associated with
such a covariance is a stationary process. In particular we know that for /3 > 0,
the function exp(—f}\t\) is the characteristic function of the Cauchy law with
parameter /6. Consequently, the function f(f, t') = cexp(-^|/ - r'|) with с > 0
is the covariance of a stationary Gaussian process called a stationary Ornstein-
Uhlenbeck process (abbreviated OU in the sequel) with parameter fi and size с If
we call this process X, it is easily seen that
E[{X,-X,.f]<2cp\t-t'l
hence, by Kolmogorov's continuity criterion, X has a continuous modification.
Henceforth, we will consider only continuous modifications of this process and
the time set will often be restricted to Ш+.
Another important example is the Brownian Bridge. This is the centered Gaus-
Gaussian process defined on T = [0, 1] and with covariance F(s, t) = s(\ — t) on
(s < t). The easiest way to prove that Г is a covariance is to observe that for
the process X, = B, - tB\ where В is a BM, E[XSX,] - s(l - t) for s < t.
This gives us also immediately a continuous version of the Brownian Bridge. We
observe that X\ = 0 a.s. hence all the paths go a.s. from 0 at time 0 to 0 at time 1;
this is the reason for the name given to this process which is also sometimes called
the tied-down or pinned Brownian motion. More generally, one may consider the
Brownian Bridge Xy between 0 and у which may be realized by setting
Xy = B,-t(B{- v) = X°, +1 v, 0 < / < 1,
where Xr° = X, is the Brownian Bridge going from 0 to 0. Exercise C.16) de-
describes how Xy may be viewed as BM conditioned to be in у at time 1. In the
sequel, the words "Brownian Bridge" without further qualification will mean the
Bridge going from 0 to 0. Naturally, the notion of Bridge may be extended to
higher dimensions and to intervals other than [0, 1].
Finally, there is a centered Gaussian process with covariance F(s,t) = 1 [.,=»],
s, t e Ш+. For such a process, any two r.v.'s Xs, X, with .v ф t are independent,
which can be seen as "great disorder". It is interesting to note that this process ¦
does not have a good version; if it had a measurable version, i.e. such that the
map (r, oo) -»¦ X,(w) were measurable (see Definition A.14) Chap. IV), then for
each /, Y, = /J Xsds would be a Gaussian r.v. and, using a Fubini argument, it is

easily shown that we would have ?[Y,2] = 0, hence Y, = 0 a.s. Consequently, we
would get X,((i>) = 0 dtP(doo)-a.s., which is in contradiction with the equality
¦DM--
# C.8) Exercise. 1°) Let В be a linear BM and A a positive number; prove that the
process
X, =e-ArficxpBA0, teR,
is a stationary Ornstein-Uhlenbeck process; compute its parameter and its size.
Conclude that the stationary OU process has a continuous version whose paths are
nowhere differentiable.
2°) Let X be a continuous OU process with parameter 1/2 and size 1 and set
P, = X, + A/2) / Xudu for r>0,
Jo
fa = X,- A/2) / Xudu for t < 0.
Jo
Prove that ft is a Gaussian process with continuous paths. In what sense is fi a
BM?
[Hint: See Exercise C.14).]
This is related to the solution of the Langevin equation in Chap. IX.
* C.9) Exercise (Fractional Brownian motion). Let d be a positive integer and p
a real number such that (d/2) — 1 < p < d/2. We set or = d — 2p.
1°) For x, у е Rd, prove that the function
fy(x) = \x- уГр - \x\-p
is in L2(M.d, m) where m is the Lebesgue measure on Rd'.
2°) Let X be the Gaussian measure with intensity m and set ZT = X(fy).
Prove that there is a constant с depending only on d and /? such that
The process Z is called the fractional Brownian motion of index or. For or = 1,
Z is Levy's Brownian motion with d parameters.
3°) Let X\ and X2 be two independent Gaussian measures with intensity m,
and set X = X{ + iX2. For v e Rd, set fy(x) = A - exp(i.y • x))/\x\d~P, where
• indicates the inner product in M.d. Show that
y= f fy(
(x)-X(dx) (уеМ')
is a constant multiple of the fractional Brownian motion of index or.
[Hint: The Fourier-Plancherel transform of the function /v is yfy, where у is
a constant independent of _y.]

& (ЗЛО) Exercise (Brownian Bridge). Let X be the Brownian Bridge (BB).
Г) Prove that the process Xi_,, 0 < t < 1 is also a BB and that
B, = (r + l)X(,/(,+i)), t>0,
isaBM. Conversely, if В is aBM, the processes (\-t)B(t/(\-t)) andrfi(r~'-l),
0 < t < 1, are BB's.
2°) If we write ? for f modulo 1, the process
Y, = XTTI-XS, 0<f<l,
where s is a fixed number of [0, 1] is a BB.
3°) (Continuation of Exercise A.21)). Prove that
sup- (|Д,| - 1) = sup(|S,| - 0 = sup X2,.
/>0 f />0 0</<l
C.11) Exercise (Brownian sheet). 1°) Prove that the function Г defined on Ж2+ x
R*.by
Г((и, s), (v, t)) = inf(M, v) x inf(s, t)
is a covariance and that the corresponding Gaussian process has a continuous
version. This process is called the Brownian sheet.
2°) Prove that the Brownian sheet may be obtained from L2AR^) as BM was
obtained from L2(l+) in Sect. 1.
Let (В(Г); Г e.S?(R2+), ffrdsdu < oo} be the Gaussian measure associ-
associated with the Brownian sheet B.
Prove that {!(#,), 0 < t < 1} is a Brownian bridge, where
R, = {(s, m);0 < s <t < и < 1}, 0<Г<1.
3°) If B(j.,) is a Brownian sheet, prove that the following processes are also
Brownian sheets:
a) ^(a2s.b2t)/ab where a and b are two positive numbers;
b) sB^-i,,) and i7B(i-i,,-ij;
c) %„+*./«+/) -B(.vA+.v.,0) -B,,o.,()+,)+B,,.o.,o) where s0, t0 are fixed positive numbers.
4C) If B(v,,) is a Brownian sheet, then for a fixed s, t —*¦ B(Si,) is a multiple of
linear BM. The process / -> B^.e-/, is a Ornstein-Uhlenbeck process.
C.12) Exercise (Reproducing Kernel Hilbert Space). Let Г be a covariance on
T x T, and X,,t e T, a centered Gaussian process with covariance Г.
1°) Prove that there exists a unique Hilbert space H of functions on T such
that
i) H is the closure of the subspace spanned by {F(t, ¦), t e Г);
ii) for every / e Я, </, Г(г. •)> = /(О-

2°) Let .Ж be the Gaussian space generated by the variables X,,t e T. Prove
that H and .%в are isomorphic, the isomorphism being given by
Ze-Ж-» (E[ZX,],t eT).
3°) In the case of the covariance of BM on [0, 1], prove that H is the space
H of Exercise A.12) in Chap. 1.
* C.13) Exercise. 1°) Let ф be a locally bounded measurable function on ]0, 1]
such that
Ш to/2)~s\<p(t)\ < +oo
no
for some S > 0. If В is the BM, prove that the integral
xf = f
Jo
defines a continuous Gaussian process on [0, 1]. (A sharper form is given in
Exercise B.31) of Chap. III). Characterize the functions ф for which В - Хф is
again a BM.
2°) Treat the same questions for the Brownian Bridge.
# C.14) Exercise. For any x e W, define the translation r,- on W = C(R+,:
by zx(w)(t) = x + w(t) and call WA the image of W by rv (thus, in particular
W° = W).
1°) Prove that under Wx the coordinate process X is a version of the BMd
started at jc. Observe that Wx is carried by the set [w : w@) = x] and conclude
that Wx and Wy are mutually singular if x ф у. On the other hand, prove that for
any e > 0, Wx and Wy are equivalent on the a -algebra cr(Xs, s > e).
2°) For any Г e a(Xs,s > 0), prove that the map x -*¦ У/*(Г) is a Borel
function. If д is a probability measure on the Borel sets of M.d, we define a
probability measure W^ on W by
-I
Under which condition is X a Gaussian process for W^? Compute its covariance
in that case.
* C.15) Exercise. Let us first recall a result from Fourier series theory. Let / be
continuous on [0. 1] with /@) = /A) = 0 and set, for к > 1,
ak = y/2 / /(/)cos2ttkt dt, bk = -Jl \ f(t)sm2nkt dt,
Jo Jo
then if J2T ak converges,
k=\

where the series on the right converges in L2([0,1]). By applying this to the paths
of the Brownian Bridge Bt —tB\, prove that there exist two sequences (&)*=<Г
and (Пк)к=Г of independent reduced Gaussian random variables such that almost
surely
B, = fg0 + V2 У" i-^T (cos2ttlet -\) + ~ sin27rfo ).
fr[\2nk Ink )
C.16) Exercise. (The Brownian Bridge as conditioned Brownian motion). We
consider ?2 = C([0, 1], R) endowed with its Borel cr-field and the Wiener measure
W. As Q is a Polish space and ./}\ = a{X() is countably generated, there exists a
regular conditional distribution P{ v, -) for the conditional expectation with respect
to.i?i.
1°) Prove that we may take P(y, •) to be the law Py of the Brownian Bridge
between 0 and y. In other words, for any Borel subset Г of Q
W(D
Ju.
2°) The function v -> Py is continuous in the weak topology (Chap. 0 and
Chap. XIII) that is: for any bounded continuous function / on П the map >• ->
fQ f(co)dPy(oo) is continuous.
C.17) Exercise. 1°) Let X be a Gaussian process with covariance Г on [0,1] and
suppose that for each t, X is differentiable at /, i.e., there exists a r.v. X't such that
lim/T1 (X,+h-X,) = X', a.s.
h—>0
Prove that d2F(s, t)/ds dt exists and is equal to the covariance of the process X'.
2°) Let В be the standard linear BM on [0, 1] and T(w) the derivative in the
sense of Schwartz distributions of the continuous map / -*¦ B,(a>). If f,g are
in C??(]0, 1[) prove that (T(w), f) and (T(co), g) are centered Gaussian random
variables with covariance equal to (U, fg) where U is the second mixed derivative
of inf(s, /) in the sense of Schwartz distributions, namely a suitable multiple of
the Lebesgue measure on the diagonal of the unit square.
§4. Filtrations and Stopping Times
In this section, we introduce some basic notations which will be used constantly
in the sequel.
D.1) Definition. A filtration on the measurable space (Q, .7") is an increasing
family (.i^"),>o, of sub-o-algebras of'.7'. In other words, for each t we have a
sub-a-algebra .7[ and .% С .7% if s < t. A measurable space (Q, .P~) endowed
with a filtration (.i^),>o, is said to be a filtered space.

D.2) Definition. A process X on (?2, .7") is adapted to the filtration (.^) ifX, is
.^-measurable for each t.
Any process X is adapted to its natural filtration .7f° = a (Xs, s < t) and
(¦7f°) is the minimal filtration to which X is adapted. To say that X is adapted to
(•>T) is to say that .7f° С -W, for each t.
It is the introduction of a filtration which allows for the parameter t to be really
thought of as "time". Heuristically speaking, the a-algebra .7, is the collection of
events which may occur before or at time / or, in other words, the set of possible
"pasts" up to time t. In the case of stationary processes, where the law is invariant,
it is the measurability with respect to .7, which places the event in time.
Filtrations are a fundamental feature of the theory of stochastic processes and
the definition of the basic objects of our study such as martingales (see the follow-
following chapter) or Markov processes, will involve filtrations. We proceed to a study
of this notion and introduce some notation and a few definitions.
With a filtration (.7^), one can associate two other filtrations by setting
We have \/1>0-7-- = V»>oi-^'= V/>o-^+ an(^ tn's ^-algebra will be denoted
.7^. The a -algebra .7^- is not defined and, by convention, we put .7Щ- = .^.
We always have .7t- с .7~, с J^V and these inclusions may be strict. If for
instance Q = D(R+, R), X is the coordinate process and the filtration is (-7^°),
the event [X, = a] is in .7^° and is not in .i^l°; a little later, we shall see an
example of an event in .7^ which is not in •^"°.
We shall encounter other examples of pairs {.7~,), (&,) of filtrations such that
¦7f С '¦?, for all t. We can think about this set-up as the knowledge that two differ-
different observers can have gained at time t, the '?, -observer being more skillful than
the .^"-observer. For instance, the ,7^+ -observer above can foresee the immediate
future.
This also leads us to remark that, in a given situation, there may be plenty of
filtrations to work with and that one may choose the most convenient. That is to
say that the introduction of a filtration is no restriction on the problems we may
treat; at the extreme, one may choose the constant filtration {.% = .7" for every
f) which amounts to having no filtration.
D.3) Definition. If .7^ = -7i+ for every t, the filtration is said to be right-
continuous.
For any filtration (.J^) the filtration (.Я+) is right-continuous.
We come now to an important definition.
D.4) Definition. A stopping time relative to the filtration (.^") is a map on Q with
values in [0, oo], such that for every t,

In particular, Г is a positive r.v. taking possibly infinite values. If (.7~t) is
right-continuous, it is equivalent to demand that [T < t] belongs to .7^ for every
/. In that case, the definition is also equivalent to: Г is a stopping time if and
only if the process X, = l]o.r]@ is adapted (X is then a left-continuous adapted
process, a particular case of the predictable processes which will be introduced in
Sect. 5 Chap. IV and in Exercise D.20)).
The class of sets A in .7^. such that А П [T < t] e .>^for all t is a a -algebra
denoted by -Щ; the sets in .Щ- must be thought of as events which may occur
before time T. The constants, i.e. T(w) = s for every w, are stopping times and
in that case .Tj- = .7~s. Stopping times thus appear as generalizations of constant
times for which one can define a "past" which is consistent with the "pasts" of
constant times.
The proofs of all the above facts are left to the reader whom we also invite to
solve Exercises D.16-19) to become acquainted with stopping times.
A stopping time may be thought of as the first time some physical event occurs.
Here are two basic examples.
D.5) Proposition. IfE is a metric space, A a closed subset ofE and X the coor-
coordinate process on W = C(M+, E), and if we set
DA(co) = inf[t > 0, Х,(ш) е A]
with the understanding that inf@) = +oo, then DA is a stopping time with respect
to the natural filtration .7^° = o(Xs,s < t). It is called the entry time of A.
Proof. For a metric d on E, we have
[DA < t} = I ш : inf d (Xs(a>), A) = 0
and the right-hand side set obviously belongs to .3^"°. ?
This is one of the rare examples of interesting stopping times with respect to
(J^°). If we remove the assumption of path-continuity of X or of closedness of
A, we have to use larger filiations. We have for instance
D.6) Proposition. If A is an open subset of E and ?2 is the space of right-
continuous paths from R+ to E, the time
TA = inf{/ > 0 : X, e A] (inf@) = +oo)
is a stopping time with respect to .7^+. It is called the hitting time of A.
Proof. As already observed, TA is a .^"-stopping time if and only if [TA < t] e
&f° for each t. If A is open and Xs(w) e A, by the right-continuity of paths,
(со) e A for every / e [s, s + e[ for some e > 0. As a result
(TA < t} = (J [Xs eA}e .7?'.
.veQ,.5</

It can be seen directly or by means of Exercise D.21) that TA is not a (.i^0)-
stopping time and that, in this setting, .^+ is strictly larger than .^"°. This is a
general phenomenon; most interesting stopping times will be (.^"+°)-stopping times
and we will often have to work with (.J^"+°) rather than (.J^"°).
We now turn to the use of stopping times in a general setting.
Let (.7,) be a filtration on (Q,.W) and T a stopping time. For a process X,
we define a new mapping XT on the set [со : T(co) < 00} by
XT(co) = X,(co) if T(co) = t.
This is the position of the process X at time T, but it is not clear that XT is
a random variable on [T < 00}. Moreover if X is adapted, we would like XT
to be .^-measurable just as X, is .^-measurable. This is why we lay down the
following definitions where .^([0, t]) is the ст-algebra of Borel subsets of [0, t].
D.7) Definition. A process X is progressively measurable or simply progressive
(with respect to the filtration (.^)) if for every t the map (s, со) -*¦ Xs(co) from
[0, t] x Q into (?, <C) is ./J(\Q, t]) ® .^-measurable. A subset Г ofR+ x Q is
progressive if the process X — \r is progressive.
The family of progressive sets is a a-field on Ш+ х Q called the progressive
a-field and denoted by Prog. A process X is progressive if and only if the map
(t, to) -*¦ X,(w) is measurable with respect to Prog.
Clearly, a progressively measurable process is adapted and we have conversely
the
D.8) Proposition. An adapted process with right or left continuous paths is pro-
progressively measurable.
Proof. Left to the reader as an exercise.
We now come to the result for which the notion was introduced.
D.9) Proposition. IfX is progressively measurable and T is a stopping time (with
respect to the same filtration (.^)) then Xj is -Щ -measurable on the set {T < 00}.
Proof. The set [T < 00} is itself in -Щ-. To say that Xj is .Щ-measurable on this
set is to say that XT ¦ l[r</] e -K for every t. But the map
T:{(T< t), (T<t)n.%)-+ ([0, /]. ./?([0. /]))
is measurable because Г is a stopping time, hence the map со -*¦ (T(co), со) from
(^2, .^) into ([0, /] x Q, ./9([0, t]) ® .7^) is measurable and XT is the composi-
composition of this map with X which is ./?([0, t]) (gi .^-measurable by hypothesis. ?
With a stopping time T and a process X, we can associate the stopped process
XT defined by Xj(со) = Х,лТ(со). By Exercise D.16) the family of a -fields
(•Ялг) is a filtration and we have the

D.10) Proposition. If X is progressive, then XT is progressive with respect to the
filtration (.5Гл7-)-
Proof. Left to the reader as an exercise.
The following remark will be technically important.
D.11) Proposition. Every stopping time is the decreasing limit of a sequence oj
stopping times taking only finitely many values.
Proof. For a stopping time T one sets
Tk = +00 if T >k%
Tk = ql~k if (q - 1J"* < T < q2~k,q < 2kk.
It is easily checked that Tk is a stopping time and that {7^} decreases to T. ?
The description we have given so far in this section does not involve any
probability measure. We must now see what becomes of the above notions when
(?2, ;V) is endowed with a probability measure or rather, as will be necessary in
the study of Markov processes, with a family of probability measures.
D.12) Definition. If Pg, в e &, is a family of probability measures on A2, -У) а
property is said to hold almost-surely if it holds P(,-a.s.for every в е (9.
With this notion, two processes are, for instance, indistinguishable if they are
indistinguishable for each Pg. If we have a filtration (.^) on (П,.У) we will
want that a process which is indistinguishable from an adapted process be itself
adapted; in particular, we want that a limit (almost-sure, in probability, in the
mean) of adapted processes be an adapted process. Another way of putting this is
that a process which is indistinguishable from an adapted process can be turned
into an adapted process by altering it on a negligible set; this demands that the
negligible sets be in.% for all t which leads to the following definition.
D.13) Definition. If for each в, we call .У~? the completion of .7^ with respect
to Pg, the filtration (,^) is said to be complete if-^j, hence every .'Ti contains all
the negligible sets ofn№.^.
Of course, as follows from Definition D.12), negligible means negligible for
every Рв, в e ft), in other words Г is negligible if for every в there exists a set
Ae in .5<oo such that Г С Ай and Рв[А0] = 0.
If (-^") is not complete, we can obtain a larger but complete filtration in the
following way. For each в we call .Цв the a -field a (.5fU. /¦'") where . / '" is
the class of Pe-negligible, .^-measurable sets; we then set .'?, = rVJ^. The
filtration (($>+) is complete and right continuous. It is called the usual augmentation
f(^
Of course, if we use the usual augmentation of (.3^") instead of {.7[) itself,
we will have to check that a process with some sort of property relative to (.i^")

retains this property relative to the usual augmentation. This is not always obvious,
the completion operation for instance is not an innocuous operation, it can alter
significantly the structure of the filtration. Evidence of that will be given later
on; in fact, all the canonical processes with the same state spaces have the same
uncompleted natural filtrations and we will see that the properties of the completed
ones may be widely different.
We close this section with a general result which permits to show that many
random variables are in fact stopping times. To this end, we will use a difficult
result from measure theory which we now recall.
D.14) Theorem. lf(E. <C) is a LCCB space endowed with its Borel о-field and
(Q. .7, P) is a complete probability space, for every set A e К (g> .7, the pro-
projection л (A) of A into ?2 belongs to .7.
If Г is a subset of Q x R+, we define the debut Dp of Г by
Dr(co) - infjr > 0 : (?, со) е Г),
with the convention that inf@) = +oo.
D.15) Theorem. If the filtration (-7/) is right-continuous and complete, the debut
of a progressive set is a stopping time.
Proof It is enough to reason when there is only one probability measure involved.
Let Г be a progressive set. We apply Theorem D.14) above to the set Г, =
ГП([0,([хй) which belongs to ./?{[0, t}) ® .7;. As a result {Dr < t} = л(Г,)
belongs to .7t.
# D.16) Exercise. Let (.7f) be a filtration and S, T be (-Tf)-stopping times.
Iе) Prove that S Л T and S V T are stopping times.
2°) Prove that the sets [S - T}, {S < T\, {S < T) are in.7^Г\.7г.
3°) If 5" < T, prove that .7S С -7Г.
# D.17) Exercise. 1°) If (Tn) is a sequence of (-7t)-stopping times, then the r.v.
supn Т„ is a stopping time.
2°) If moreover (.7/) is right-continuous, then
infT,,, limT1,,. lim Т„
П n n
are stopping times. If Т„ | T, then .7j = f]lt .7т„.
# D.18) Exercise. Let (-7) be a filtration. If T is a stopping time, we denote by
.7j - the о -algebra generated by the sets of .7§ and the sets
{T > t\nr
where Г е .7,.
1J) Prove that .Щ- С -7i. The first jump time of a Poisson process (Exercise
A.14) Chapter II) affords an example where the inclusion is strict.

2°) If 5 < T prove that .Vs- С .Щ-. If moreover 5 < T on {5 < оо}П{Т > О},
prove that .Щ С -Щ--.
3°) Let (Г„) be a sequence of stopping times increasing to a stopping time T
and such that 7), < T for every n; prove that \Jn .Щп = .Щ-.
4°) If (rn) is any increasing sequence of stopping times with limit T, prove
# D.19) Exercise. Let T be a stopping time and Г е .^". The random variable Tr
defined by Tr = T on Г, 7> = +oo on Г< is a stopping time if and only if
Г е &.
D.20) Exercise. Let (.5O be a right-continuous filtration.
1°) Prove that the a -fields generated on Q x Ж+ by
i) the space of adapted continuous processes,
* ii) the space of adapted processes which are left-continuous on ]0, oo[,
are equal. (This is solved in Sect. 5 Chap. IV). This a -field is denoted --^(-Ю or
simply 0° and is called the predictable a-field (relative to (JO)- A process Z on
п is said to be predictable if the map (ш, t) -*¦ Z,(co) is measurable with respect
to ЙР(&\). Prove that the predictable processes are progressively measurable.
2°) If S and T are two .^-stopping times and S <T, set
]S, T] = {(«, f) : S(w) < t < T(co)}.
Prove that .^>(,F') is generated by the family of sets ]5, T].
3°) If S is a positive r.v., we denote by .&s- the ст-field generated by all the
variables Zs where Z ranges through the predictable processes. Prove that, if 5 is
a stopping time, this a -field is equal to the ст-field .Щ- of Exercise D.18).
[Hint: For A e .Ws-, consider the process Z,(cd) — 1л(&>I]о.5(Ш)]@-]
Prove that S is .^--measurable.
4°) In the general situation, it is not true that S < T entails .%- С -Щ-. Give
an example of a variable 5 < 1 such that.%- = .Рьо.
*# D.21) Exercise (Galmarino's test). Let П = ?>(Ж+, W) or C(R+, Rd) and use
the notation at the beginning of the section.
Iе) Prove that T is a (.i^°)-stopping time if and only if, for every r, the
properties T(a>) < t and Xs(a>) = Xs(od') for every s < t, imply T(od) = T{co').
Prove that the time TA of Proposition D.6) is not a (.^°)-stopping time.
2°) If T is a (,5f°)-stopping time, prove that A e .^° if and only if cd g A,
T(w) = т(и') and Xs{a>) = Xx{a>') for every s < T(a>) implies w e A.
3°) Let (oT be the point in S2 defined by <wr(.v) = w(s л Г(а))). Prove that /
is .5^°-measurable if and only if f(w) = f(o)T) for every w.
4°) Using the fact that .%? is the union of the a -fields generated by the
countable sub-families of coordinate mappings, prove that .^° = a (xj, s > 0).
5°) Deduce from 4°) that .Pf° is countably generated.

D.22) Exercise. A positive Borel function ф on Ж+ is said to have the property
(P) if for every stopping time T in any filtration (.3^") whatsoever, ф(Т) is a
СЯ")-stopping time. Show that ф has the property (P) if and only if there is a
to < +00 such that фA) > t for t < t0 and 0@ = '0 for ' > {o-
Notes and Comments
Sect. 1 There are many rigorous constructions of Brownian motion, some of which
are found in the following sections and in the first chapter of the book of Knight
[5]. They are usually based on the use of an orthonormal basis of L2(R+) or on a
convergence in law, a typical example of which is the result of Donsker described
in Chap. XIII. The first construction, historically, was given by Wiener [1] which
is the reason why Brownian motion is also often called the Wiener process.
Follmer [3] and Le Gall [8] give excellent pedagogical presentations of Brow-
Brownian motion. The approach we have adopted here, by means of Gaussian measures,
is a way of unifying these different constructions; we took it from the lecture course
of Neveu [2], but it goes back at least to Kakutani. Versions and modifications
have long been standard notions in Probability Theory (see Dellacherie-Meyer
[1]).
Exercise A.17) is from Levy, Exercise A.18) from Durrett [1] and Exercise
A.19) from Hardin [1]. The first study of polar functions for BM2 appears in
Graversen [1]; this will be taken up more thoroughly in Chap. V. Exercises A.21)
and A.22) are from Barlow et al. [1], Song-Yor [1] and Yor [14].
Sect. 2 Our proof of Kolmogorov's criterion is borrowed from Meyer [8] (see
also Neveu's course [2]). Integral type improvements are found in Ibragimov [1];
we also refer to Weber [1]. A very useful Sobolev type refinement due to Garsia
et al. [1] is found in Stroock-Varadhan [1] (see also Dellacherie-Maisonneuve-
Meyer [1]): it has been used in manifold contexts, for instance in Barlow-Yor [2],
to prove the BDG inequalities in Chap. IV; see also Barlow ([3] and [5]) and
Donati-Martin ([1]).
The rest of this section is due to Levy. The proof of Theorem B.7) is borrowed
from Ito-McKean [ 1 ] which contains additional information, namely the "Chung-
Erdos-Sirao" test.
Exercise B.8) is due to Levy and Exercise B.9) to Dvoretzky et al. [1].
Sect. 3 The material covered in this section is now the common lore of probabilists
(see for instance Dellacherie-Meyer [1] Vol. I). For Gaussian processes, we refer
to Neveu [1]. A direct proof of the existence of the Wiener measure is given in
Ito [4].
Exercise C.9) deals with fractional Brownian morion; a number of references
about this family of Gaussian processes, together with original results are found
in Kahane [1], Chap. 18. The origin of question 2°) in this exercise is found
in Albeverio et al. [1], p. 213-216, Stoll [1] and Yor [20]. Fractional Brownian
motions were introduced originally in Mandelbrot and Van Ness [1]; they include

Levy's Brownian motions with several parameters, and arise naturally in limit
theorems for intersection local times (Weinryb-Yor [1], Biane [3]).
Exercise C.11) is due to Aronszajn [1] (see Neveu [1]).
Sect. 4 Filtrations and their associated notions, such as stopping times, have, since
the fundamental work of Doob, been a basic feature of Probability Theory. Here
too, we refer to Dellacherie-Meyer [1] for the history of the subject as well as for
many properties which we have turned into Exercises.

Chapter II. Martingales
Martingales are a very important subject in their own right as well as by their
relationship with analysis. Their kinship to BM will make them one of our main
subjects of interest as well as one of our foremost tools. In this chapter, we describe
some of their basic properties which we shall use throughout the book.
§1. Definitions, Maximal Inequalities and Applications
In what follows, we always have a probability space (Q, .P', P), an interval T of
N or K+ and an increasing family .Щ, t e T, of sub-er-algebras of .7. We shall
call it a filtration as in the case of R+ introduced in Sect. 4 Chap. I, the results of
which apply as well to this case.
A.1) Definition. A real-valued process X,, t € T, adapted to (.!%) is a submartin-
gale (with respect to.%) if
i) E [Xf] < oo for every t € T;
ii) E \Xt | ..Д"] > Xs a.s. for every pair s, t such that s < t.
A process X such that —X is a submartingale is called a supermartingale and
a process which is both a sub and a supermartingale is a martingale.
In other words, a martingale is an adapted family of integrable random vari-
variables such that
/ XsdP = f X,dP
a Ja
for every pair s, t with s < t and A e .%.
A sub(super)martingale such that all the variables Xt are integrable is called
an integrable sub(super)martingale.
Of course, the filtration and the probability measure P are very important in this
definition. When we want to stress this fact, we will speak of (.^)-submartingales,
v*i, P)-supermartingales, ...). A (..^-martingale X is a martingale with respect
to its natural filtration cr(Xs, s < t). Conversely, if #, э .%, there is no reason why
a C^")-martingale should be a (.'?,)-martingale. Obviously, the set of martingales
with respect to a given filtration is a vector space.

A.2) Proposition. Let В be a standard linear BM; then the following processes
are martingales with respect to a (Bs, s < /):
i) B, itself, ii) Bj - t, Hi) M? = exp (aB, - ft^j for a e K.
Proof. Left to the reader as an exercise. This proposition is generalized in Exercise
A.18). П
These properties will be considerably generalized in Chap. IV. We notice that
the martingales in this proposition have continuous paths. The Poisson process of
Exercise A.14) affords an example of a martingale with cadlag paths. Finally, if
(•>*Г) is a given filtration and Y is an integrable random variable, we can define
a martingale Y, by choosing for each ; one random variable in the equivalence
class of E [Y I .J*j"]. Of course, there is no reason why the paths of this martingale
should have any good properties and one of our tasks will precisely be to prove
the existence of a good version.
Another important remark is that if X, is a martingale then, because of Jensen's
inequality, \X,\P is a submartingale for p > 1 provided ?[|X,|P] < oc for
every t.
We now turn to the systematic study of martingales and of submartingales.
Plainly, by changing X into —X any statement about submartingales may be
changed into a statement about supermartingales.
A.3) Proposition. Let (Х„), n — 0, 1, ... be a (sub)martingale with respect to a
discrete filtration (.7^) and Hn, n = 1,2,..., a positive bounded process such that
Н„ € -Уп-\ for n > \; the process Y defined by
Yq = Xq, Yn = F,,_i + Н„ (Х„ — Х„_])
is a (sub)martingale. In particular, ifT is a stopping time, the stopped process X
is a (sub)martingale.
Proof The first sentence is straightforward. The process Y thus defined is the
discrete version of the stochastic integral we will define in Chap. IV. It will be
denoted ИХ.
The second sentence follows from the first since Hn = 1[,,<7] is -Kt-\-
measurable being equal to 1 - \[т<„-\]. ?
We use this proposition to obtain a first version of the optional stopping the-
theorem which we will prove in Sect. 3. The setting is the same as in Proposition
A.3).
A.4) Proposition. If S and T are two bounded stopping times and S < T, i.e.
there is a constant M such that for every to,
S(co) < T(u>) < M < oc,
then

[Z] a.s.,
with equality in case X is a martingale. Moreover, an adapted and integrable
process X is a martingale if and only if
E[XS] = E[XT]
for any such pair of stopping times.
Proof. Suppose first that X is a martingale. If Hn — 1[„<г] - 1[»<S], for n > M,
we have
(H ¦ X),, -X0 = XT- Xs,
but since E [(H ¦ X)n] = E[X0] as is easily seen, we get E[XS] = E[XT].
If we apply this equality to the stopping times SB = S 1д + М 1д< and TB =
j \B -\- M\b> where йе,^ (see Exercise D.19) Chap. I) we get
whence it follows that Xs = E[XT\.9s\ a.s. In particular, the equality E[XS] =
E[Xt] for every pair of bounded stopping times is sufficient to insure that X is a
martingale.
If X is a submartingale, max(X, a) is an integrable submartingale to which we
can apply the above reasoning getting inequalities instead of equalities. Letting a
tend to —oc, we get the desired result. ?
We derive therefrom the following maximal inequalities.
A.5) Proposition. If (Xn) is an integrable submartingale indexed by the finite set
@, 1, ..., N), then for every X > 0,
XP I sup Х„ > X < E [XN1 (suPn Х„>Я)] <E[\XN\\ (suPii х„>л)].
Proof. Let T = inf{« : Xn > X} if this set is non empty, T = N otherwise. This
is a stopping time and so by the previous result
E[XN] > ?[Хг] = ?[Хт1(8ир„л„>л)] + ?[Хт1Eирл^<л)]
> XP supX,, > X \ + E [XN\{,UPnX:i<X)]
because XT > X on (sup,, X,, > X). By subtracting E [Х^1Eир„ х„<ю] from the
two extreme terms, we get the first inequality, while the second one is obvious. ?
A.6) Corollary. If X is a martingale or a positive submartingale indexed by the
finite set @, 1, ..., N), then for every p > 1 and X > 0
sup|X,,| > k\ < E[\XN\P].
and for any p > \,
E[\XN\r] < fTsuplXJ'1! < (-Е-Л E[\XN\"].
'

Proof. By Jensen's inequality, if XN is in Lp, the process |Xn|p is a submartingale
and the first part of the corollary follows from the preceding result.
To prove the second part, we observe that the left-hand side inequality is trivial;
to prove the right-hand side inequality we set X* = supn \Xn\. From the inequality
kP[X* >k]<E [\XN\llx->k)], it follows that, for a fixed к > О,
E[(X*Ak)p] = e\J Л /rt'-'rf*"] = ? Г/"
= / рк?'1 Р[Х* > k]dk < f [
Jo Jo
= pEUXN\j \p-2d\\ = -^E[\XN\(X*
Holder's inequality then yields
E [(X* A k)"] < -^E [(X* A k)p]{p~X)lp E [\XN\rfp
and after cancellation
E[(X* AkY]<(-^—^P E[\XN\P].
The proof is completed by making к tend to infinity. Q
These results carry over to general index sets. If X is a martingale indexed by
an interval T of Ш, we can look at its restriction to a countable subset D of T.
We can then choose an increasing sequence Dn of finite subsets of D such that
[jDn = D and apply the above results to Dn. Since E[|X,|P] increases with t,
we get, by passing to the limit in n, that
XpP |sup|X,| > к 1 < supE[\Xt\p]
LtcD J t
and for p > 1
LteD J \P ~
We insist that this is true without any hypothesis on the filtration (.%) which may
be neither complete nor right continuous.
A.7) Theorem (Doob's /^-inequality). ifX is a right-continuous martingale or
positive submartingale indexed by an interval T ofR, then if X* — sup, \Xt\,for
kpP[X* > k] < sup E[\Xt\r]
t
and for p > 1,
P ~ p- 1 r ' P'

Proof. If D is a countable dense subset of T, because of the right-continuity
X* = sup,eO |X,| and the results follow from the above remarks. Q
If td is the point on the right of T, we notice that sup, ||X, ||p is equal to \\X,d \\p
[ftdeT and to lim,^ ||X,||p if T is open on the right.
рог р = 2, the second inequality reads
||X*||2<2sup||X,||2
r
and is known as Doob's L2-inequality. Since obviously ||X,||2 < ||X*||2 for every
t, we see that X* is in L2 if and only if sup, ||X,||2 < +00, in other words if
the martingale is bounded in L2. We see that in this case the martingale, i.e. the
family of variables [Xt, t e T] is uniformly integrable. These remarks are valid
for each p > \. They are not for p — 1 and a martingale can be bounded in
L1 without being uniformly integrable and a fortiori without X* being integrable.
This has been the subject of many studies. Let us just mention here that
E[X*] < -^ (l + sup E [X, log+ X,] J
(see Exercise A.16)).
In the following section, we will apply these inequalities to establish the con-
convergence theorems for martingales. We close this section with some important
applications to Brownian motion. The first is known as the exponential inequality.
It will be considerably generalized in Exercise C.16) of Chap. IV. We recall that
5, = sups<, Bs.
A.8) Proposition. For a > 0,
P[S, > at] < exp(-a2r/2).
Proof. For a > 0, we use the maximal inequality for the martingale Ma of
Proposition A.2) restricted to [0, t]. Since exp (aS, - ^-\ < supv<, Mf, we get
P[S, >at] < p\ sup Aff > exp (atat - a2t/2)
< exp (-aat + a2t/2) E[M?]\
but ?[M,"] = E[M%] = 1 and infa>0 [-aat + ^f) = -a2t/2, whence the result
follows.
Remark. This inequality is also a consequence of the equality in law: S, = \B,\
valid for fixed t, which may be derived from the strong Markov property of BM '
(see Exercise C.27) in Chap. III).

A.9) Theorem (Law of the iterated logarithm). For the linear standard Brow-
nian motion B,
° Brlog2(l/0I/2
where log2 x = log(logx) for x > 1.
Proof. Let h(t) = y/2t Iog2(l/r) and pick two numbers в and <5 in ]0, 1[. We set
By the same reasoning as in the preceding proof
P I sup(B.s - atts/2) > pa] < е~а"^ = Кп^-&
for some constant К. Thus, by the Borel-Cantelli lemma, we have
П—*ЭС \.V<
P lim sup(B.v - a,,s/2) < Д,
= 1.
If we restrict s to [0, в" '] we find that, a fortiori, for almost every со, there is an
integer no(to) such that for n > no(co) and s e [0. 0"~'[
1 l +<5
But the function h is increasing on an interval [0, a[, with a > 0, as can easily
be checked. Therefore, for n sufficiently large and s in the interval ]0", #""'], we
have, for these o/s,
Г1 +.5 1
As a result, Wmsi0BJh(s) < A + <5)/26l + 1/2 a.s. Letting 9 tend to 1 and then
<5 tend to zero, we get that hmxioBx/h(s) < 1 a.s.
We now prove the reverse inequality. For 9 e]0, 1[. the events
г„ =
are independent; moreover (see the proof of Theorem B.7) Chap. I)
f
J a
e-2ndu > -^e-2':
1 +a2
with a = ('l-^B1og20-7(l-0))'/2.Thisisoftheorderof«-A-2^+f')/(|-fl)
= n~a with a < 1. As a result, ]Г^° Р[Г„] = +oc and by the Borel-Cantelli
lemma,
Be» > (\ - Ve^j h (9") + Вв„^ infinitely often a.s.

_B is also a Brownian motion, we know from the first part of the proof
that Вв"^(ш) < 2/?(#"+1) from some integer n0(o)) on. Putting the last two
inequalities together yields, since h(9n+x) < 2\1вк(в") from some n on, that
> h(9") (\ - s/в - 4sle\ infinitely often,
and consequently
UmB,/h(t) > Ш\Вв,/к(в") > 1 - 5-Je a.s.
no
It remains to let 9 tend to zero to get the desired inequality. П
Using the various invariance properties of Brownian motion proved at the end
of Sect. 1 in Chap. I, we get some useful corollaries of the law of the iterated
logarithm.
A.10) Corollary. P []untlQBt/ij2t\og2(\/t) = -l] = 1.
Proof. This follows from the fact that — В is a BM. П
Since the intersection of two sets of probability 1 is a set of probability 1, we
actually have
г i
P\\imBl/J2t\og2(\/t) = \ and Y\m B,/J2t\og,(\/t) = -1=1
L'i° no " J
which may help to visualize the behavior of В when it leaves zero. We see that
in particular 0 is a.s. an accumulation point of zeros of the Brownian motion, in
other words В takes a.s. infinitely many times the value 0 in any small interval
[0,fl[.
By translation, the same behavior holds at every fixed time. The reader will
compare with the Holder properties of Sect. 2 in Chap. I.
A.11) Corollary. For any fixed s,
Г
P lfim(#r+.v - 5s)/V2?log2(l/0 = 1 and
lim(S,+v - fi.s.)/v/2flog2(l/f) = -1=1.
'Ю J
Proof. (Bl+S - Bs, t > 0) is also a BM.

Finally using time inversion, we get
A.12) Corollary.
P \\imBl/v/2t\og2t - 1 and lim Bt/j2t log7 r = -1=1.
Remark. This corollary entails the recurrence property of BM which was proved
in Exercise A.13) of Chap. I, namely, for every x e R the set {r : B, — x) is a.s.
unbounded.
# A.13) Exercise. If X is a continuous process vanishing at 0, such that, for every
real a, the process M" = expjaX, — y?| is a martingale with respect to the
filtration (.ЯП, prove that X is a (.>^")-Brownian motion (see the Definition B.20)
in Chap. III).
[Hint: Use the following two facts:
i) a r.v. X is . / @, 1) if and only if E[ekx] = eA'/2 for every real A.,
ii) if X is a r.v. and ./} is a sub-ст-algebra such that
E[e'x\./J} = E[exx] < +oo
for X in a neighborhood of 0, then X and ./} are independent.]
# A.14) Exercise (The Poisson process). Let (X,,) be a sequence of independent
exponential r.v.'s of parameter c. Set S,, = ^" X/. and for t > 0, N, = ^^ l[s,,<(].
Г) Prove that the increments of Л', are independent and have Poisson laws.
2°) Prove that N, — ct is a martingale with respect to o(N,, s < t).
3 ) Prove that (N, — ctJ — ct is a martingale.
A.15) Exercise (Maximal inequality for positive supermartingaies). If X is a
right-continuous positive supermartingale, prove that
P supX, > X \< k~[E[X0]
A.16) Exercise (The class L logL). 1°) In the situation of Corollary A.6), if ф
is a function on R+, increasing, right-continuous and vanishing at 0, prove that
Е[ф(Х*)] <e\\Xn\ f
X"
2:) Applying Г) with ф{Х) = (X — 1)+, prove that there is a constant С such
that
E[X*]<c(\+supE[\Xn\\og+(\Xn\)]\
[Hint: For a, b > 0, we have alogb < a\og+ a + e~lb.]
The class of martingales for which the right-hand side is finite is called the
class LlogL. With the notation of the following exercise, we have LlogL С Я1.

A.17) Exercise (The Space Hp). The space of continuous martingales indexed
by K+. such *at x* = SUP/ l^'l 's 'n ^ Z7 — U is a Banach space for the norm
[Hint: See Proposition A.22) in Chap. IV.]
Remark. In this book, we focus on continuous processes, which is the reason for
limiting ourselves to continuous martingales. In fact, the same result holds for the
space Шр of cadlag martingales such that X* is in Lp; the space Hp above is a
closed subspace of Шр.
# A.18) Exercise. Retain the notation of Exercise A.14) of Chap. 1 and prove that
f f(.s)dBs, ( [ f(s)dBt) - [' f(sJds,
Jo \Ja / Jo
f(s)dBs- ~j f{sJds\
are continuous martingales. This will be considerably generalized in Chap. IV.
A.19) Exercise. Let X and Y be two positive supermartingales with respect to
the same filtration (.3%) and T a stopping time such that Xr > YT on {T < oo}.
Prove that the process Z defined by
Zn(o) = Х„(а)) if и<Г(а»), Z,,(co) = Yn(co) if n>T(co),
is a supermartingale.
** A.20) Exercise. Iе) For any measure д on M+, prove that
P [gn|B,+* - B,IA/2Alog2(l/A) = 1 for д-ае. Л = I.
The following questions deal with the exceptional set.
2°) Using Levy's modulus of continuity theorem (Theorem B.7) of Chap.
I) prove that for a.e. со and any pair a < b, one can find si and t\ such that
a < sx < r, < b and
\BU -Btl\ > -d(ti -si)
where d(h) = ,/2hlog(l/h).
3°) Having chosen 5, sn, t\,...,tn such that \BU - BSi \ > -^d{tj - s,),
choose sj, € ]s,,,tn[, s'n < sn + 2"" and \B,n - B,\ > ^d\tn -'s) for every
s_€\sn, s'n]. Then, choose j,1+i, f,,+) in ]л„, s'n[ and so on and so forth. Let {.v0} =
MJsn,fn]; prove that
iimlB^+A -BsJ/d(h)>\.
4°) Derive from 3C) that for a.e. со there is a set of times t dense in R+ and
such that

\im\Bl+h - Bt\/J2h\og2(\/h) = +00.
5°) Prove that the above set of times is even uncountable.
[Hint: Remove from ]sn,s'n[ the middle-third part in Cantor set-like fashion
and choose two intervals ].v,,+i, f,,+i[ in each of the remaining parts.]
* A.21) Exercise. If В is the BM'', prove that
[Hint: Pick a countable subset (en) of the unit sphere in W1 such that |x| =
sup,, \(x.e,,)\.]
Using the invariance properties, state and prove other laws of the iterated
logarithm for ^
A.22) Exercise. If В is the Brownian sheet, for fixed s and t.
lim (B(i+A.,, - В„.„) /sjlh log2(l/A) = V7 a.s.
A.23) Exercise. Prove that if В is the standard BM''
P sup|fiv| >S\ < 2de\p(-S2/2dt).
s<t
[Hint: Use Proposition A.8) for (в. В,) where 9 is a unit vector.]
§2. Convergence and Regularization Theorems
Let us first recall some facts about real-valued functions. Let / be a function
which maps a subset T of M into M. Let t\ < t2 < ¦ ¦ ¦ < hi be a finite subset F
of T. For two real numbers a, b with a < b, we define inductively
л-, = inf {u : f(ti) > b). s2 - inf [t, > s{ : /(r,) < a].
¦?2n+i = inf Ui > S2n ¦ f(ti) > b\. s2n+2 = inf {tj > л'2,1+1 : /(?,) < a},
where we put inf@) = tj. We set
D(/, F, [a, b]) = sup {n : s2l, < td}.
and we define the number of downcrossings of [a, b] by / as the number
D(f. T, [a, b]) = sup {D(f, F, [a, I?]) : F finite . F С Т].
One could define similarly the number U( f. T. [a. b]) of upcrossings. The function
/ has no discontinuity of the second kind, in particular / has a limit at the
boundaries of T whenever T is an open interval, if and only if D(f, T, [a, b]) (or
?/(/. T, [a, b])) is finite for every pair [a. b] of rational numbers.
We now consider the case where / is the path of a submartingale X; if T is
countable, D(X, T, [a, b]) is clearly a random variable and we have the

B.1) Proposition. If X is a submartingale and T is countable, then for any pair
(a,b),
(b - a)E[D(X, T, [a, b])] < sup E [(X, - b)+].
Proof. It is enough to prove the inequality when T is finite and we then use the
notation above. The sk's defined above are now stopping times with respect to the
discrete filtration (-Vtl). We are in the situation of Proposition A.4) which we can
apply t0 tne stopping time sk. Set Ak = {sk < td)\ then Ak e -Kk and Ak D Ak+\.
On Агя-ь we nave Xs,,,-, > b, on A2n we have Х.,,„ < a and therefore
0 < f (XSbt_,-b)dP<( (XS2u-b)dP
< (a-b)P(A2n)+ f {XS2n -b)dP.
JA2,,-i\A2,,
Consequently, since s2n = U on Al2n,
(b-a)P(A2n)< f (Xnn-b)+dP=[ (X,d-b)+dP.
J A2ll-\\A2n J А2„_1\А2„
But P{A2n) — P[D(X, T, [a, b]) > n] and the sets A2n_\\A2n are pairwise dis-
disjoint so that by summing up the above inequalities, we get
(b - a)E[D(X, T, [a, b])] < E [(X,d - b)+],
which is the desired result. ?
We now apply this to the convergence theorem for discrete submartingales.
B.2) Theorem. //(X,),neN,/sa submartingale such that
sup?[X+] < +oo,
n
then (X,,) converges almost-surely to a limit which is < +oo a.s.
Proof. Fatou's lemma ensures that HmX,, < +oo a.s. So if our claim were false,
there would be two real numbers a and b such that limX,, < a < b < limX,, with
positive probability; thus, we would have D(X, N, [a, b]) = +oo with positive
probability, which, by the foregoing result is impossible. ?
It is also useful to consider decreasing rather than increasing families of a-
algebras, or in other words to "reverse"the time in martingales. Let (.3^),,<o, be
a sequence of sub-ст-fields such that .7~n С -Vm if и < m < 0. A submartingale
with respect to {.Wn) is an adapted family (X,,) of real-valued r.v.'s such that ,
?[^,t] < oo for every n and Х„ < Е\Хт\-Ул\ for л < w < 0. We then get the
following

B.3) Theorem. If(Xn), n e —N, is a submartingale, then \\т„^_оо Х„ exists a.s.
If moreover supn ?[|Х„|] < оо, then (Xn) is uniformly integrable, the convergence
holds in L' and, for every n
lim Xk <E[Xn\.Kx]
where .*_х = f|,, -K-
Proof. It is easily seen that sup,, E[X+] < E[X%] < +00, so that the first state-
statement is proved as Theorem B.2). To prove the second, we first observe that the
condition supn ?[|X,,|] < +00 is equivalent to lim^.oc E[Xn] > -00. Now, for
any с > О and any и, we have
f \X,,\dP=[ XndP-E[Xn)+f X,,dP.
For e > 0, there is an integer n0 such that E[Xn] > E[Xllg] — e for и < и0; using
this and the submartingale inequality yields that for и < и0,
f \X,,\dP < f XllodP-E[X,Ht]+ f XnodP + s
J{\X,,\>c) J{Xn>-c) J{X,,>c)
(*) - I XHodP + e
J{\X,,\>c)
As P[|X,,| > c] < c~l supH ?[|X,,|], the uniform integrability of the family Xn,
n e —N, now follows readily from (*) and implies that the convergence holds in
L1. Finally, if Г е .^ю, for m < n,
f XmdP < f X,,dP
Jr Jr
and we can pass to the limit, thanks to the L1-convergence, to get
f (}imXm)dP < f XndP
Jr '" Jr
which ends the proof. D
The following corollary to the above results is often useful.
B.4) Corollary. Let X,, be a sequence of r.v. 's converging a.s. to a r.v. X and
such that for every n, \X,,\ < Y where Y is integrable. If (.7f,) is an increasing
(resp: decreasing) sequence of sub-a-algebras, then Е[Х„ | .РЦ] converges a.s. to
E[X I ->H where .T = o{\j.%) (resp. .7 = f),, -Ю-

Proof- Pick e > 0 and set
U = ЫХ„, V
where m is chosen such that E[V — U] < e. Then, for и > т we have
e[u\.K]<e[х„ \.K]<e[v\.Щ;
the left and right-hand sides of these inequalities are martingales which satisfy the
conditions of the above theorems and therefore
E [U | .7\ < limE [Х„ \ <Щ < lim? [х„ \ Щ < E[V | .T].
We similarly have
E[U | ,9"\ < E[X | .T\ < E[V | .V\.
It follows that E [fiinE [Х„ \ Щ - ЦгпЕ [Х„ \ ,Щ\ < e, hence E [х„ | .Щ con-
converges a.s. and the limit is E[X \ Ж\. D
We now turn to the fundamental regularization theorems for continuous time
(sub)martingales.
B.5) Theorem. If X,, t e №.+, is a submartingale, then for almost every w, for
each t e]0, oo[, limr1-,,reQ Xr(co) exists and for each t e [0, oo[, limr|,,reQ Xr (<u)
Proof. It is enough to prove the results for t belonging to some compact sub-
interval /. If tj is the right-end point of /, then for any re/,
E [(X, - b)+] < E [X+] + b~.
It follows from Proposition B.1) that there exists a set Qq С Q such that
P(i20) = 1 and for w e i20
D (Х(ш), / П Q, [a, b]) < oo
for every pair of rational numbers a < b. The same reasoning as in Theorem B.2)
then proves the result. D
We now define, for each t e [0, oo[,
Xt+ = lim Xr
and for t e]0, oo[,
X,- = Tim Xr.
By the above result, these upper limits are a.s. equal to the corresponding
lower limits. We study the processes thus defined.

B.6) Proposition. Suppose that E[\X,\] < +00 for every t, then E[\Xl+\] < 00
for every t and
X, < E[X,+ I .Щ a.s.
This inequality is an equality if the function t —* E[X,] is right-continuous, in
particular if X is a martingale. Finally, (X,+) is a submartingale with respect to
(.i^+) and it is a martingale ifX is a martingale.
Proof. We can restrict ourselves to a compact subinterval. If (tn) is a sequence
of rational numbers decreasing to t, then (X,J is a submartingale for which we
are in the situation of Theorem B.3). Thus, it follows immediately that X,+ is
integrable and that X,a converges to X,+ in L1. Therefore, we may pass to the
limit in the inequality X, < E [X,n \ .7f\ to get
X, < E [Xl+ I .ft] ¦
Also, the L1-convergence implies that E[X,+] = lim,, E[X,n] so that iff —> E[X,]
is right-continuous, E[X,] - E[X,+] hence X, — E[X,+ \ .Щ a.s.
Finally, let s < t and pick a sequence (sn) of rational numbers smaller than t
decreasing to s. By what we have just proved,
XSll <E[X,\ J^,] < E [X,+ I .rt I .Ъ„] = E [Xl+ I .^,],
and applying Theorem B.3) once again, we get the desired result.
Remark. By considering X, v a instead of X,, we can remove the assumption that
X, is integrable for each t. The statement has to be changed accordingly.
The analogous result is true for left limits.
B.7) Proposition. If E[\X,\] < 00 for each t, then E[|X,_|] < +00 for each
t > 0 and
X,- < E [X, I .^L] a.s.
This inequality is an equality ift —*¦ ?[X,] is left-continuous, in particular ifX is
a martingale. Finally, X,-,t > 0, is a submartingale with respect to (-^-) and a
martingale ifX is a martingale.
Proof. We leave as an exercise to the reader the task of showing that for every
a e R, [X, v a], t e /, where / is a compact subinterval, is uniformly integrable.
The proof then follows the same pattern as for the right limits. D
These results have the following important consequences.
B.8) Theorem. IfX is a right-continuous submartingale, then
1) X is a submartingale with respect to (.%+). and also with respect to the com-
completion of(.&i'+),
2) almost every path of X is cadlag.
Proof. Straightforward. D

B.9) Theorem. Let X be a submartingale with respect to a right-continuous and
complete filtration (.Wt); ij't —> E[X,] is right-continuous (in particular, ifX is a
martingale) then X has a cadlag modification which is a (-Wt)-submariingale.
Proof- We go back to the proof of Theorem B.5) and define
X,(a)) = X,+ (to) if сое Go, X,(co) = 0 if со ? Qq.
The process X is a right-continuous modification of X by Proposition B.6). It is
adapted to (-Я"), since this filtration is right-continuous and complete and Qq is
negligible. Thanks again to Proposition B.6), X is a submartingale with respect
to (-3%) and finally by Theorem B.5) its paths have left limits. D
These results will be put to use in the following chapter. We already observe
that we can now answer a question raised in Sect. 1. If (.Pf) is right-continuous
and complete and Y is an integrable random variable, we may choose Y, within
the equivalence class of E[Y | .i^"] in such a way that the resulting process is a
cadlag martingale. The significance of these particular martingales will be seen in
the next section.
From now on, unless otherwise stated, we will consider only right-continuous
submartingales. For such a process, the inequality of Proposition B.1) extends at
once to
(b-a)E [D (X, R+, [a, b])] < sup E [(X, - b)+]
t
and the same reasoning as in Theorem B.2) leads to the convergence theorem:
B.10) Theorem. If sup, E[X+] < oo, then lim,-^ X, exists almost-surely.
A particular case which is often used is the following
B.11) Corollary. A positive supermartingale converges a.s. as t goes to infinity.
In a fashion similar to Theorem B.3), there is also a convergence theorem as
t goes to zero for submartingales defined on ]0, oo[. We leave the details to the
reader.
The ideas and results of this section will be used in many places in the sequel.
We close this section by a first application to Brownian motion. We retain the
notation of Sect. 2 in Chap. I.
B.12) Proposition. If {A,,} is a sequence of refining (i.e. А„ С An+\) subdivisions
o/[0, t] such that \A,,\ -+ 0. then
limV(B, -B,) =t almost-surely .

Proof. We use the Wiener space (see Sect. 3 Chap. 1) as probability space and
the Wiener measure as probability measure. If 0 = to < t\ < ... < tk — t is
a subdivision of [0, t], for each sequence e = (ei,..., sk) where e,- = ±1, we
define a mapping 6S on Q by
вешф) = 0,
eeco(s) = 9ew(t,-i) + Si (u)(s) - eo(ti-i)) if s e [f,_i, t,],
ee<a(s) - esa)(tk) + (o(s) - w(tk) if s>tk.
Let ¦/? be the a-field of events left invariant by all 9e's. It is easy to see that
W is left invariant by all the 6e's as well. For any integrable r.v. Z on W, we
consequently have
hence E [(fi,, - B,,.,) (Btj - B,;_,) \./t] = 0 for г ^ y. If ./}„ is the or-field
corresponding to A,,, the family ./?„ is decreasing and moreover
By Theorem B.3), X], (^r, — #?,_,) converges a.s. and, as we already know that
it converges to t in L2. the proof is complete. D
B.13) Exercise. 1°) Let {Q..7'. P) be a probability space endowed with a fil-
filtration (.Wn) such that a(U.5^) = .W. Let Q be another probability measure on
.7" and Xn be the Radon-Nikodym derivative of the restriction of Q to .7~n with
respect to the restriction of P to .^".
Prove that (Х„) is a positive (.^, P)-supermartingale and that its limit Xt»
is the Radon-Nikodym derivative dQ/dP. If Q «. P on .F", then (Хя) is a
martingale and Х„ = E[Xoo | .3%].
More on this matter will be said in Sect. 1 Chap. VIII.
2") Let P be a transition probability (see Sect. 1 Chap. Ill) on a separable
measurable space (E, %) and Я is a probability measure on r>. Prove that there
is a bimeasurable function / on E x E and a kernel N on (?, #) such that for
each x, the measure N(x, •) is singular with respect to Я and
P(x, A) = N(x, A)+ f ff{x, y)k(dy).
Ja
B.14) Exercise (Dubins' inequality). If (X,,), n = 0, 1,... is a positive super-
martingale, prove, with the notation of the beginning of the section, that
P[D(X, N, [a, b]) >p]< ap-lb-pE[X0 A b].
State and prove a similar result for upcrossings instead of downcrossings.

B.15) Exercise. Let (J2,.X , P) be a probability space and (?/>,,и > 0) be a
sequence of sub-ет-fields of .W such that ^n с (/„„ if 0 < m < n. If ?" is
another sub-er -field of .j^ independent of C6§, prove that
up to P-negligible sets.
[Hint: Show that, if С e К, D e ^, then Iim,,^ /> (CD | fcT v C/n) belongs
to^v(a^)-]
B.16) Exercise. For the standard BM, set #, — a(Bu,u > 0- Prove that for
every real k, the process exp {(XB,/t) - Q?/2t)\ J > 0, is a martingale with
respect to the decreasing family (#,).
[Hint: Observe that Bs - {.s/t)B, is independent of •?, for s < t or use time-
inversion.]
B.17) Exercise. Suppose that we are given two nitrations (.^"°) and (.^") such
that ^"° ^ .^for each r and these two a -fields differ only by negligible sets of
J^j. Assume further that (.3^"°) is right-continuous.
1°) Show that every (..^-adapted and right-continuous process is indistin-
indistinguishable from a (.^"°)-adapted process.
2°) Show that a right-continuous (.^")-submartingale is indistinguishable from
a cadlag (.S^"o)-submartingale.
B.18) Exercise (Krickeberg decomposition). A process X is said to be Lx-
bounded or bounded in L1 if there is a finite constant К such that for every
t>0,E[\Xt\]<K.
1°) If M is a Ll-bounded martingale, prove that for each t the limits
exist a.s. and the processes Л/(±) thus defined are positive martingales.
2°) If the filtration is right-continuous and complete, prove that a right-
continuous martingale M is bounded in L1 iff it can be written as the difference
of two cadlag positive martingales M{+) and M{~\
3°) Prove that M(+) and Ml~} may be chosen to satisfy
supi
in which case the decomposition is unique (up to indistinguishability).
4°) The uniqueness property extends in the following way: if Y and Z are two
Positive martingales such that M = Y - Z, then Y > M{+) and Z > М{~] where •
M(±) are the martingales of 3°).

§3. Optional Stopping Theorem
We recall that all the (sub, super)martingales we consider henceforth are cadlag.
In the sequel, we shall denote by .K^ the ст-algebra \/rK- In Theorem B.9) of
last section, the limit variable X^ is measurable with respect to .7^. We want to
know whether the process indexed by R+ U {+00} obtained by adjoining X^ and
•>^c is still a (sub)martingale. The corresponding result is especially interesting
for martingales and reads as follows.
C.1) Theorem. For a martingale X,, t e R+, the following three conditions are
equivalent,
i) lim,_»oc X, exists in the Ll-sense;
ii) there exists a random variable X^ in Ll, such that X, = E [Хж \ .Щ;
Hi) the family {X,,t e R+} is uniformly integrable.
If these conditions hold, then Xoc = lim^oc X, a.s. Moreover, if for some
p > 1, the martingale is bounded in Lp, i.e. sup, ?[|X,|/'] < 00, then the equivalent
conditions above are satisfied and the convergence holds in the Lp-sense.
Proof. That ii) implies iii) is a classical exercise. Indeed, if we set Г, =
{\Е[Хоо\.Щ\>а},
[
Jr,
On the other hand, Markov's inequality implies
< f E[\Xx\\.%\dP= f \Xx\dP.
Jr, Jr,
It follows that, by taking a large, we can make a, arbitrarily small independently
off.
If iii) holds, then the condition of Theorem B.10) is satisfied and X, converges
to a r.v. Xoc a.s., but since {X,,t e R+} is uniformly integrable, the convergence
holds in the L1-sense so that i) is satisfied.
If i) is satisfied and since the conditional expectation is an О -continuous
operator, passing to the limit as h goes to infinity in the equality
yields ii).
Finally, if sup, ftlXJ'1] < 00, by Theorem A.7), sup, \X,\ is in Lp, and
consequently the family {|X/|'\ t e R+} is uniformly integrable. ?
It is important to notice that, for p > 1, a martingale which is bounded in
L1' is automatically uniformly integrable and its supremum is in V. For p — 1,
the situation is altogether different. A martingale may be bounded in L' without
being uniformly integrable, and may be uniformly integrable without belonging

to И1, where H1 is the space of martingales with an integrable suprerrium (see
Exercise A.17)). An example of the former is provided by exp{fi, — t 12} where В
is the BM; indeed, as B, takes on negative values for arbitrarily large times, this
martingale converges to zero a.s. as t goes to infinity, and thus, by the preceding
theorem cannot be uniformly integrable. An example of the latter is given in
Exercise C.15).
The analogous result is true for sub and supermartingales with inequalities in
ii); we leave as an exercise to the reader the task of stating and proving them.
We now turn to the optional stopping theorem, a first version of which was
stated in Proposition A.4). If X is a uniformly integrable martingale, then Xx
exists a.s. and if 5 is a stopping time, we define Xs on {5 = oo} by setting
Xs — -^oc-
C.2) Theorem. If X is a martingale and 5, T are two bounded stopping times
with S<T,
\]
a.s.
IfX is uniformly integrable, the family {Xs} where S runs through the set of all
stopping times is uniformly integrable and ifS<T
= E [Хт [Щ = E
a.s.
Remark. The two statements are actually the same, as a martingale defined on an
interval which is closed on the right is uniformly integrable.
Proof. We prove the second statement. We recall from Proposition A.4) that
if S and T take their values in a finite set and S < T. It is known that the family U
of r.v.'s ?[X(x;|.>?] where./? runs through all the sub-a-fields of .У is uniformly
integrable. Its closure 0 in L1 is still uniformly integrable. If S is any stopping
time, there is a sequence 5* of stopping times decreasing to 5 and taking only
finitely many values; by the right-continuity of X, we see that Xs also belongs
to U, which proves that the set {Xs, S stopping time} is uniformly integrable.
As a result, we also see that XSt converges to Xs in Ll. If Г е .Ws, it belongs a
fortiori to .Щк and we have
XxdP\
ir
I XSkdP = f
Jr Jr
passing to the limit yields
[ XsdP = [ X^dP :
Jr Jr
in other words, Xs = Е[ХЖ \ .7^\ which is the desired result.

We insist on the importance of uniform integrability in the above theorem. Let
X be a positive continuous martingale converging to zero and such that Xo = 1,
for instance X, = expE, — t/2); if for a < 1, T = inf[t : X, < a) we have
XT = a, hence E[XT] = a, whereas we should have E[XT] — E[X0] — 1 if
the optional stopping theorem applied. Another interesting example with the same
martingale is provided by the stopping times d, = inf[s > t : Bs = 0}. In this
situation, all we have is an inequality as is more generally the case with positive
supermartingales.
C.3) Theorem. If X is a positive right-continuous supermartingale and if we set
Xao = 0,for any pair S, T of stopping times with S <T,
Proof. Left to the reader as an exercise as well as analogous statements for sub-
martingales. ?
Before we proceed, let us observe that we have a hierarchy among the processes
we have studied which is expressed by the following strict inclusions:
supermartingales D martingales D uniformly integrable martingales D H1.
We now turn to some applications of the optional stopping theorem.
C.4) Proposition. IfX is a positive right-continuous supermartingale and
T(co) = inf [t : X,(a)) = 0} л inf [t > 0 : X,- (a>) = 0}
then, for almost every со, XXto) vanishes on [T(co), oo[.
Proof. Let Т„ = inf{f : X, < l/л}; obviously, Г„_1 < Tn < T. On [Tn = oo), a
fortiori T = oo and there is nothing to prove. On [Tn < oo}, we have Хт„ < \/п.
Let q e Q+; T + q is a stopping time > Т„ and, by the previous result,
1/n > E[XTiiliT,,<cc)] > E[XT+ql(Ti,<0O)].
Passing to the limit yields
E [Xr+Cj\(Tn<:X! vn)J = 0.
Since {T < oo} с {Т„ < oo, V«}, we finally get XT+Cj = 0 a.s. on [T < oo}. The
proof is now easily completed. ?
C.5) Proposition. A cadlag adapted process X is a martingale if and only if for
every bounded stopping time T, XT is in L' and
E[XT] = E[XQ].

Proof- The "only if part follows from the optional stopping theorem. Conversely,
jf s <. t and A e .5^the r.v. T = t\A.¦ + slA is a stopping time and consequently
E[X0] = E[XT] = E[X,\Ar] + E[X,\A].
On the other hand, t itself is a stopping time, and
E[X0] = E[X,] = E[XtlA,] + E[Xt\A].
Comparing the two equalities yields Xs = E[X, | .Щ.
C.6) Corollary. If M is a martingale and T a stopping time, the stopped process
MT is a martingale with respect to {.^).
Proof The process MT is obviously cadlag and adapted and if S is a bounded
. stopping time, so is S л Т; hence
E [Mj] = E [MSAT] = E [Mo] = E [<] .
Remarks. 1°) By applying the optional stopping theorem directly to M and to
the stopping times T As and T л t, we would have found that MT is a martingale
but only with respect to the filtration {-Щм)- But actually, a martingale with
respect to (.3^Л/) is automatically a martingale with respect to .Vt.
2°) A property which is equivalent to the corollary is that the conditional
expectations E [• | .9s\ and E [• | .^] commute and that E [¦ | .У% I -^t\ —
E [• I -^лг]- The proof of this fact, which may be obtained also without referring
to martingales, is left as an exercise to the reader.
Here again, we close this section with applications to the linear BM which we
denote by B. If a is a positive real number, we set
Ta = inf{r > 0 : B, = a), fa = inf {t > 0 : |s,| = a);
thanks to the continuity of paths, these times could also be defined as
Ta = inf{t > 0 : B, > a), fa = inf [t > 0 : \B,\ > a):
they are stopping times with respect to the natural filtration of B. Because of the
recurrence of BM, they are a.s. finite.
C.7) Proposition. The Laplace transforms of the laws ofTa and fa are given by
E [exp (~XTa)] = exp (-ау^Щ . E Гехр (-^u)l = (cosh (а^\\ '

Proof. For s > 0, Mst = exp (sB, — s2t/2) is a martingale and consequently,
Л/;5л7-я is a martingale bounded by e*a. A bounded martingale is obviously uni-
uniformly integrable, and therefore, we may apply the optional stopping theorem to
the effect that
which yields E Гехр (- у 7"«I = e~sa, whence the first result follows by taking
к = s2/2.
For the second result, the reasoning is the same using the martingale N? ~
(Mf + M~s) /2 = cosh (sB,) exp (- jt\ as Nj f is bounded by cosh(sa). ?
Remark. By inverting its Laplace transform, we could prove that Ta has a law
given by the density аBлх3)~1/2 exp(—a2/2x), but this will be done by another
method in the following chapter. We can already observe that
(law of Ta) * (law of Tb) = law of Ta+b.
The reason for that is the independence of Ta and (Та+ь — Ta), which follows from
the strong Markov property of BM proved in the following chapter.
Here is another application in which we call Px the law of x + B.
C.8) Proposition. We have, for a < x < b,
p, [Ta < ть] = h-—^, px [Tb < та] = X-^-.
о — a b — a
Proof. By the recurrence of BM
P,[Ta <Tb]+Ps[Tb<Ta]=l.
On the other hand, Вт"лТ" is a bounded martingale to which we can apply the
optional stopping theorem to get, since Вт„ = a, BTh = b,
aPx[Ta <Tb] + bPx[Tb< Ta] = x.
We now have a linear system which we solve to get the desired result.
C.9) Exercise. If X is a positive supermartingale such that ?[lim,, Xn] =
E[XQ] < oo, then X is a uniformly integrable martingale.
C.10) Exercise. Let с and d be two strictly positive numbers, В a standard linear
BM and set T = Tv л T_d.
V) Prove that, for every real number s,
E \e'lsl/2)T 1G=7;.)] = sinh(sd)/ sinh (s(c + d)),
and derive therefrom another proof of Proposition C.8). Prove that

+
Е Гехр (- S— Tj 1 = cosh (s(c - rf)/2)/cosh (j(c
compare with the result in Proposition C.7).
[Hint: Use the martingale exp (s (B, - ^) - ^tj.]
2°) Prove that for 0 < s < тт(с + d)~l,
E Гехр (у^I = cos (s(c - d)/2)/ cos (s(c + d)/2).
[Hint: Either use analytic continuation or use the complex martingale
)
C.11) Exercise. 1°) With the notation of Proposition C.7), if В is the standard
linear BM, by considering the martingale B] - t, prove that fa is integrable and
compute E[Ta].
[Hint: To prove that fa e L], use the times fa л п.]
2°) Prove that Ta is not integrable.
[Hint: If it were, we would have a — E [Вт„] — 0.]
3°) With the notation of Proposition C.8), prove that
Ex[TaATb] = {x-a)(b-x).
This will be taken up in Exercise B.8) in Chap. VI.
[Hint: This again can be proved using the martingale Bj — t, but can also be
derived from Exercise C.10) 2°).]
# C.12) Exercise. Let M be a positive continuous martingale converging a.s. to
zero as t goes to infinity. Put M* — sup, M,.
l°)Forx > 0, prove that
Р[М*>х\.Щ= \A(M0/x).
[Hint: Stop the martingale when it first becomes larger than x.]
2°) More generally, if X is a positive .^-measure r.v. prove that
P[M* >Х\.Щ= 1 л(М0/Х).
Conclude that Mo is the largest .^-measurable r.v. smaller than M* and that
M* = Mq/U where U is independent of Mo and uniformly distributed on [0. 1].
3°) If В is the BM started at a > 0 and 7 = inf {t : B, = 0}, find the law of
the r.v. r = sup,<ro?,.
4°) Let В be the standard linear BM; using M, = exp B/i (B, - /it)), ц > 0,
Prove that the r.v. Y — sup, {B, — /xt) has an exponential density with parameter
2д. The process B, — (it is called the Brownian motion with drift (—/z) and is
farther studied in Exercise C.14).
5C) Prove that the r.v. Ji of Exercise A.21) Chap. I is integrable and compute
*e constant C2(X).
N.B. The questions 3°) through 5C) are independent from one another.

C.13) Exercise. Let В be the standard linear BM and / be a locally bounded
Borel function on Ш.
Г) If f(B,) is a right-continuous martingale with respect to the filtration
(^°) = (<т (Bs, s < t)), prove that / is an affine function (one could also make
no assumption on / and suppose that f{B,) is a continuous (.^"°)-martingale).
Observe that the assumption of right-continuity is essential; if / is the indicator
function of the set of rational numbers, then f(B) is a martingale.
2C) If we suppose that /(B,) is a continuous (.^°)-submartingale, prove that
/ has no proper local maximum.
[Hint: For с > 0, use the stopping times T = Tt л 711 and
5 = jnf{? > T : B, ~ -1 or c + s or c-e}.]
3C) In the situation of 2°), prove that / is convex.
[Hint: A continuous function is convex if and only if f(x) + ax + fi has no
proper local maximum for any a and /J.]
* C.14) Exercise. Let В be the standard linear BM and, for a > 0, set
aa = inf{t : B, < t - a).
\°) Prove that au is an a.s. finite stopping time and that Нт,,-^ аа = +oo a.s.
2C) Prove that E [exp (|<rfl)] = exp(a).
[Hint: For к > 0, use the martingale exp(-(v/l +2X- 1)(B, - t) - Xt)
stopped at aa to prove that E [e~Xa"] = exp (—a (Vl +2Л — l)). Then, use ana-
analytic continuation.]
3°) Prove that the martingale exp (в, — |f) stopped at aa is uniformly inte-
grable.
4°) For a > 0 and b > 0, define now
aab = [nf{t : B, < bt — a);
in particular, cra =cra.\- Prove that
E exp ( -b2(ja,b j = exp(ab).
[Hint: Using the scaling property of BM, prove that ста,л = b 2erab.i-]
5°) For b < 1, prove that ? [exp (^i.fc)] = +oo.
[Hint: Use 2=).]
* C.15) Exercise. Let {Q,.T,P} be ([0, l]../?([0, 1]). dco) where c/tu is the
Lebesgue measure. For 0 < t < 1, let .3^" be the smallest sub-a-field of .7"
containing the Borel subsets of [0, t] and the negligible sets of [0. 1].
1°) For / e Ll([0, 1]. dco), give the explicit value of the right-continuous
version of the martingale

2°) Set Hf{t) = yL- ftl f(u)du and, for p > 1, prove Hardy's Lp-inequality
II Up ~ p - 1 ''
[Hint: Use Doob's Lp-inequality.]
3°) Use the above set-up to give an example of a uniformly integrable mar-
martingale which is not in H'.
4°) If fo \f(co)\\og+ \f(co)\dco < oc, check directly that Hf is integrable.
Observe that this would equally follow from the continuous-time version of the
result in Exercise A.16).
C.16) Exercise (BMO-martingales). 1°) Let У be a continuous uniformly inte-
integrable martingale. Prove that for any p e[\, oo[, the following two properties are
equivalent:
i) there is a constant С such that for any stopping time T
С a.s.;
ii) there is a constant С such that for any stopping time T
[Hint: Use the stopping time Tr of Exercise D.19) Chap. I.]
The smallest constant for which this is true is the same in both cases and
is denoted by ||K||BM0/j. The space \Y : НПвмо,, < ool is called BMO,, and
II Немо, is a semi-norm on this space. Prove that for p < q, BMOg ? BMOp.
The reverse inclusion will be proved in the following questions, so we will write
simply BMO for this space.
2°) The conditions i) and ii) are also equivalent to
iii) There is a constant С such that for any stopping time T there is an .Щ-
measurable, L''-r.v. aT such that
E[\Y0C-aT\p\.^]<C".
3°) If Y, = E [Уос | Щ for a bounded r.v. Y^, then Y e BMO and || Y ||BM0, <
2II Уос II ос. Examples of unbounded martingales in BMO will be given in Exercise
C.30) of Chap. III.
4°) If Y € BMO and T is a stopping time, YT e BMO and ||Kr|L.,, <
5°) (The John-Nirenberg inequality). Let Y e BMO and ||K||BM0| < 1. Let
a > 1 and T be a stopping time and define inductively
Ro = 7\ Rn = infjr > Д„_, : \Y, - Уд„_, | > a};
Prove that P [Rn < oo] > aP [Rn+i < oo]. Prove that there is a constant С such
that for any T

P [sup \Y, - YT\ > A1 < Ce~k/eP[T < oo];
in particular, if Y* = sup, |K,|,
P [Y* >X]< Ce~k/e.
As a result, K* is in Lp for every p.
[Hint: Apply the inequality E [\YS - YT\] < Ш1вмо, p[T < °°] which is
valid for S > 7", to the stopping times /?„ and Rn+i.]
6°) Deduce from 5°) that BMO,, is the same for all p and that all the semi-
norms || Y || BMOp are equivalent.
**
C.17) Exercise (Continuation of Exercise A.17)). [The dual space of Hl\.
1°) We call atom a continuous martingale A for which there is a stopping time T
such that
i) A, — 0 for f < 7"; ii) |A,| < P[T < oo] for every t. Give examples of
atoms and prove that each atom is in the unit ball of #'.
2°) Let X € H1 and suppose that Xo = 0; for every peZ, define
and Cp = 3 ¦ г^Г,, < oo]. Prove that A? = (Xr"+1 - X7"'') /Cp is an atom for
each p and that X = ?+^ C^A? in tf,. Moreover, ?+~ |CP| < 6 ||X||Hi.
3°) Let У be a uniformly integrable continuous martingale. Prove that
and deduce that the dual space (#')* of Я1 is contained in BMO.
[Hint: For the last step, use the fact that the Hilbert space H2 (Sect. 1 Chap. IV)
is dense in #'.]
4°) If X and Y are in H2, prove Fefferman's inequality
|Е[(ХэсКос)]|<6||Х||я, IUIbMO,
and deduce that (Я1)* = BMO.
[Hint: Use 2°) and notice that
АР
< 2X*.]
The reader will notice that if X is an arbitrary element in #' and Y an
arbitrary element in BMOb we do not know the value taken on X by the linear
form associated with Y. This question will be taken up in Exercise D.24) Chap. IV.
C.18) Exercise (Predictable stopping). A stopping time T is said to be pre-
predictable if there exists an increasing sequence (Г,,) of stopping times such that
i) lim,, Tn = T
ii) T,, < T for every и on {T > 0}. (See Sect. 5 Chap. IV)
If X,J e Ш+, is a uniformly integrable martingale and if S < T are two
predictable stopping times prove that
XS- = E[XT-\.?s-] = Е[ХТ\.Щ-]
[Hint: Use Exercise D.18) 3°) Chap. I and Corollary B.4).]

Notes and Comments
Sect. 1. The material covered in this section as well as in the following two is
classical and goes back mainly to Doob (see Doob [1]). It has found its way in
books too numerous to be listed here. Let us merely mention that we have made
use of Dellacherie-Meyer [1] and Ikeda-Watanabe [2].
The law of the iterated logarithm is due, in varying contexts, to Khintchine
[1], Kolmogorov [1] and Hartman-Wintner [1]. We have borrowed our proof from
McKean [1], but the exponential inequality, sometimes called Bernstein's inequal-
inequality, had been used previously in similar contexts. In connection with the law of
the iterated logarithm, let us mention the Kolmogorov and Dvoretzky-Erdos tests
which the reader will find in Ito-McKean [1] (see also Exercises B.32) and C.31)
Chap. III).
Most exercises are classical. The class L logL was studied by Doob (see Doob
[1]). For Exercise A.20) see Walsh [6] and Orey-Taylor [1].
Sect. 2. The proof of Proposition B.12) is taken from Neveu [2] and Exercise
B.14) is from Dubins [1]. The result in Exercise B.13) which is important in
some contexts, for instance in the study of Markov chains, comes from Doob [1];
it was one of the first applications of the convergence result for martingales. The
relationship between martingales and derivation has been much further studied;
the reader is referred to books centered on martingale theory.
Sect. 3. The optional stopping theorem and its applications to Brownian motion
have also been well-known for a long time. Exercise C.10) is taken from Ito-
McKean [1] and Lepingle [2].
The series of exercises on H' and BMO of this and later sections are copied
on Durrett [2] to which we refer for credits and for the history of the subject. The
notion of atom appears in the martingale context in Bernard-Maisonneuve [1]. The
example of Exercise C.15) is from Dellacherie et al. [1].

Chapter III. Markov Processes
This chapter contains an introduction to Markov processes. Its relevance to our
discussion stems from the fact that Brownian motion, as well as many processes
which arise naturally in its study, are Markov processes; they even have the strong
Markov property which is used in many applications. This chapter is also the
occasion to introduce the Brownian filtrations which will appear frequently in the
sequel.
§1. Basic Definitions
Intuitively speaking, a process X with state space (E,tC) is a Markov process
if, to make a prediction at time s on what is going to happen in the future, it is
useless to know anything more about the whole past up to time л- than the present
state Xs.
The minimal "past" of X at time 5 is the ст-algebra .?^ — a(Xu, и < s). Let
us think about the conditional probability
P[XteA\a{Xu,u<s)]
where A e К, s < t. If X is Markov in the intuitive sense described above, this
should be a function of Xs, that is of the form g(Xs) with g an #-measurable
function taking its values in [0, 1]. It would better be written gSJ to indicate its
dependence on 5 and t. On the other hand, this conditional expectation depends on
A and clearly, as a function of A, it ought to be a probability measure describing
what chance there is of being in A at time t, knowing the state of the process at
time s. We thus come to the idea that the above conditional expectation may be
written gSJ(Xs, A) where, for each A, x -> gs.t{x. A) is measurable and for each
x, A —> gSJ{x, A) is a probability measure. We now give precise definitions.
A.1) Definition. Let (E, ft) be a measurable space. A kernel N on E is a map
from E x % into R+ UR°°} such that
i) for every x e E, the map A -> N(x, A) is a positive measure on К ;
ii) for every A e К, the map x -> N(x, A) is К-measurable.

A kernel n is called a transition probability if тт(х, E) = 1 for every x e E. In
a Markovian context, transition probabilities are often denoted P, where < ranges
through a suitable index set.
If f e K+ and N is a kernel, we define a function N/ on E by
= / N(x,dy)f(v).
JE
It is easy to see that Nf is also in K+. If M and N are two kernels, then
MN(x,A) = / M(x,dy)N(y. A)
is again a kernel. We leave the proof as an exercise to the reader.
A transition probability л provides the mechanism for a random motion in E
which may be described as follows. If, at time zero, one starts from jc, the position
x\ at time 1 will be chosen at random according to the probability тг(х, ¦), the
position xi at time 2 according to л(х\, ¦), and so on and so forth. The process
thus obtained is called a homogeneous Markov chain and a Markov process is a
continuous-time version of this scheme.
Let us now suppose that we have a process X for which, for any s < t, there
is a transition probability PSJ such that
P [X, e А|ст(Х„, и < s)] = PSAX,, A) a.s.
Then for any / e K+, we have E [f(Xt)\a(Xu, и < s)] - /^/(Xsi.as is proved
by the usual arguments of linearity arid monotonicity. Let s < t < v be three
numbers, then
p[Xve A\o(Xu,u <s)] = P[XveA\o(Xu,u<t)\o(Xinu<s)]
= E[P,.v(Xl,A)\o(Xu.u<s)]
= f P,AXs,dy)P,.v{y,A).
But this conditional expectation should also be equal to P,.,.(X,, A). This leads
us to the
A.2) Definition. A transition function (abbreviated t.f.) on (?, K) is a family PSJ,
0 < s < t of transition probabilities on (E, /J) such that for every three real
numbers s < t < v, we have
for every x e E and A e <C. This relation is known as the Chapman-Kolmogorov
equation. The t.f. is said to be homogeneous ifPs.t depends on s and t only through
the difference t—s.ln that case, we write P,for Pq., and the Chapman-Kolmogorov
equation reads
Pt+,{x.A) = J P,[x,dy)P,(y\A)
for every .v, / > 0; in other words, the family {Pt,t> 0} forms a semi-group.

The reader will find in the exercises several important examples of transition
functions. If we refer to the heuristic description of Markov processes given above,
we see that in the case of homogeneous t.f.'s, the random mechanism by which the
process evolves stays unchanged as time goes by, whereas in the non homogeneous
case, the mechanism itself evolves.
We are now ready for our basic definition.
A.3) Definition. Let (?2, .7", (•?,), Q) be a filtered probability space; an adapted
process X is a Markov process with respect to (&,), with transition Junction Pst
if for any f e K+ and any pair (s, t) with s < t,
E[f(X,)\Vs] = PSJf(Xs) 6-a.s.
The probability measure Xq(Q) is called the initial distribution ofX. The process is
said to be homogeneous if the t.f. is homogeneous in which case the above equality
reads
Let us remark that, if X is Markov with respect to (•?,), it is Markov with
respect to the natural filtration (-5^°) = (a(Xu, и < t)). If we say that X is Markov
without specifying the filtration, it will mean that we use 0^"°). Let us also stress
the importance of Q in this definition; if we alter Q, there is no reason why
X should still be a Markov process. By Exercise A.16) Chap. I, the Brownian
motion is a Markov process, which should come as no surprise because of the
independence of its increments, but this will be shown as a particular case of a
result in Sect. 2.
Our next task is to establish the existence of Markov processes. We will need
the following
A.4) Proposition. A process X is a Markov process with respect to ^
(a(Xu,u < t)) with t.f. PSJ and initial measure v if and only if for any 0 =
to <t\ < ... <tkand fi € K+,
с г J с
Je Je Je
Proof. Let us first suppose that X is Markov. We can write
"i-l
I, \ Л.1 I E. fl, I Л.. I
= E
,'=0
k-\
1=0

this expression is the same as the first one, but with one function less and fk-\
replaced by Д_1 Ph._utkfk; proceeding inductively, we get the formula of the state-
statement.
Conversely, to prove that X is Markov, it is enough, by the monotone class
theorem, to show that for times tx < t2 < ¦ ¦ ¦ < h < t < v and functions
/i ft.g
\t\f'
,) \ = E П // (X,,) P,.vg(X,) ;
but this equality follows readily by applying the equality of the statement to both
sides.
Remark. The forbiddingly looking formula in the statement is in fact quite intu-
intuitive. It may be written more loosely as
Q [X,o e dxQ, X,, €dxi,..., Xh e dxk] =
= v(dxo)POjl (*o, dxO ... P,k_,,h{xk-\,dxk)
and means that the initial position jc0 of the process is chosen according to the
probability measure v, then the position x\ at time t\ according to POtl (jr0, •) and
so on and so forth; this is the continuous version of the scheme described after
Definition A.1).
We now construct a canonical version of a Markov process with a given t.f.
Indeed, by the above proposition, if we know the t.f. of a Markov process, we
know the family of its finite-dimensional distributions to which we can apply the
Kolmogorov extension theorem.
From now on, we suppose that (E, %) is a Polish space endowed with the
cr-field of Borel subsets. This hypothesis is in fact only used in Theorem A.5)
below and the rest of this section can be done without using it. We set Q = ER+,
.У? — ft^ and .J^"° = а (Х„, и < t) where X is the coordinate process.
A.5) Theorem. Given a transition function PSJ on (E, K), for any probability
measure v on (?\ fC), there is a unique probability measure Pv on (J2, -У^) such
that X is Markov with respect to (.^°) with transition function Ps,, and initial
measure v.
Proof. We define a projective family of measures by setting
Pi! '"(Aox A, x ...x Ая) =
/ v(dx0) P0Jl(x0,dxi) I Ptuf,{x\,dx2)... / Ptn_t.tSxn-Udxn)
Ja0 Jai Ja? Ja,,
and we then apply the Kolmogorov extension theorem. By Proposition A.4), the
coordinate process X is Markov for the resulting probability measure Pv. ?

From now on, unless otherwise stated, we will consider only homogeneous
transition functions and processes. In this case, we have
(eq. A.1)) PV[XO e Д0,Х„ е Д, Xtn e An] =
j v(dx) [ Ph{x.dxx) [ Ph_lt(xudx2)... I P,,-,._Mn-\,dxn).
Jaii Ja, j a2 j a,,
Thus, for each x, we have a probability measure Ptx which we will denote
simply by Px. If Z is an .3^-measurable and positive r.v., its mathematical ex-
expectation with respect to Px (resp. Pv) will be denoted by EA[Z] (resp. EV[Z]). If,
in particular, Z is the indicator function of a rectangle all components of which
are equal to E with the exception of the component over t, we get
PX[X, e Д] = P,{x.A).
This reads: the probability that the process started at x is in A at time t is given
by the value P,(x, A) of the t.f. It proves in particular that x —> PX[X, e A] is
measurable. More generally, we have the
A.6) Proposition. IfZ is .^F^-measurable and positive or bounded, the map x —>
EX[Z] is К -measurable and
E,,[Z]= f v{dx)Ex[Z].
Proof. The collection of sets Г in .j^0 such that the proposition is true for Z = \ г
is a monotone class. On the other hand, if Г = {Xo e До, Xu eA\, ..., Х,н еД„),
then РХ[Г] is given by eq.(l.l) with у = e,- and it is not hard to prove inductively
that this is an ^-measurable function of x; by the monotone class theorem, the
proposition is true for all sets Г e -J^. It is then true for simple functions and,
by taking increasing limits, for any Z e (.J^) + . ?
Remark. In the case of BMd, the family of probability measures Pv was already
introduced in Exercise C.14) Chap. I.
In accordance with Definition D.12) in Chap. I, we shall say that a property
of the paths ш holds almost surely if the set where it holds has P,,-probability 1
for every v; clearly, it is actually enough that it has P, -probability 1 for every x
in E.
Using the translation operators of Sect. 3 Chap. I, we now give a handy form
of the Markov property.
A.7) Proposition (Markov property). If Z is .7^'-measurable and positive (or
bounded), for every t > 0 and starting measure v,
Ev[Zo9,\.yr°] = EXi[Z] P,,-a.s.

The right-hand side of this formula is the r.v. obtained by composing the two
measurable maps w —*¦ Xt(a>) and x —> EX[Z], and the formula says that this r.v.
is within the equivalence class of the left-hand side. The reader will notice that,
by the very definition of в,, the r.v. Z о 9, depends only on the future after time
t; its conditional expectation with respect to the past is a function of the present
state X, as it should be. If, in particular, we take Z = 1|л\едь the above formula
reads
Pv [Xt+, e A\.%°] = PX,[XS e A] = PAX,, A)
which is the formula of Definition A.3).
Moreover, it is important to observe that the Markov property as stated in
Proposition A.7) is a property of the family of probability measures Px, x e E.
Proof of Proposition A.7). We must prove that for any .j^-measurable and posi-
positive Y,
Ev[Zo9t-Y] = Ev[Ex,[Z]-Y].
By the usual extension arguments, it is enough to prove this equality when Y —
Flf=i MX*) with /¦ e %+ and t,< t and Z = flj=i 8jiXtj) where gj e <C+, but
in that case, the equality follows readily from Proposition A.4). ?
We now remove a restriction on P,. It was assumed so far that Pt(x, E) — I,
but there are interesting cases where Pt(x, E) < 1 for some x's and /'s. We will
say that Pt is Markovian in the former case, submarkovian in the general case
i.e. when Pt(x, E) may be less than one. If we think of a Markov process as
describing the random motion of a particle, the submarkovian case corresponds to
the possibility of the particle disappearing or dying in a finite time.
There is a simple trick which allows to turn the submarkovian case into the
Markovian case studied so far. We adjoin to the state space E a new point A
called the cemetery and we set Ел = E U {A} and Кй — a()C, [A]). The point
A is isolated in Ед. We now define a new t.f. P on (Ea, %a) by
P,(x.A) = P,(x,A) if ACE,
= \-P,(x,E), P,(A,{A})=\.
In the sequel, we will not distinguish in our notation between P, and P, and in
the cases of interest for us A will be absorbing, namely, the process started at A
will stay in A.
By convention, all the functions on E will be extended to E& by setting
f(A) — 0. Accordingly, the Markov property must then be stated
Ev [Z о в,\.Т;0] = EX,(Z) P,,-a.s. on the set {Xr ф A}.
because the convention implies that the right-hand side vanishes on {X, = A] and
there is no reason for the left-hand side to do so.
Finally, as in Sect. 1 of Chap. I, we must observe that we cannot go much
further with the Markov processes thus constructed. Neither the paths of X nor
the filtration (.3^°) have good enough properties. Therefore, we will devote the

following section to a special class of Markov processes for which there exist
good versions.
# A.8) Exercise. Prove that the following families of kernels are homogeneous t.f.'s
(i) (Uniform translation to the right at speed v) E — R, К = .s?(R);
/>,(*••) = e.x+vt.
(ii) (Brownian motion) E = R, '6 = ./9(R); P,(x, •) is the probability measure
with density
g,(y - x) = Bл7)/2 exp ( - (y - xf/2t).
(iii) (Poisson process). E = R, К = ./9(R);
P,(x, dy) = J2 (е-'1я/п\) sx+n(dy).
о
This example can be generalized as follows: Let n be a transition probability on
a space (E, K); prove that one can define inductively a transition probability n"
by
tt"(x,A)= / л-^.^
Then
P,(x, rfv) - ? (<Г'*7и!) w"(x, rfy)
0
is a transition function. Describe the corresponding motion.
A.9) Exercise. Show that the following two families of kernels are Markovian
transition functions on (M+r ../?(IR+)):
(i) P,f(x) = exp(-r/x)/(x) + f™ty~2 exp(-t/y)f(y)dy
- (x/(x + r))/(x + 0 + /* r(r + y)-2/(? + y)dy.
# A.10) Exercise (Space-time Markov processes). If X is an inhomogeneous
Markov process, prove that the process (t, X,) with state space (R+ x E) is a
homogeneous Markov process called the "space-time" process associated with X.
Write down its t.f. For example, the heat process (see Sect. 1 Chap. I) is the
space-time process associated with BM.
Я AЛ1) Exercise. Let X be a Markov process with t.f. (P,) and / a bounded Borel
function. Prove that (/*,_.,/(X.,). 5 < t) is a Pt-martingale for any x.
A.12) Exercise. Let X be the linear BM and set X, = /()' Bsds. Prove that X
is not a Markov process but that the pair (Z?, X) is a Markov process with state
space R2. This exercise is taken up in greater generality in Sect. 1 of Chap. X.

A.13) Exercise (Gaussian Markov processes). 1°) Prove that a centered Gaus-
Gaussian process X,, t > 0, is a Markov process if and only if its covariance satisfies
the equality
r(s. u)T(t. t) = T(s, t)T{t, u)
for every s < t < u.
If F(t, t) = 0, the processes (Xv> s < t) and (X,. .9 > t) are independent. The
process B, — tB\, t > 0 (the restriction of which to [0, 1] is a Brownian Bridge)
is an example of such a process for which F(t, t) vanishes at t — 1. The process
Y of Exercise A.14) Chap. I is another example of a centered Gaussian Markov
process.
2°) If Г is continuous on Ш2+ and > 0, prove that F(s, t) = a(s)a(t)p(mf(s, t))
where a is continuous and does not vanish and p is continuous, strictly positive
and non decreasing. Prove that (X,/a(t), t > 0) is a Gaussian martingale.
3C) If a and p are as above, and В is a BM defined on the interval [p@), p(oo)[.
the process Y, = a(t)BP(,) is a Gaussian process with the covariance Г of
2°). Prove that the Gaussian space generated by Y is isomorphic to the space
L2(R+, dp), the r.v. Y, corresponding to the function a(f)l[o,r]-
4") Prove that the only stationary Gaussian Markov processes are the stationary
OU processes of parameter /3 and size с (see Sect. 3 Chap. I). Prove that their
transition functions are given by the densities
p,(x, y) - Bnc A - e^<)Y112 exp (- (>• - e^xf /2c (l - e'
Give also the initial measure m and check that it is invariant (Sect. 3 Chap. X) as
it should be since the process is stationary. Observe also that lim^oc P,(x, A) —
m(A).
5°) The OU processes (without the qualifying "stationary") with parameter fi
and size с are the Markov processes with the above transition functions. Which
condition must satisfy the initial measure v in order that X is still a Gaussian
process under /*,,? Compute its covariance in that case.
6°) If и and v are two continuous functions which do not vanish, then
r(s, t) = u(mf(s, t))v(sup(s, 0)
is a covariance if and only if u/v is strictly positive and non decreasing. This
question is independent of the last three.
A.14) Exercise. 1°) If В is the linear BM, prove that |B| is, for any probability
measure Pv, a homogeneous Markov process on [0. oo[ with transition function
given by the density
1 Г / 1 Л / 1
exp I - — (v - xY I + exp I - —(v + x)
This is the BM reflected at 0. See Exercise A.17) for a more general result.
2°) More generally, prove that, for every integer d, the modulus of BMd is a
Markov process. (This question is solved in Sect. 3 Chap. VI).

* 3°) Define the linear BM reflected at 0 and 1; prove that it is a homogeneous
Markov process and compute its transition function.
[Hint: The process may be defined as X, = \Bt — 2n on {\B, — 2n < l}.]
The questions 2°) and 3°) are independent.
# A.15) Exercise. 1") Prove that the densities
exp [ - —(у - x)z ) - exp [ - —(у + x)z ) \ , х > О, у > О,
define a submarkovian transition semi-group Q, on ]0, oo[. This is the transition
function of the BM killed when it reaches 0 as is observed in Exercise C.29).
2°) Prove that the identity function is invariant under Q,, in other words,
/0°° Qi(x, dy)y = x. As a result, the operators H, defined by
H,f(x) = - f Q,(x,dy)yf(y)
x Jo
also form a semi-group. It may be extended to [0, oo[ by setting
H,@.dy) = B/7rf3I/2y2 exp(-y2/2t)dy.
This semi-group is that of the Bessel process of dimension 3, which will be
studied in Chap. VI and will play an important role in the last parts of this book.
# A.16) Exercise (Transition function of the skew BM). Let 0 < a < 1 and g,
be the transition density (i.e. the density of the t.f. with respect to the Lebesgue
measure) of BM. Prove that the following function is a transition density
p?@. y) = 2ag,(y)\iy>0) + 2A - a)g,(}')l(.v<0).
P?(x.y)
= lu>0)[(gr(v -x) + Ba - \)g,(y +x))l(v>0) +2A - a)g,(y - x)l(v<0)]
+ hx<0)[(g,(y ~ x) + (\ - 2a)g,(y + x))\iY<0) + 2ag,(y - x)\{y>0)].
What do we get in the special cases a — 0, a = 1 and a — 1/2 ?
# A.17) Exercise. 1°) Let X be a Markov process with t.f. (P,) and ф a Borel
function from (E, К) into a space (?", К') such that ф(А) e 16' for every A e *6.
If moreover, for every t and every A' e K'
Р,(х,ф'\А')) = Р,(у,ф~\А')) whenever ф(х)=ф(у).
then the process X't = ф(Х,) is under P,, x e E, a Markov process with state
space (?", У/). See Exercise A.14) for the particular case of BM reflected at 0.
2C) Let X = BMrf and ф be a rotation in W' with center x. For со е П, define
ф(со) by Х,(ф(со)) = ф(Х,(со)). Prove that ф is measurable and for any Г е .K^

3°) Set Т, = inf \t > 0 : \Х, - Хо\ > г} and prove that Tr and XT, are inde-
independent. Moreover, under Px, the law of Xjr is the uniform distribution on the
sphere centered at x of radius r.
[Hint: Use the fact that the uniform distribution on the sphere is the only
probability distribution on the sphere which is invariant by all rotations.]
These questions will be taken up in Sect. 3 Chap. VIII.
§2. Feller Processes
We recall that all the t.f.'s and processes we consider are time-homogeneous. Let
E be a LCCB space and Cq(E) be the space of continuous functions on E which
vanish at infinity. We will write simply Co when there is no risk of mistake. We
recall that a positive operator maps positive functions into positive functions.
B.1) Definition. A Feller semi-group on Cq(E) is a family Tt, t > 0, of positive
linear operators on Cq(E) such that
i) To = Id and || T, \\ < 1 for every t;
H) T,+s — T, о Ts for any pair s,t > 0/
Hi) limno \\T,f - /|| = Ofor every f e C0(E).
The relevance to our discussion of this definition is given by the
B.2) Proposition. With each Feller semi-group on E, one can associate a unique
homogeneous transition function P,, t > 0 on (E, rC) such that
T,f(x) = P,f(x)
for every f € Q, and every x in E.
Proof For any x e ?, the map / —> T,f(x) is a positive linear form on Co; by
Riesz's theorem, there exists a measure P, (x, •) on К such that
•h
T,f(x)= I P,(x,dy)f(y)
for every / e Co. The map x -> / P,(x, dy)f(y) is in Q, hence is Borel, and,
by the monotone class theorem, it follows that x —> Pt(x, A) is Borel for any
A € К. Thus we have defined transition probabilities P,. That they form a t.f.
follows from the semi-group property of T, (Property ii)) and another application
of the monotone class theorem. D
B.3) Definition. A t.f. associated to a Feller semi-group is called a Feller transition
function.

With the possible exception of the generalized Poisson process, all the t.f.'s
of Exercise A.8) are Feller t.f.'s. To check this, it is easier to have at one's
disposal the following proposition which shows that the continuity property iii) in
Definition B.1) is actually equivalent to a seemingly weaker condition.
B.4) Proposition. A t.f is Feller if and only if
i) P,Co С Со for each t;
ii) V/ e Co, Vx e E. limuo P,f(x) = f(x).
Proof. Of course, only the sufficiency is to be shown. If / e Co, P,f is also
in Co by i) and so limy|() P,+Sf(x) — P,f(x) for every x by ii). The function
(t, x) —> P,f(x) is thus right-continuous in t and therefore measurable on K+ x E.
Therefore, the function
e~ptP,f(x)dt, p > 0,
is measurable and by ii),
lim pUpf(x) = f(x).
Moreover, Upf e Co, since one easily checks that whenever xn -*¦ x (resp. Л),
then Upf(x,,) -> Upf(x) (resp. 0). The map / -» Upf is called the resolvent of
order p of the semi-group P, and satisfies the resolvent equation
VPf ~ Uqf = {q- p)UpUqf = {q - p)UqUpf
as is easily checked. As a result, the image D = UP(CO) of Up does not depend
onp> 0. Finally \\pUpf\\ < 11/11.
We then observe that D is dense in Co; indeed if ц is a bounded measure
vanishing on D, then for any / 6 Co, by the dominated convergence theorem,
/ fdfi = lim / pUpfdfi = 0
so that /i = 0. Now, an application of Fubini's theorem, shows that
PlVpf(x) = ei" f e-psPsf{x)ds
hence
WP.Upf-UpfW <(f'"-l)|E/,,/||+/||/||.
It follows that lim,|0 II P,f - /II = 0 for / e D and the proof is completed by
means of a routine density argument. ?

90 Chapter ГП. Markov Processes
By Fubini's theorem, it is easily seen that the resolvent Up is given by a kernel
which will also be denoted by Up that is, for / e Co,
Upf(x) = J Up(x,dy)f(y).
For every x e E, Up(x, E) < 1 /p and these kernels satisfy the resolvent equation
Up(x. A) ~ U4(x, A) = (q - p) f Up(x. dy)Uq(y, A)
= (q - p) / Uq(x.dy)U,,(y.A)
One can also check that for / e Co, lim,,^^ \\pUpf — /|| = 0. Indeed
\\pUpf-f\\ = sap\pUpf(x)-f(x)\
X
< sup/ pe-'"\P,f(x)~ f(x)\dt
л JO
which converges to 0 by the property iii) of Definition B.1) and Lebesgue's
theorem. The resolvent is actually the Laplace transform of the semi-group and
therefore properties of the semi-group at 0 translate to properties of the resolvent
at infinity.
Basic examples of Feller semi-groups will be given later on in this section and
in the exercises.
B.5) Definition. A Markov process having a Feller transition function is called a
Feller process.
From now on, we work with the canonical version X of a Feller process for
which we will show the existence of a good modification.
B.6) Proposition. For any a and any f e Cj~, the process e~a'Uaf(X,) is a
supermartingale for the filtration (-У^0) and any probability measure Pv.
Proof. By the Markov property of Proposition A.7), we have for s < /
But it is easily seen that e~a{l~s)P,-sUaf < Uaf everywhere so that
Ev [e-a'Uaf(X,)\.^0] < e-atU
which is our claim.

We now come to one of the main results of this section. From now on, we
always assume that Ел is the one-point compactification of E, the point Л being
the point at infinity. We recall (Sect. 3 Chap. I) that an Ea-valued cadlag function
is a function on Ш+ which is right-continuous and has left limits on ]0, oo[ with
respect to this topology on Ед.
B.7) Theorem. The process X admits a cadlag modification.
Since we do not deal with only one probability measure as in Sect. 1 of
Chap. I but with the whole family Pv, it is important to stress the fact that the
above statement means that there is a cadlag process X on (?2,.У) such that
X, = X, P,,-a.s. for each t and every probability measure />,..
To prove this result, we will need the
B.8) Lemma. Let X and Y be two random variables defined on the same space
(?2, .3^". P) taking their values in a LCC В space E. Then, X = Y a.s. if and only
if
E[f{X)g(Y)] = E[f(X)g(X)]
for every pair (/, g) of bounded continuous functions on E.
Proof Only the sufficiency needs to be proved. By the monotone class theorem,
it is easily seen that
E[f(X.Y)] = E[f(X.X)]
for every positive Borel function on Ex E. But, since E is metrizable, the indicator
function of the set \(x, у) : x ф у} is such a function. As a result, X = Y a.s.
Proof of Theorem B.7). Let (/„) be a sequence on C^ which separates points,
namely, for any pair (x, y) in EA, there is a function /„ in the sequence such
that fn(x) ф /,,(y). Since aUafn converges uniformly in x to /„ as a —> oo, the
countable set .W; = {[/„/„, a e N. « e Nj also separates points.
Let S be a countable dense subset of R+. By Proposition B.6) and Theorem
B.5) in Chap. II, for each h e .Ж\ the process h(Xt) has a.s. right limits along
S. Because .Ж separates points and is countable, it follows that almost-surely the
function t —* X, (со) has right limits in Ед along S.
For any со for which these limits exist, we set X, (со) = lim .;> Xs and for an со
for which the limits fail to exist, we set X (со) = x where x is an arbitrary point
in E. We claim that for each t, X, = X, a.s. Indeed, let g and h be two functions
of С(Ед); we have
] = lmEv[g(X,)h(Xt)]
= lim Ev [g(X,)P^MX,)] = Ev
since Ps-,h converges uniformly to h as л- | t. Our claim follows from Lemma
B.8) and thus X is a right-continuous modification of X.

92 Chapter III. Markov Processes


This modification has left limits, because for h E JC, the processes h (X 1) are
now right-continuous supennartingales which by Theorem (2.8) of Chap. II, have
a.s. left limits along JR+. Again, because .711 separates points, the process X has
a.s. left limits in E,1 along JR+. 0


Remark. ln almost the same way we did prove X, = X, a.s., we can prove that
for each t, X, = X , - a.s., in other words, X , - is a left continuous modification
of X. It can also be said that X has no fixed time of discontinuity i.e. there is no
fixed time t such that P [X , - i= X,] > O.


From now on, we consider only cadlag versions of X for which we state


(2.9) Proposition. lfl;(w) = inf{t :::: 0 : X,-(w) = ,1 or X,(w) = L\}, we
have almost-surely X. = ,1 on [1;,00[.
Proof Let cp be a strictly positive function of Co. The function g = U 1 cp is also
strictly positive. The supennartingale Zt = e- t g(X t ) is cadlag and we see that
Z,- = 0 if and only if X t - = ,1 and Zt = 0 if and only if Xt = ,1. As a result


nw) = inf{t:::: 0: Zt-(w) = 0 or Z,(W) = O};
we then conclude by Proposition (3.4) in Chap. II.


o


With a slight variation from Sect. 3 in Chap. 1, we now cali D the space
of functions w from JR+ to E,1 which are cadlag and such that w(t) = ,1 for
t > s whenever w(s-) = ,1 or w(s) = ,1. The space Dis contained in the space
il = E
+ and, by the same reasoning as in Sect. 3 of Chap. 1, we can use it as
probability space. We still cali Xt the restrictions to D of the coordinate mappings
and the image of Pv by the canonical mapping cP will still be denoted Pv. For
each Pv, X is a cadlag Markov process with transition function Pro we cali it the
canonical cadlag realization of the semi-group Pt.
For the canonical realization, we obviously have a family et of shift operators
and we can apply the Markov property under the fonn of Proposition (1.7). We will
often work with this version but it is not the only version that we shall encounter
as will be made clear in the following section. Most often however, a problem
can be carried over to the canonical realization where one can use freely the shift
operators. The following results, for instance, are true for all cadlag versions. It
may nonetheless happen that one has to work with another version; in that case,
one will have to make sure that shift operators may be defined and used if the
necessity arises.
So far, the filtration we have worked with, e.g. in Proposition (1.7), was the
natural filtration (.
o). As we observed in Sect. 4 of Chap. 1, this filtration is not
right-continuous and neither is it complete; therefore, we must use an augmentation
of (.
 0).
We shall denote by .
 the completion of .r
 with respect to Pv and by
(.
v) the filtration obtained by adding to each .
o all the Pv-negligible sets in
.
. Finally, we will set




B.10) Proposition. The filtrations {<Wtv} and (.Jf) are right-continuous.
Proof. Plainly, it is enough to prove that (.^"') is right-continuous and, to this
end, because .Pfv and .Wt+ are Pv-complete, it is enough to prove that for each
J^-measurable and positive r.v. Z,
By the monotone class theorem, it is enough to prove this equality for Z =
IT=i fi(xti) where /• e Co and h <h< ...tn. Let us observe that
Ev [Z\.^v] = Ev [Z\.9)°] P,-a.s. for each t.
Let t be a real number; there is an integer к such that tk-\ < t < tk and for h
sufficiently small
A:— 1
Ev [Z\.^h] = П // (Xf( )gh(Xi+h) Pv-a.s.
where
gh(x) = I P,k-,-h(x,dxk)Mxk) I Ptk+t-,k(xk,dxk+i)...
If we let h tend to zero, gh converges uniformly on E to
g(x) = I P,k-t(x,dxk)fk(xk) I P,k+l-lk(xk,dxk+i)fk+i(xk+i)...
...J Л,-|,_, x,,-i,
Moreover, Xt+i, converges to Xt as h decreases to 0, thanks to the right-
continuity of paths and therefore, using Theorem B.3) in Chap. II,
t-i
Ev \Z\.T,r\ = ton Ev rzl.%1 = Г\МХ„)8{Х,) = Ev [Zl^] p,-a.s.,
which completes the proof. ?
It follows from this proposition that (.^") is the usual augmentation (Sect. 4
Chap. I) of (.J^°) and so is (.^"") if we want to consider only the probability
measure />,,. It is remarkable that completing the filtration was also enough to
make it right-continuous.
The filtrations (-^) and (-^v) are those which we shall use most often in the
sequel; therefore, it is important to decide whether the properties described so far
for (.^°) carry over to (.^")- There are obviously some measurability problems
which are solved in the following discussion.

B.11) Proposition. If Z is .^-measurable and bounded, the map x -+ Et[Z] is
<6 *-measurable and
EV[Z]= I E,[Z]v(dx).
Proof. For any v, there are, by definition of the completed ст-fields, two -im-
-immeasurable r.v.'s Z\ and Z2 such that Zx < Z < Z2 and EV[Z2 - Zx\ = 0.
Clearly, ?\[Z,] < EX[Z] < EX[Z2] for each x, and since x -> ?V[Z,], / = 1.2,
is ^-measurable and f (EX[Z2] - Ex[Zx])dv(x) = ?r[Z2 - Z-,] = 0, it follows
that E [Z] is in fCv. As v is arbitrary, the proof is complete.
B.12) Proposition. For each t, the r.v. X, is in .T
Proof. This is an immediate consequence of Proposition C.2) in Chap. 0. ?
We next want to extend the Markov property of Proposition A.7) to the
a-algebras -W,. We first need the
B.13) Proposition. For every t and h > 0, B^x (.Wt) С .^+„.
Proof As вi, e -^/-5^+,,, the result will follow from Proposition C.2) in Chap. 0
if we can show that for any starting measure v, there is a starting measure /j.
such that eh(Pv) — Р/л. Define /л = Xi,(Pv); then using the Markov property of
Proposition A.7) we have, for Г е .W^
Pv [\ro0h] = Ev
which completes the proof. D
We may now state
B.14) Proposition (Markov property). If Z is .T^-measurable and positive (or
bounded), then, for every t > 0 and any starting measure v,
?\ = Ex,[Z] /Va.s.
on the set {X, ф Л]. In particular, X is still a Markov process with respect to
Proof. By Propositions B.11) and B.12), the map Ex,()[Z] is .^"-measurable, so
we need only prove that for any Ae,/|,
We may assume that Z is bounded; by definition of .Kc, there is a .X^-measurable
r.v. Z' such that {Z ф Z'\ С Г with Г е .7^ and Р„[Г] = 0 where /x = X,(PV)
as in the preceding proof. We have {Z о в, ф Z' о в,} С 6~](Г) and as in the
above proof, Pv [#,"'(-0] — Рц[Г] — 0. Since it was shown in the last proof that
?",, [Ex, [ • ]] =?¦,,[• ], it now follows that

EW[EX,[|Z-Z'|]] = EM[|Z-Z'|] = O
so that EX,[Z] = ?x,[Z'] Pv-a.s. Therefore, we may replace Z by Z' on both
sides of (*) which is then a straightforward consequence of Proposition A.7). о
Feller processes are not the only Markov processes possessing good versions,
and actually they may be altered in several ways to give rise to Markov processes in
the sense of Sect. 1, which still have all the good probabilistic properties of Markov
processes but no longer the analytic properties of Feller transition functions. The
general theory of Markov processes is not one of the subjects of this book; rather,
the Markov theory is more something we have to keep in mind when studying
particular classes of processes. As a result, we do not want to go deeper into
the remark above, which would lead us to set up axiomatic definitions of "good"
Markov processes. In the sequel, if the necessity arises, we will refer to Markov
processes with values in (?, %) as collections X — {Q,.W,."%, Px,x e E,9,);
these symbols will then have the same meaning and can be used in the same
manner as for Feller processes. For instance, the maps t -» X, are supposed to be
a.s. cadlag. This may be seen as a sad departure from a rigorous treatment of the
subject, but we shall make only a parcimonious use of this liberty, and the reader
should not feel uneasiness on this count. Exercise C.21) gives an example of a
Markov process which is not a Feller process.
We proceed to a few consequences of the existence of good versions. The
following observation is very important.
B.15) Theorem (Blumenthal's zero-one law). For any x e E and Г € .^',
either РК[Г] = 0or РХ[Г] = 1.
Proof. If Г e ct(Xo), then РХ[Г] = 0 or 1 because PX[XO = x] - 1. Since one
obtains .^e" by adding to ст(Хо) sets of Px-measure zero, the proof is complete.
?
B.16) Corollary. IfT is a (.^e')-stopping time, then either PX[T = 0] = 1 or
PX[T > 0] = 1.
This corollary has far-reaching consequences, especially in connection with the
following result (see Exercise 2.25). If A is a set, we recall from Sect. 4 Chap. I
that the entry and hitting times of A by X are defined respectively by
DA = inf {t > 0 : X, e A}, TA = inf [t > 0 : X, e A}
where as usual, inf@) = +oo. For any s,
s + DAo9s=s + inf [t > 0 : Xl+S e A] = inf{t > s : X, e A}.
It follows that s + DA о 9, = DA on {DA > s] and also that
TA = lim | (s + DAo 9,).
.v|0
Similarly, one proves that t + TA о 9, — TA on {TA > t).

B.17) Theorem. If A is а В or el set, the times DA and TA are (.'A )-stopping times.
Proof. Since X is right-continuous, it is clearly progressively measurable and,
since (.Wt) is right-continuous and complete, Theorem D.15) of Chap. I shows
that DA which is the debut of the set Г = {(?, со) : X,(co) e A] is a (.W,) -stopping
time.
The reader will now check easily (see Proposition C.3)) that for each s, the
time s + DA о Bs is a (.>^")-stopping time. As a limit of (.^-stopping times, TA
is itself a (.7~t)-stopping time. ?
We will next illustrate the use of the Markov property with two interesting
results. For the first one, let us observe that a basic example of Feller semi-groups
is provided by convolution semi-groups i.e. families (fi,,t > 0) of probability
measures on Ш'1 such that
i) fi, * fis = /x,+, for any pair (л, t);
ii) /io — ?q and lim,j,o fit = ?q in the vague topology.
If we set
P,(x,A) = / \A(x + y)fi,(dy)
J
we get a Feller t.f. as is easily checked by means of Proposition B.4) and the
well-known properties of convolution. Most of the examples of Exercise A.8), in
particular the t.f. of BM';, are of this type. A Feller process with such a t.f. has
special properties.
B.18) Proposition. If the transition function of X is given by a convolution semi-
semigroup (fit), then X has stationary independent increments. The law of the increment
X, - Xs is fi,-s.
The word stationary refers to the fact that the law of the increment X, — Xs
depends only on f- s, hence is invariant by translation in time. The process X
itself is not stationary in the sense of Sect. 3 Chap. I.
Proof. For any f e X+ and any t we have, since Px[Xq — x] = 1
Ex [f(X, - Xo)] = Ex [f(X, - x)] = ,i,(f)
which no longer depends on x. Consequently, by the Markov property, for s < t,
Ev [f(X, - Xs)\.7;} = EXi [fiX,-, - Xo)] - fi,-Af) /Va.s..
which completes the proof. ?
Conversely, if a Feller process has stationary independent increments, it is
easily checked that its t.f. is given by a convolution semi-group having property ii)
above Proposition B.18). These processes will be called processes with stationary
independent increments or Levy processes. Some facts about these processes are
collected in Sect. 4.
We now turn to another result which holds for any Markov process with good
versions.

B.19) Proposition. Let x e E and ax = inf {t > 0 : X, ф х); there is a constant
a e [0, oo] depending on x such that
Proof. The time ax is the hitting time of the open set {xf and therefore a stopping
time (see Sect. 4 Chap. I). Furthermore ax = t+axod, on {стл > t} as was observed
before Theorem B.17); thus, we may write
Px [ax >t+s] = Px [(ax > f) П (av > t 4- s)] = Ex [l((Tv>n 1((Тл>л) о в,]
and by the Markov property, since obviously X, ф Л on [ax > t}, this yields
Px [<*x > t + s] = Ex [l(ff[>()?x, [o-.v > s]];
but, on [ax > t}, we have X, = x, so that finally
Px [ax > t + s] = Px [ax > t] Px [ax > s]
which completes the proof. ?
Finally, this proposition leads to a classification of points. If a = +oo, ax is
/\-a.s. zero; in other words, the process leaves x at once. This is the case for all
points if X is the BM since in that case P, (x. {x}) —0 for every t > 0. If a = 0,
the process never leaves x which can be said to be a trap or an absorbing point.
If a e]0, oo[, then ax has an exponential law with parameter a; we say that x is a
holding point or that the process stays in x for an exponential holding time. This
is the case for the Poisson process with a = 1 for every x, but, in the general
case, a is actually a function of x. Let us further observe that, as will be proved
in Proposition C.13), X can leave a holding point only by a jump; thus, for a
process with continuous paths, only the cases a = 0 and a = oo are possible.
We close this section by a few remarks about Brownian motion. We have now
two ways to look at it: one as the process constructed in Chap. I which vanishes at
time zero and for which we consider only one probability measure; the other one
as a Markov process which can be started anywhere so that we have to consider
the whole family of probability measures Pv. The probability measure of the first
viewpoint, which is the Wiener measure in the canonical setting, identifies with
the probability measure Pq = Pta of the second viewpoint. Any result proved for
Po in the Markov process setting will thus be true for the Wiener measure.
In the sequel, the words Brownian motion will refer to one viewpoint or the
other. We shall try to make it clear from the context which viewpoint is adopted
at a given time; we shall also use the adjective standard to mean that we consider
only the probability measure for which Bo = 0 a.s., i.e. the Wiener measure.
B.20) Definition. //('¦%) is a filtration, an adapted process В is called a ('¦?,)-
Brownian motion if
i) it is a Brownian motion.
ii) for each t > 0, the process B,+x — B,, s > 0. is independent of'(¦?,).

It is equivalent to say that В is a Markov process with respect to (¦'?-,) with
the t.f. of Exercise A.8) ii).
In this definition, the notion of independence may refer to one or to a family of
probability measures. We want to stress that with the notation of this section, В is
a (.J*j")-Brownian motion if we consider the whole family of probability measures
Pv or а (-У, '')-Brownian motion if we consider only one probability measure Pfl.
Each of these filtrations is, in its context, the smallest right-continuous complete
filtration with respect to which В is a BM.
B.21) Definition. Let X be a process on a space (?2,.7~) endowed with a fam-
family Pg, в С 0, of probability measures. We denote by (-^x) the smallest right-
continuous and complete filtration with respect to which X is adapted. A ->^x-
stopping time is said to be a stopping time of X.
In the case ofBM, we have .^~s = .У, or .i^"M according to the context. These
filtrations will be called the Brownian filtrations.
B.22) Exercise. Prove that the transition functions exhibited in Exercises A.8),
A.14), A.15) are Feller t.f.'s. Do the same job for the OU processes of Exercise
A.13).
# B.23) Exercise. Show that the resolvent of the semi-group of linear BM is given
by Up(x, dy) = up{x, y)dy where
и (x, v) = -= exp y-y/2p\x - y\j .
B.24) Exercise. If X is a Markov process, ep and eq two independent exponential
r.v.'s with parameters p and q, independent of X prove that for a positive Borel
function /
pUpf{x) = Ex [f (Xep)] , pqUpUqf{x) = Ex [f (Xep+eJ]
and derive therefrom the resolvent equation.
* B.25) Exercise. 1°) A subset Л of ? is called nearly Borel if, for every v, there
are twi Borel sets A\, Ai such that A\ С А С Aj and Pv [Оддд, < oo] — 0.
Prove that the family of nearly Borel sets is a sub-гт-algebra of the universally
measurable sets. Prove that, if A is nearly Borel, then Dr\ and 7д are .^"-stopping
times.
2') If A is nearly Borel and x e E, prove that either PX[TA = 0] = 1 or
PX[TA — 0] = 0. In the former (latter) case, the point x is said to be regular
(irregular) for A.
3°) A set О is said to be finely open if, for every x e O, there is a nearly
Borel set G such that x e G С О and X is irregular for G'. Prove that the finely
open sets are the open sets for a topology which is finer than the locally compact
topology of E. This topology is called the fine topology.
4°) If a nearly Borel set A is of potential zero, i.e. j^ P,(¦, A)dt = 0 (see
Exercise B.29)). then A1 is dense for the fine topology.

5°) If / is universally measurable and t -*¦ f(X,) is right-continuous,.then /
is finely continuous.
6°) Prove the converse of the property in 5°).
[Hints: Pick s > 0 and define To = 0 and for any ordinal a of the first kind
define
Ta+i = inf jf > Ta : \f(X,) - f(XTu)\ > s}
and if a is a limit ordinal
Ta = sup Ta.
w<ct
Prove that Ta < Ta+i a.s. on {Ta < oo} and that, as a result, there are only
countably many finite times Ta.]
* B.26) Exercise. Prove that for a Feller process X, the set {Xs(a>), 0 < s < t,
t < ?(a>)} is a.s. bounded.
• [Hint: Use the quasi-left continuity of Exercise B.33) applied to exit times of
suitable compact sets.]
* B.27) Exercise (A criterion for the continuity of paths). Г) Let d be a metric
on E, and / a function from [0, 1] to E& with left (right) limits on ]0, l]([0,1[).
Then, / is not continuous if and only if there is an s > 0 such that
N,,(f) = max d(f(k/n), f((k + \)/n)) > s
0<k<n-l
for all и sufficiently large.
2°) Let B(x, s) = {>¦ : d(x, y) < s}. For s > 0 and a compact set K, define
Men = {a>: Nn{X,{(o)) > e\ Xs(o>) e К for every 5 e [0, 1]}.
Prove that Pv (Men) < и supveA: Ру„(х, В(х, е)с).
3е) Using the result in the preceding exercise, prove that if X satisfies the
condition
limsup-P,(x,B(x.s)c) =0
liO xeK t
for every s > 0 and compact set K, then a.s. X has continuous paths.
4°) Check that the condition in 3°) is satisfied for BM. Thus, the results of this
section together with 3°) give another construction of BM independent of Chap. I.
* B.28) Exercise. Let В be the BM'', . /, = cr(fi,,s > t) and . /¦ = f),- ^ its
asymptotic cr-field.
Iе) Use the time inversion of Sect. 1 Chap. 1 and Blumenthal's zero-one law to
prove that. ? is Pq-zs. trivial i.e. for any Ae./ either Pq{A) = 0 or Pq(A) = 1.
2°) If A is in . /, then for any fixed t, there is an event Be./ such that
1A = 1B o0,. Prove that
P<[A] = j P,(x.dy)Py(B)
and conclude that either P\A] = 0 or P [A] = 1.

3") Prove that for any initial distribution v and Ге./%
lim sup \PV(A П Г) - Pv(A)Pv(r)\ = 0.
[Hint: Use Theorem B.3) of Chap. II.]
4') If vi and V2 are two starting measures, show that
Um || (v, -v2)P,|| =0
where the norm is the variation norm on bounded measures.
[Hint: Use a Jordan-Hahn decomposition of (uj — иг).]
B.29) Exercise. Let X be a Feller process. For x e E and A e К, set
U(x.A) = / P,(x,A)dt.
Jo
Г) Prove that this integral is well defined, that U is a kernel on (E. K) and
that if / e rC+, Uf(x) = E, [f™ f(X,)dt]. The kernel U is called the potential
kernel of X.
2) Check that Uf — lim;40 Ukf and that for every A. > 0
и = uA + xuAu = uk + xuuk.
3C) Prove that for X — BM'', d < 2, the potential kernel takes only the values
0 and +oo on yJ+. This is linked to the recurrence properties of BM in dimensions
1 and 2 (see Sect. 3 Chap. X).
4°) Prove that for BM'', d > 3, the potential kernel is the convolution kernel
associated with (\/2ndl2) Г {{d /2)—Y)\x\2~d i.e. the kernel of Newtonian potential
theory. In particular, for d = 3,
2л- J
x - y\
5C) Compute the potential kernel of linear BM killed when it reaches 0 (Exer-
(Exercise C.29)) and prove that it has the density 2(x Л v) with respect to the Lebesgue
measure on №.+ .
6C) Prove that g, is a density for the potential kernel of the heat process.
B.30) Exercise. Let Л be a Borel set.
Г ) Prove that for every ,v, t > 0,
2 ) Let v be the probability measure carried by the complement of the closure
A of A. Prove that under Py, the process Y defined by
Y, = X, on {t < Тл), Y, = A on {t > TA]
is a Markov process with respect to (.7\). One says that Y is the process X killed
when entering A. See Exercise C.29) for a particular case.

* B.31) Exercise. Let X be the standard linear BM and set \(r(t) = t~a, a > 0.
Prove that the following three properties are equivalent
i) Нт^о/е B,y//(t)dt exists on a set of strictly positive probability;
ii) a < 3/2;
iii) /0' \j/(t)\B,\dt < oo a.s.
[Hint: Use Blumenthal's zero-one law to prove that i) is equivalent to a stronger
property. Then, to prove that i) entails ii) use the fact that for Gaussian r.v.'s
almost-sure convergence implies convergence in L2, hence convergence of the
L2-norms.]
The assertion that for, say, any positive continuous function \}r on ]0, 1], the
properties i) and iii) are equivalent is false. In fact, it can be shown that if тД е
L]]oc{]0, 1]), iii) is equivalent to
* iv) /0 ir(t)ti/2dt < 00,
and there exist functions \jj satisfying i) but not iv).
This subject is taken up in Exercise C.19) Chap. IV.
* B.32) Exercise. Let В be the standard linear BM and h a continuous function on
]0, 1 [. Let Г be the event
{a>: B,{w) < h(t) on some interval ]0, 7"(<u)[c]0, 1[}.
Prove that either Р(Г) = 0 or Р(Г) = 1; in the former (latter) case, h is said to
belong to the lower (upper) class. For every e > 0, h(t) = A + ?-)^/2/ Iog2(l/f)
belongs to the upper class and h(t) = A — e)^/2t Iog2(l/f) to the lower class.
* B.33) Exercise. 1°) (Quasi-left continuity). If X is a Feller process and (Т„) а
sequence of (..^-stopping times increasing to T, prove that
\\mXT — XT a.s. on [T < 00}.
и
[Hint: It is enough to prove the result for bounded T. Set Y = Нт„ Хт„ (why
does it exist?) and prove that for continuous functions / and g
E, [f(Y)g(XT)] = HmlimE, [f(XT,,)g(XTl,+t)] = Ex [f(Y)g(Y)].].
This result is of course totally obvious for processes with continuous paths.
For processes with jumps, it shows that if Xj_ ф Хт on {0 < T < 00}, then
a sequence (Т„) can increase to T only in a trivial way: for a.e. a>, there is an
integer n(a>) such that Г„(а>) — Г(а>) for и > н(а>). Such a time is said to be
totally inaccessible as opposed to the predictable times of Sect. 5 Chap. IV, a
typical example being the times of jumps of the Poisson process.
2°) Using only 1°) and Proposition D.6) in Chap. I, prove that if A is a closed
set, then TA is a (.>f )-stopping time.

§3. Strong Markov Property
Stopping times are of constant use in the study of Markov processes, the reason
being that the Markov property extends to them as we now show. We must first
introduce some notation.
We shall consider the canonical cadlag version of a Feller process. We use
the results and notation of §2. For a (.3^")-stopping time T, we define Xj on the
whole space Q by putting Xj — A on {T = oo}. The r.v. XT is ../r-measurable
as follows from Sect. 4 in Chap. 1 and is the position of the process at time T.
We further define a map 0j from J2 into itself by
вт(ш)=в,(ш) if T(w) = t. Вт(ш) = шл if T(w) = +oo.
where сол is the path identically equal to A. Clearly, X, о 0T = Xr+, so that
вт[(-7?)Со{Хт+,.1 >0).
We now prove the Strong Markov property of Feller processes.
C.1) Theorem. IfZ is a .^-measurable and positive (or bounded) random vari-
variable and T is a slopping time, for any initial measure v.
Ev[ZoeT\.?r] = EXl[Z]
Pv-a.s. on the set {Хт ф Л}.
Proof. We first prove the formula when T takes its values on a countable set D.
We have
— / ,
which proves our claim.
To get the general case, let us observe that by setting
we define a sequence of stopping times taking their values in countable sets and
decreasing to T. For functions /-,/ = 1.2 к in C^ and times t\ < ti < ... <
h, let
gix) = j P,,(.v.u?x1)/1U1) J P,1_,,(.xi.dx2)... j Л,-i,-,(**-
Because X is Feller the function g is in C,+ and by the special case.
.rTn =g(xTj.

Because of the right-continuity of paths, and by Corollary B.4) Chap. II, we get
the result for the special case ]~[( fi(Xti). By an application of the monotone class
Theorem, we get the result for every positive Z in -Я^'.
It remains to prove the theorem for Z e (.Kx)+. By working with P[, =
Pv (' C\(%T Ф <4)) / PV(XT ф A) for which the conditional expectation given .7j-
is the same as under Pv, we may assume that Xj ф A a.s. and drop the corre-
corresponding qualification. Call \x the image of Pv by XT i.e. ц(А) = PV[XT e A].
By definition of .7~x, there are two .^"-measurable r.v.'s Z' and Z" such that
Z' < Z < Z" and P,,[Z" - Z' > 0] = 0. By the first part of the proof
/>,.[Z" о 6»r - Z' с 6)r > 0] = ?„ [?*, [Z" - Z' > 0]] = 0.
Since v is arbitrary, it follows that Z o0T is .^-measurable.
The conditional expectation EV[Z о 9T .7j\ is now meaningful and
EV[Z' овт\.Щ < Ev[Zc0T\.7r] < EV[Z" овт\.7т].
By the foregoing, the two extreme terms are P,.-a.s. equal to Ex, [Z], which ends
the proof. П
Remark. The qualification (X/- ф Л} may be forgotten when ? — +oo a.s. and
T < oo a.s.. in which case we will often drop it entirely from the notation.
In the course of the above proof, we saw that 9T is a .Xjc-measurable mapping.
We actually have
C.2) Lemma. For any t > 0. T + t is a stopping time and 9~l(.^) с -7}-+,.
Proof. By a monotone class argument, it is easily seen that 0~x (.7^°) С .Pj-+i and
the reasoning in the above proof yields the result. ?
We shall use this lemma to prove
C.3) Proposition. IfS and T are two (.7^)-stopping times, then S + T о % is an
(-Ti')-stopping time.
Proof. Since (¦/]) is right-continuous, it is enough to prove that {S + T о % <
t} e . 7", for every /. But
{S + T o6s <t} = \J{S < t -q}n{T o0s <q].
«ей
By the lemma, the set {T о 0S < q) is in -7v+v; by definition of .7$+сп the set
{S < t — q) П {T о 9s < q\ = {S + q < t) П \Т о 9S < q] is in ,7i which proves
our claim. П

If we think of a stopping time as the first time some physical event occurs,
the stopping time S + T о 9S is the first time the event linked to T occurs after
the event linked to 5 has occured. For instance, using the notation of §2, if A and
В are two sets in К, the stopping time 7д + Тв ° 9тл is the first time the process
hits the set В after having hit the set A. This will be used in the sequel of this
section.
We now give a first few applications of the strong Markov property. With a
stopping time 7", we may also associate a kernel Pj on (?, fC) by setting
A]
or more generally for f e K+,
The following result tells us how to compose these kernels (see Definition A.1)).
C.4) Proposition. ffS and T are two stopping times, then
Proof. By definition
Ps(Pt f)(x) = Ex[EXs[f(XT)\},
so that, using the strong Markov property, we have
Ps(PTf)(x) = E,[\tX^u)E,[f(XT)o
Now f(XT) о 9S = f(Xs+Tc,H,) and f(XT) о 0s = 0 on {Xs = A], so that the
result follows. D
Remark. We have thus generalized to stopping times the fundamental semi-group
property. Indeed if T = t a.s., then PT = P, and S+To9s = S + t.
We now prove that, if we start to observe a Markov process at a stopping time,
the resulting process is still a Markov process with the same t.f.
C.5) Proposition. //' T is a stopping time, the process Y, — XT+, is a Markov
process with respect to (.St+1) and with the same transition junction.
Proof. Let / e У,+; for every v,
E,. [f(XT+I+s)\.7i+,] = E, [f(Xs)oeT+t\.Sr+l] = Psf(XT+t)
on the set {XT+t ф A]. But, on the set {Xj+t = A], the equality holds also so
that Xr+t satisfies the conditions of Definition A.3). D

Remarks. 1) The process Y is another version of the Feller process X. This shows
that non canonical versions arise naturally even if one starts with the canonical
one.
2) The above property is in fact equivalent to the strong Markov property as
is stated in Exercise C.16).
In the case of processes with independent increments, the above proposition
can be stated more strikingly.
C.6) Corollary. If X has stationary' independent increments, the process
(Xj+i — Xt- t > 0) is independent oj -7j and its law under Pv is the same as
that ofX under Po.
Proof. For fi € X+.t, G №.+ , i - 1.2 n,
=EXl
П •/'
and, as in Proposition B.18), this is a constant depending only on /)¦ and /,-. n
In particular, in the case of BM, BT+, - Вт is a (.7^+,)-Brownian motion
independent of.^. Another proof of this fact is given in Exercise C.21) Chap. IV.
We devote the rest of this section to an application of the Strong Markov property
to linear BM.
We recall that the continuous increasing process S, = supy<, Bs and the stop-
stopping time Ta introduced in Sect. 3 Chap. II are inverses of one another in the
sense that
Ta = infjf : S, > a]. S, = inf[a : TB > /}.
The map a —>¦ Ta is increasing and left-continuous (see Sect. 4 Chap. 0 and Sect. 1
in Chap. V).
In the next result, P is the probability measure for the BM started at 0. i.e.
the Wiener measure if we use the canonical version.
C.7) Proposition (Reflection principle). For every a > 0 and t > 0,
P[S, >a]= P[Ta <t] = 2P[B, > a] = P(\B,\ > a).
The name for this proposition comes from the following heuristic argument.
Among the paths which reach a before time t, "half will be above a at time /;
indeed, if we consider the symmetry with respect to the line у = a (in the usual
representation with the heat process paths) for the part of the path between Tu and
/, we get a one-to-one correspondence between these paths and those which are
under a at time /. Those which are exactly in a at time / have zero probability
and therefore
P[St>a] = P[St >a.B, > a] + P[S, >a. B, < a]
= 2P[S, >a.B, >a] = 2P[B, > a]

b,l
s,
b
a
gr
/ity /i i
Fig. 3. Reflection in /;
since {В, > а) С (S, > a). This argument which is called the reflection principle
can be made rigorous but it is easier to use the Strong Markov property. See also
Exercise C.14).
Proof. Indeed, since Bj: = a,
P [S, >a.B,<a] = P(Ta < t. Вт„+и-т„) ~ % < 0].
and since Btu+s — Вти is a BM independent of .7jt this is further equal to
I
Since
-P[Ta<t] = -
P[S, >a]= P[B, >a] + P[S, >a,B, < a].
the result follows.
Remarks. 1°) For each f, the random variables S, and \B,\ have thus the same
law. Of course, the processes S and \B\ do not have the same law (S is increasing
and \B\ is not). More will be said on this subject in Chap. VI.
2C) As an exercise, the reader may also prove the above result by showing,
with the help of the strong Markov property, that the Laplace transforms in t of
P(Ta < t) and 2P(B, > a) are equal.
The preceding result allows to derive from the law of B, the laws of the other
variables involved. As we already pointed out, S, has the same law as \B,\ namely
the density 2Bл7)~|/2 ехр(-у2/2/) on [0. oc[. As for the law of Tlt, it could have
been obtained by inverting its Laplace transform f-"v'2v found in Proposition C.7)
Chap. II, but we can now observe that

P[Ta < t] = 2 f -L= exp (-.v2/2/) dy.
Ja \l2nt
Upon differentiation with respect to /, the density /„ of Ta is found to be equal
on [0, 90[ to
fjs) = -i= (--L f exp (-.v2/2i-) <V + 4г f >2 exP (--v2/2i-)
and integrating by parts in the integral on the right yields
fu(s) = яBтг.гУ'/2ехр(-я2/2.:)-
The reader will find in Proposition C.10) another proof based on scaling properties.
The densities fa form a convolution semi-group, namely /„ * j\ = ,/«+/,; this is an
easy consequence of the value of the Laplace transform, but is also a consequence
of the following proposition where we look at Ta as a function of a.
C.8) Proposition. The process Ta, a > 0, is a left-continuous increasing process
with stationary independent increments and is purely discontinuous, i.e. there is
a.s. no interval on which a —*¦ Ta is continuous.
Proof. It is left-continuous and increasing as already observed. To prove that
P [{a> : a —* Ta(a>) is continuous on some interval!] = 0.
we only need to prove that for any pair {p. q) of rational numbers with p < q
P [{a> : a —»¦ Ти{ш) is continuous on [p, q]}] = 0.
We remark that a —> Ta(u>) is continuous on [p, q] iff 5 is strictly increasing on
[Tp, T4] (see Sect. 4 Chap. 0), but, since BTl,+, — BTr is a Brownian motion, this
is impossible by the law of the iterated logarithm.
To prove the independence of the increments, pick two real numbers 0 < a <
b. Since T/, > Ta a.s. we have Tb — Ta + Th о вт„ a.s., hence, for f & K+,
E[f(Th-T(,)\.7;(i] = E[f{TboeTi)\.TTi]
= EBra [f (Tb)] a.s.
But Bj: = a a.s. and because of the translation invariance of BM the last displayed
term is equal to the constant E [/ G},_a)] which shows that Th - Ta is independent
of .Я7;, thus completing the proof.
Thus, we have proved that Tit, a > 0, is a process with stationary independent
increments, hence a Feller process (see Sect. 2). It is of course not the canonical
version of this process since it is defined on the probability space of the Brownian
motion. It is in fact not even right-continuous. We get a right-continuous version
by setting T,r — Пт/;|„ 7), and proving
C.9) Proposition. For any fixed a, Ta = TaP-a.s.

Proof. We also have Ta+ = inf{/ : S, > a} (see Sect. 4 Chap. 0 and Sect. 1
Chap. V). The strong Markov property entails that for every / > 0, we have
5У„+, > a whence the result follows immediately. D
Remark. This could also have been proved by passing to the limit as h tends to
zero in the equality
E exp ( - у (Ta+h - Ta) I = exp(-Xft)
which is a consequence of results in Chap. II.
Furthermore, since Ta, a > 0, is a Feller process, Proposition C.9) is also a
consequence of the remark after Theorem B.8).
The above results on the law of Ta which can in particular be used to study
the Dirichlet problem in a half-space (see Exercise C.24)) may also be derived
from scaling properties of the family Т„ which are of intrinsic interest and will be
used in Chap. XI. If a is a positive real, we denote by B\a) the Brownian motion
a~x Ba2, and adorn with the superscript (a) anything which is defined as a function
of B{a). For instance
With this notation, we obtain
C.10) Proposition. We have Ta = a2T{x"] and consequently Ta = u2Tx. Moreover,
Proof. By definition
Ta = inf{/ :a
= inf a2/ :/?;"'= l|=fl27-,(l"
As for the second sentence, it is enough to prove that T\ = 5, 2. This follows
from the scaling property of St, namely: S, = s/tS]. Indeed
P[Ti >u] = P[Stt < l] =
Knowing that S\ = \B\ |, it is now easy to derive anew the law of T\. We will
rather use the above results to prove a property of importance in Chap. X (see
also Exercise C.24) in this section, of which it is a particular case).
C.11) Proposition. // /6 iv another standard linear BM independent of B, then
Рт„ = a ¦ С where С is a Cauchy random variable with parameter I.

Proof. Because of the independence of p and В and the scaling properties of p\
a ^ FF в ^ а й ^ a в
Рт„ — V '« ¦ P\ — tt ¦ pi — tttt • Pi
Ji \tS\\
which ends the proof, since pVl^i I is known to have the Cauchy distribution. ?
Remarks, (i) The reader may look in Sect. 4 for the properties of the processes
Ta+ and р>„, a > 0.
(ii) Proposition C.11) may also be seen as giving the distribution of BM2 when
it first hits a straight line.
We now give a first property of the set of zeros of the linear BM.
C.12) Proposition. The set Z = {t : B, = 0J is a.s. closed, without any isolated
point and has zero Lebesgue measure.
Proof. That Z is closed follows from the continuity of paths. For any x,
E, \ f \zU)ds\ = f Ps(x. {0})ds = 0,
since Ps(x, @J) = 0 for each 5 and x. It follows that Z has a.s. zero Lebesgue
measure, hence also empty interior.
The time zero belongs to Z and we already know from the law of the iterated
logarithm that it is not isolated in Z. We prove this again with the techniques of
this section. Let 7b = inf{/ : B, — 0}; the time t 4- To о 0, is the first point in Z
after time /; by the Markov property and the explicit value of ?o[exP ~°tTtl] we
have
E0[exp{-a(t + T0oe,)}] = exp(-at)E0[EBl[exp(-aT0)]]
= exp(-a/)?0[exp{-|Br|v/2aJ]
and this converges to 1 as / goes to zero. It follows by Fatou's lemma that
Po [lim,jo (t + To о в,) = 0] — I, namely, 0 is a.s. the limit of points in Z. Now
for any rational number q, the time dq = q + T{) о Вц is the first point in Z
after q\ by Corollary C.6), Bdii+I is a standard linear BM and therefore dq is a.s.
the limit of points in Z. The set ^V = (J e~ {dq is not a limit of points in Z}
is negligible. If h e Z(<w) and if we choose a sequence of rational numbers q,,
increasing to h, either h is equal to some d4n or is the limit of the d4i/s. Thus,
if w <? N, in either case h is the limit of points in Z(«) which establishes the
proposition. ?
Thus Z is a perfect set looking like the Cantor "middle thirds" set. We will see
in Chap. VI that it is the support of a random measure without point masses and
singular with respect to the Lebesgue measure and which somehow accounts for
the time spent at 0 by the linear BM. Moreover, the Complement of Z is a countable
union of disjoint intervals /„ called the excursion intervals; the restriction of В to

such an interval is called an excursion of B. Excursions will be studied in great
detail in Chap. XII.
Finally, we complete Proposition B.19) by proving that X can leave a holding
point only by a jump, namely, in the notation of Proposition B.19):
C.13) Proposition. /fO <a < oo. then Px [XOt = x] = 0.
Proof. If a < oc, then Px[ax < oc] = 1. On \Xa = x\ we have ax ов„х = 0 and
by the strong Markov property
Px [ax < oo; Х„х = x] = Px [ax < ос; ХПк — x, а, с #„ = о]
= Px [ax = 0] Px [ax < oc: XOl = x].
Thus if Px [ХПх = x] > 0 we have Px [ax = 0] = 1 which completes the proof.
# C.14) Exercise (More on the reflection principle). We retain the situation and
notation of Proposition C.7). Questions 3°) and 4) do not depend on each other.
Iе) Prove that the process B" defined by
Я," = B, on {/ < Ta\. S," =2a- B, on {/ > Ttl)
has the same law as B.
2C) For a < b, b > 0, prove that
P [S, > b.B, < a] = P[B, <a- 2b] = P2h [B, < a].
and that the density of the pair (B,, 5,) is given on {(a. b)\ a < b, b > 0} by
B/nt2I'2 Bb - я)ехр (-Bb- aJ/2t).
This can also be proved by computing the Laplace transform
e-a'P[S, > b, B, <a]dt.
/
Jo
3°) Prove that for each /, the r.v. 5, - B, has the same law as \B,\ and that
25, — B, has the same law as |BM^|; prove further that, conditionally on 25, — B,,
the r.v.'s 5, and S, — B, are uniformly distributed on [0.25, — B,]. Much better
results will be proved in Chap. VI.
4C) Let .v, = infs<, Bs and a > 0; prove that under the probability measure Pa
restricted to {s, > 0|, the r.v. B, has a density equal to
Bnn [exp (- -^-j - exp \- -^)\ • A > 0.
Compare with Exercise C.29).

* C.15) Exercise. 1°) Let a < 0 < b; prove that for F C] — <x>, a] and / > 0,
P [Th < Tu. B,e F] = P [B, eshF]-P [Ta < Tb, В, е shF]
where shF = [2b - y\ у e F].
2°) In the notation of Exercise C.14), prove that for every Borel subset E of
[a.b],
P[a < s, < S, < b, B, e E]= / k(x)dx
Je
where
= Bjt/)^1/2 ^ exp(- — (x+2k(b-a)J
1
- exp ( - — (x - 2b + 2k(b - a))
V 2'
[Hint: P [Ta < Tb. Т„ <t. B, e E] = P [Ta < Th, В, е saE]. Apply repeat-
repeatedly the formula of 1°) to the right-hand side.]
3°) Write down the laws of B* — sup(<, \BS\ and fa = inf|/ : \B,\ > a). This
can be done also without using 2")).
C.16) Exercise. Let X be the canonical version of a Markov process with transi-
transition semi-group P,. If for every stopping time 7", every initial distribution v and
every / e <C+,
E, [f(XT+l)\.7r] = P,f(XT) />,.-a.s.
on {XT ф A] then X has the strong Markov property.
C.17) Exercise. Prove that (B,, S,) is a Markov process with values in E =
{{a,b)\a < b, b > 0) and using 2 ) in Exercise C.14) compute its transition
function.
C.18) Exercise. For the standard linear BM, prove that
lim -ftP[B, < l,Vs < /] = ,/-.
# C.19) Exercise. Г) Let X be a Feller process, T a finite stopping time. Prove
that any .V-*.-measurable and positive r.v. Z may be written ф(со. 0T(w)) where ф
is .Я7 ® .3^c-measurable. Then
?,, [Z|.J^] (w) = / 0(w, w')PXr(a;)(^w') Pv-a.s.
2) Let 5 be > 0 and .^-measurable. For a positive Borel function /, prove
that
Ev [f(XT+s)\-?r] [со) - Ех,ш [f {XS(W)(-))] Pv-a.s.
This can be proved using 1°) or directly from the strong Markov property.
3°) Write down the proof of Proposition C.7) by using 2°) with T = Ta.

# C.20) Exercise (First Arcsine law). The questions 3°) through 5°) may be solved
independently of 1") and 2').
Г) For a real number и, let J1( = и 4- To ° #» as in the proof of Proposition
C.12). Using the BM (B,+l< - В„, t > 0), prove that du = и + В2 ¦ T\ where T\
is independent of Bu. Hence du = u{\ + C2) where С is a Cauchy variable with
parameter 1.
[Hint: du = и + T_Bu where T refers to Bl+Il - B,t.]
2) Prove that the r.v. g\ = sup{/ < 1 : B, = 0} has the density
(jtV.vO -v)) on [0. 1].
[Hint: [gi <u} = [du > 1}.]
3°) Use the strong Markov property and the properties of hitting times recalled
before Theorem B.17) to give another proof of 2").
4 ) Let d\ = inf{/ > 1 : B, = 0); by the same arguments as in 3 ), prove that
the pair (gi. d\) has the density j^y~i/2{z - y)~3/2 on 0 < у < 1 < ;.
5') Compute the law of^i — g\. In the language of Chap. XII, this is the law
of the length of the excursion straddling 1 (see Exercise C.7) in Chap. XII). For
more about g\ and d\, see the following Exercise C.23).
C.21) Exercise. Let ц be a Bernoulli r.v. Define a family Xх of processes on
RU{4j by
Xх =x + t if x < 0 and x + I < 0,
Xх =x + t if x < 0. x + I > 0 and ц = 1.
Xх = Л if x <0.x + t >0 and rj = -l.
Xх =x + t if л > 0.
Let Px be the law of Xх. Prove that under Px. x € K, the canonical process is
a Strong Markov process which is not a Feller process.
C.22) Exercise. Let E be the union in K2 of the sets [x < 0, у = 0}, [x > 0: у =
x) and {x > 0: у = —л'}; define a transition function on E by setting:
forx<0. P,((x.0).-) = eu+,.o, if.v+/<0.
P,((x,0).-) = ^i.r+M+n + ^ц+л-л-и ib"+/>0.
forA->0. P, ((.v.x).-) = %+,.,+,).
P, ((X,-X). •) = ?,,+,._,_,,.
Construct a Markov process X on E with t.f. P, and prove that it enjoys neither
the Blumenthal zero-one law nor the Strong Markov property.
[Hint: For the latter, consider the time T = inf {/ : X, € {x > 0. у = л}}.]
C.23) Exercise. For the standard BM and / > 0, let
g, = sup (i- < t : B, = 0}. d, = inf {s > t : BK = 0}.
1°) By a simple application of the Markov property, prove that the density of
the pair (B,.d,) is given by

|x[(?(i-/K) 1/2exp(-.vx2/2/(s - t)) 1,,>„.
2) By using the time-inversion invariance of Proposition A.10) in Chap. 1,
derive from Г ) that the density of the pair (B,, g,) is given by
Bjr)-l\x\(s(t -,vK)~'/2exp(-x2/2(/ -.v)) 1(л<„.
Sharper results along these lines will be given in Sect. 3 Chap. Xll.
# C.24) Exercise. Г) Denote by (X,.Y,) the Brownian motion in R" x К started
at @, a) with a > 0. Let Su = inf{/ : Y, = 0} and prove that the characteristic
function of Xs,, is exp(—я|и|). In other words, the law of XSu is the Cauchy
law with parameter a, and this generalizes Proposition C.11); the corresponding
density is equal to
2C) If (X,, Y,) is started at (x. a), write down the density P{yM){z) of XSil- This
density is the Poisson kernel. If / e Q(K"), prove that
g(x.a)= / Pu.t,)(z)f(z)dz
is a harmonic function in W x]0, oo[.
# C.25) Exercise. Let (X, Y) be the standard planar BM and for a > 0 let fa =
inf{/ : |X,| = a]. Show that the r.v. Yf has the density
Bacosh(jrx/2a))"' .
* C.26) Exercise (Local extrema of BM). Let В be the standard linear BM.
Г) Prove that the probability that a fixed real number x be a local extremum
of the Brownian path is zero.
2°) For any positive real number r, prove that
P [{со : S,(co) is a local extremum of / —>• B,(a>), t > r}] = 0.
3°) Prove that consequently a.e. Brownian path does not have two equal local
extrema. In particular, for every /", there is a.s. at most one ,v < r such that
Bs = Sr.
4) Show that the set of local extrema of the Brownian path is a.s. countable.
C.27) Exercise. Derive the exponential inequality of Proposition A.8) in Chap.
11 from the reflection principle.
C.28) Exercise. 1 ') Prove that the stopping time ст„.,, of Exercise C.14) in Chap.
11 has a density equal to
aBjn3r'/2exp(-(« - btJ/2t).
[Hint: Use the scaling property of Exercise C.14) in Chap. 11 and the known
forms of the density and Laplace transform for Ta.]
2') Derive therefrom another proof of 5 ) in the above mentioned exercise.

C.29) Exercise. Г) Let В be the linear BM and T = inf {/ > 0 : B, = 0}. Prove
that, for any probability measure Pv such that v is carried by ]0, oo[, the process
X defined by
X, = B, on {t < 7"}, X, = A on {/ > 7"}
is a Markov process on ]0. oof with the transition function Q, of Exercise
A.15). This process can be called the BM killed at 0. As a result for a > 0,
Q, (a. ]0. oo[) = PoG"« < /); check this against Proposition C.10).
[Hint: See Exercise B.30). To find the transition function, use the joint law of
(B,, S,) found in Exercise C.14).]
2) Treat the same question for the BM absorbed at 0, that is X, = В,лт-
* C.30) Exercise. (Examples of unbounded martingales of BMO). 1") If В is
the standard linear BM, prove that fi1 (i.e. the BM stopped at time 1) is in BMO
and that Цй'Ьмо, = B/n)l/2. Prove that ?[fif|.^] is not in BMO.
2') If В is the standard linear BM and 5 is a positive r.v. independent of В
then X, = B,j,s is a martingale of BMO for the filtration ¦'!/> — a(S..^) if and
only if 5 is a.s. bounded.
** C.31) Exercise. If h is a real-valued, continuous and increasing function on ]0. 1[.
if h{t)/sfi is decreasing and /0+ t~3/2h(t) exp (-h2(t)/2t) dt < oo, prove that
P[B, > h(t) for some / e]0.b[] < / exp (-h2(t)/2t)dt.
Jo+ \Z2nt}
Show that, as a result, the function h belongs to the upper class (Exercise B.32)).
[Hint: For 0 < a < b and a subdivision (tn) of ]a, b[.
P[B, > h(t) for some t e]a.b[] < P [TlHa) < a]
It is also true, but more difficult to prove, that if the integral diverges, h belongs
to the lower class. The criterion thus obtained is known as Kolmogorov \s test.
§4. Summary of Results on Levy Processes
In Sect. 2, we defined the Levy processes which include Brownian motion anc
the Poisson process. In Sect. 3, we found, while studying BM, another example
of a Levy process, namely the process 7"u+. This is just one of the many examples
of Levy processes cropping up in the study of BM. Thus, it seems worthwhile
to pause a little while in order to state without proofs a few facts about Lev}
processes which may be used in the sequel. Levy processes have been widel)
studied in their own right; if nothing else, their properties often hint at properties

of general Markov processes as will be seen in Chap. VII about infinitesimal
generators.
In what follows, we deal only with real-valued Levy processes. We recall that
a probability measure ц on R, or a real-valued r.v. Y with law /л, is said to be
infinitely divisible if, for any n > 1, there is a probability measure fi,, such that
fi = ц*" or equivalently if Y has the law of the sum of и independent identically
distributed random variables. It is easy to see that Gaussian, Poisson or Cauchy
variables are infinitely divisible.
Obviously, if A' is a Levy process, then any r.v. X, is infinitely divisible.
Conversely, it was proved by Levy that any infinitely divisible r.v. Y may be
imbedded in a unique convolution semi-group, in other words, there is a Levy
process X such that Y = X\. This can be proved as follows. By analytical
methods, one can show that д is infinitely divisible, if and only if, its Fourier
transform /i is equal to ехр(^) with
where j3el,G>0 and v is a Radon measure on К — (OJ such that
„2
< oo.
This formula is known as the Levy-Khintchine formula and the measure v as the
Levy measure. For every t e K+, exp(?^) is now clearly the Fourier transform of
a probability measure \xt and plainly /x, * д5 = fj.,+s and lim,joA<r = ?o which
proves Levy's theorem.
The different terms which appear in the Levy-Khintchine formula have a prob-
probabilistic significance which will be further emphasized in Chap. VII. If a = 0 and
v = 0, then д = sp and the corresponding semi-group is that of translation at
speed p\ if p = 0 and v = 0, the semi-group is that of a multiple of BM and
the corresponding Levy process has continuous paths; if p — 0 and a — 0, we
get a "pure jump" process as is the case for the process Ta- of the preceding
section. Every Levy process is obtained as a sum of independent processes of the
three types above. Thus, the Levy measure accounts for the jumps of X and the
knowledge of v permits to give a probabilistic construction of X as is hinted at in
Exercise A.18) of Chap. Xll.
Among the infinitely divisible r.v.'s, the so-called stable r.v.'s form a subclass
of particular interest.
D.1) Definition. A r.v. Y is stable if, for every k, there are independent r.v.'s
У],..., У* with the same law as Y and constants «a > 0. bk such that
Y+ + Y

It can be proved that this equality forces a/< = kl/a where 0 < a < 2. The
number a is called the index of the stable law. Stable laws are clearly infinitely
divisible. For a = 2, we get the Gaussian r.v.'s; for 0 < a < 2 we have the
following characterization of the corresponding function \j/.
D.2) Theorem. If Y is stable with index a e]0. 2[, then a — 0 and the Levy
measure has the density (m\ 1 (л<0) + mi 1 (x>0)) \x\~{'+a) with m \ and тг > О.
With each stable r.v. of index a, we may, as we have already pointed out,
associate a Levy process which will also be called a stable process of index a.
The process 7> of the last section is thus a stable process of index 1/2. If, in the
above result, we make /9 = 0 and ni\ =И2 we get the symmetric stable process
of order a. In that case, \j/(u) = — c\u\a where с is a positive parameter. Among
those are the linear BM and the Cauchy process which is the symmetric stable
process of index 1 such that jx, = exp(—t\u\). These processes have a scaling
invariance property germane to that of BM, namely, for any с > 0, c"'X,.., has
the same law as X,.
Another interesting subclass of stable processes is that of stable subordinators.
Those are the non-decreasing stable processes or equivalently the stable processes
X such that X, is a.s. > 0. The corresponding stable law is thus carried by [0, oc[
and may be characterized by its Laplace transform. It turns out that for the index a,
this Laplace transform is equal to exp( —cXa) for 0 < a < 1; indeed, for a €] 1, 2]
this function is not a Laplace transform and there is no stable subordinator of
index a (the case a = 1 is obvious). Once again, the process Ta+ provides us with
an example of a stable subordinator of index 1/2.
If tu is a stable subordinator of index a €]0, 1 [ vanishing at 0 and X is a Levy
process independent of the process г„, the map a —* XTa makes sense and is again
a Levy process as the reader can prove as an exercise. If X is the linear BM,
an easy computation shows that XTa is a symmetric stable process of index 2a.
If, in particular, a = 1/2, XT<I is a symmetric Cauchy process, which generalizes
Proposition C.11).
D.3) Exercise. If X is a Levy process, prove that exp (iuX, - tyj/{u)) is a complex
martingale for every real u.
D.4) Exercise. Derive the scaling invariance property of the process Г„+ from the
scaling invariance property of BM.
* D.5) Exercise. Let Т„, а > 0 be a right-continuous stable subordinator of index
a vanishing at zero. Since its paths are right-continuous and increasing for every
positive Borel function / on R+, the Stieltjes integral
/'
Jo
f(a)dTa
makes sense and defines a r.v. T(f),.
1°) What is the necessary and sufficient condition that / must satisfy in order
that T(f), < a.s. for every t'l What is the law of T(f), in that case?

2°) Prove that there is a constant ca such that the process S defined by
S,=ca f a-[<JTt,
Jo
has the same law as Т„, а > 0.
Notes and Comments
The material covered in this chapter is classical and is kept to the minimum neces-
necessary for the understanding of the sequel. We have used the books by Blumenthal-
Getoor [1], Meyer [1] and Chung [2]; the latter is an excellent means of getting
more acquainted with Markov processes and their potential theory and comple-
complements very nicely our own book. The reader may also use volume 4 of Dellacherie-
Meyer [1]. For a more advanced and up-to-date exposition of Markov processes
we recommend the book of Sharpe [3].
Most results of this chapter may be found in the above sources. Let us merely
mention that Exercise A.16) is taken from Walsh [3] and that Exercise A.17)
1°) is from Dynkin [1] whereas the other questions were taken in Chung [2].
In connection with this exercise, let us mention that Pitman-Rogers [1] contains
another useful criterion for a function of a Markov process to still be a Markov
process.
Kolmogorov's test of Exercise A.31) may be found in Ito-McKean [1]. The
equivalence between iii) and iv) in Exercise B.31) may be found in Jeulin-Yor [2]
and Jeulin [4]. Exercise C.15) is borrowed from Freedman [1].

Chapter IV. Stochastic Integration
In this chapter, we introduce some basic techniques and notions which will be used
throughout the sequel. Once and for all, we consider below, a filtered probability
space (Q,.V ,,7[, P) and we suppose that each .V, contains all the sets of P-
measure zero in .7". As a result, any limit (almost-sure, in the mean, etc.) of
adapted processes is an adapted process; a process which is indistinguishable
from an adapted process is adapted.
§1. Quadratic Variations
A.1) Definition. A process A is increasing (resp. o/finite variation,) if it is adapted
and the paths I —> А, (со) are finite, right-continuous and increasing (resp. of finite
variation) for almost every со.
We will denote by . V!+ (resp.. /,) the space of increasing (resp. of finite variation)
processes. Plainly, . ?<+ С ¦ У- and conversely, it is easily seen from Sect. 4
Chap. 0 that any element A e . / can be written A, = A+ - A~ where A +
and A~ are in . ^+. Moreover, A+ and A~ can be chosen so that for almost
every со, А* (со) - Ay (со) is the minimal decomposition of А, (со). The process
f0 \dA\, = A,+ + A~ is in . ^+ and for a.e. со the measure associated with it is
the total variation of that which is associated with A(co); it is called the variation
of A.
One can clearly integrate appropriate functions with respect to the measure
associated to A(co) and thus obtain a "stochastic integral". More precisely, if X
is progressively measurable and - for instance - bounded on every interval [0. /]
for a.e. со, one can define for a.e. со, the Stieltjes integral
= f ХЛ
Jo
(X ¦ A),(to) = / Xs(co)dA,(co).
Jo
If со is in the set where А. (со) is not of finite variation or X.(co) is not locally
integrable with respect to dA(co), we put (X • A) = 0. The reader will have no
difficulty in checking that the process X ¦ A thus defined is in . /. The hypothesis
that X be progressively measurable is precisely made to ensure that X ¦ A is
adapted. It is the "stochastic integral" of X with respect to the process A of. /,.

Our goal is now to define a "stochastic integral" with respect to martingales. A
clue to the difficulty, already mentioned in the case of BM, is given by the
A.2) Proposition. A continuous martingale M cannot be in . /, unless it is con-
constant.
Proof. We may suppose that Mo = 0 and prove that M is identically zero if it is
of finite variation. Let V, be the variation of M on [0. t] and define
Sn = inf{s : Vs > n}:
then the martingale Ms" is of bounded variation. Thus, it is enough to prove the
result whenever the variation of M is bounded by a number K.
Let Л — {to — 0 < t\ < ... < tk — t) be a subdivision of [0, /]; we have
-1
E[Mf] = E
= E
since M is a martingale. As a result,
? «, -
i=0
i=0
E[Mf] < E Г V, ( sup \M,^ - Mti И] < KE sup
/,„, - Mu
when the modulus of A goes to zero, this quantity goes to zero since M is
continuous, hence M = 0 a.s. ?
Remark. The reader may find more suggestive the proof outlined in Exercise
A.32).
Because of this proposition, we will not be able to define integrals with respect
to M by a path by path procedure. We will have to use a global method in which
the notions we are about to introduce play a crucial role. We retain the notation
of Sect. 2 Chap. 1. If Л — {t0 = 0 < t\ < ...} is a subdivision of K+ with only a
finite number of points in each interval [0, /] we define, for a process X,
4-1
'- - Х>У
/=0
where к is such that tk < t < tt+\\ we will write simply Tta if there is no risk of
confusion. We recall from Sect. 2 Chap. I that X is said to be of finite quadratic
variation if there exists a process (X, X) such that for each /, Tt& converges in
probability to (X. X), as the modulus of Л on [0, t] goes to zero. The main result
of this section is the
A.3) Theorem. A continuous and bounded martingale M is of finite quadratic
variation and (M, M) is the unique continuous increasing adapted process vanish-
vanishing at zero such that M2 — (M, M) is a martingale.

Proof. Uniqueness is an easy consequence of Proposition A.2), since if there were
two such processes A and B, then A — В would be a continuous martingale of
. / vanishing at zero.
To prove the existence of {M, M), we first observe that since for /, < s < ti+\,
E [(A/,i+1 - Muf \.К] = Е [(A/,i+, - MsJ | .ЯГ] + (A/., - M,f
it is easily proved that
A.1)
= E [M? - M2 I .Щ ¦
As a result, M2 — T^(M) is a continuous martingale. In the sequel, we write TA
instead of TtA(M).
We now fix a > 0 and we are going to prove that if {A,,} is a sequence of
subdivisions of [0. a] such that |Д,| goes to zero, then {Taa"} converges in L2.
If Л and A' are two subdivisions we call AA' the subdivision obtained by
taking all the points of A and A'. By eq. A.1) the process X = T& — T& is a
martingale and, by eq. A.1) again, applied to X instead of M, we have
E [Xfi = E [(Tf - Tffj = E [TfA(X)] .
Because (x + yJ < 2(.v2 + y2) for any pair (.v, y) of real numbers,
and to prove our claim, it is enough to show that E [Тайй (Г4)] converges to 0
as \A\ + \A'\ goes to zero.
Let then s* be in AA' and tt be the rightmost point of A such that tj < sk <
sk+\ S ti+\\ we have
T.t-K = (мч^ - m,,J - (мч - M,f
and consequently,
By Schwarz's inequality,
sup
< E Lp \
4 - 2M,,
MSi - 2M,,
Whenever \A\ + \A'\ tends to zero, the first factor goes to zero because M is
continuous; it is therefore enough to prove that the second factor is bounded by a
constant independent of A and A'. To this end, we write with a = ?„,

A=l
Because of eq. A.1), we have E [T,f - T,f \ .\ ] = E [(Mu - Mhf \ .тЛ and
consequently
E[(TAf] =
< E | f 2 sup \Ma - Mh |2 + sup \Mh - Mh
Let С be a constant such that \M\ < C; by eq. A.1), it is easily seen that
E[TaA] < AC2 and therefore
E[(TfJ]<\2C2E[TaA]<4%C4.
We have thus proved that for any sequence \An) such that |Д,| ->• 0, the
sequence {Tfn} has a limit (M. M)a in L2 hence in probability. It remains to
prove that (M, M)a may be chosen within its equivalence class in such a way that
the resulting process (M, M) has the required properties.
Let {A,,} be as above; by Doob's inequality applied to the martingale TA" —
Since, from a sequence converging in L2, one can extract a subsequence con-
converging as., there is a subsequence [Л„к} such that T, "' converges a.s. uniformly
on [0, a] to a limit {/Vf, /Vf), which perforce is a.s. continuous. Moreover, the orig-
original sequence might have been chosen such that An+\ be a refinement of An and
[Jn А„ be dense in [0, a]. For any pair (s, t) in (J/( A/ such that 5 < ?, there is
an «o such that s and I belong to А„ for any « > «q. We then have TSA" < TtA"
and as a result {/Vf, /Vf) is increasing on [Jn A,,\ as it is continuous, it is increasing
everywhere (although the Тй" are not necessarily increasing!).
Finally, that M2 — (M. M) is a martingale follows upon passing to the limit
in eq. A.1). The proof is thus complete.
To enlarge the scope of the above result we will need the
A.4) Proposition. For every stopping time T,
(MT,MT) = (M,M)T.

Proof. By the optional stopping theorem, (MTJ - (M, M)T is a martingale, so
that the result is a consequence of the uniqueness in Theorem A.3).
Much as it is interesting, Theorem A.3) is not sufficient for our purposes; it
does not cover, for instance, the case of the Brownian motion В which is not a
bounded martingale. Nonetheless, we have seen that X has a "quadratic variation",
namely t, and that Bj - t is a martingale exactly as in Theorem A.3). We now
show how to subsume the case of BM and the case of bounded martingales in a
single result by using the fecund idea of localization.
A.5) Definition. An adapted, right-continuous process X is an (.7^, P)-local mar-
martingale if there exist stopping times Т„, n > 1, such that
i) the sequence [Tn] is increasing and limn Tn = +oo a.s.;
ii) for every n, the process XT" 1 [7;>o] is a uniformly integrable (.^", P)-martingale.
We will drop (.7~t, P) when there is no risk of ambiguity. In condition ii) we can
drop the uniform integrability and ask only that Хт" 1[г„>0] be a martingale; indeed,
one can always replace Т„ by Tn A n to obtain a u.i. martingale. Likewise, if X is
continuous as will nearly always be in this book, by setting Sn = inf{/ : \X,\ = n]
and replacing Tn by Tn л Sn, we may assume the martingales in ii) to be bounded.
This will be used extensively in the sequel. In Sect. 3 we Will find a host of
examples of continuous local martingales.
We further say that the stopping time T reduces X if Хг1[Г>о] is a u.i. martin-
martingale. This property can be decomposed in two parts if one introduces the process
Y, = X,- Xo: T reduces X if and only if
i) Xo is integrable on {T > 0}; ii) YT is a u.i. martingale.
A common situation however is that in which Xo is constant and in that case
one does not have to bother with i). This explains why in the sequel we will often
drop the qualifying l[r>n]. As an exercise, the reader will show the following
simple properties (see also Exercise A.30)):
i) if T reduces X and S < T, then 5 reduces X;
ii) the sum of two local martingales is a local martingale;
iii) if Z is a .^-measurable r-v. and X is a local martingale then, so is ZX; in
particular, the set of local martingales is a vector space,
iv) a stopped local martingale is a local martingale;
v) a positive local martingale is a supermartingale.
Brownian motion or, more generally, any right-continuous martingale is a
local martingale as is seen by taking Tn = n, but we stress the fact that local
martingales are much more general than martingales and warn the reader against
the common mistaken belief that local martingales need only be integrable in order
to be martingales. As will be shown in Exercise B.13) of Chap. V, there exist
local martingales possessing strong integrability properties which nonetheless, are
not martingales. However, let us set the

A.6) Definition. A real valued adapted process X is said of class (D) if the fam-
family of random variables Хт\(т<ъ) where T ranges through all stopping times is
uniformly integrable. It is of class DL if for every a > 0, the family of random
variables X;-, where T ranges through all stopping times less than a, is uniformly
integrable.
A uniformly integrable martingale is of class (D). Indeed, by Sect. 3 in
Chap. 11, we then have XT\^T<^ = E [X^ | .Я^] \[т<ъ] and it is known that if
Y is integrable on (Q.. /,. P), the family of conditional expectations E[Y \ ./?]
where ./? ranges through the sub-<7 -fields of. /, is uniformly integrable. But other
processes may well be uniformly integrable without being of class (D). For local
martingales, we have the
A.7) Proposition. A local martingale is a martingale if and only if it is of class
(DL).
Proof. Left to the reader as an exercise. See also Exercise A.46).
We now state the result for which the notion of local martingale was introduced.
A.8) Theorem. If M is a continuous local martingale, there exists a unique in-
increasing continuous process (A/, M), vanishing at zero, such that M~ — (M, M) is
a continuous local martingale. Moreover, for every t and for any sequence {AH\
of subdivisions o/'[0, /] such that \A,,\ —>• 0, the r.v. 's
- (М,М)А
converge to zero in probability.
Proof. Let {T,,} be a sequence of stopping times increasing to +00 and such that
Xn = Мт" 1[г„>о] is a bounded martingale. By Theorem A.3), there is, for each
w, a continuous process An in . /,+ vanishing at zero and such that X* - A,,
is a martingale. Now, (xfl+] — A,, + \) " 1[г„>о] is a martingale and is equal to
XJ, ~ An"+\ l[7;>0]- By the uniqueness property in Theorem A.3), we have An'+X =
An on [Т„ > 0] and we may therefore define unambiguously a process (M, M)
by setting it equal to An on [Т„ > 0]. Obviously, (Мт"J\[Т„>о] - (M. M)T" is
a martingale and therefore (M, M) is the sought-after process. The uniqueness
follows from the uniqueness on each interval [0. Tn].
To prove the second statement, let S.s > 0 and t be fixed. One can find a
stopping time 5 such that Ms\[S>0] is bounded and P[S < t] < 8. Since TA(M)
and (M. M) coincide with TA(MS) and (Ms. Ms) on [0. S], we have
- (M.M)S\ > e)<S+ p\sup\TA(Ms)~ (MS.MS),\ > el
and the last term goes to zero as \A\ tends to zero. D
Theorem A.8) may still be further extended by polarization.

A.9) Theorem. If M and N are two continuous local martingales, there exists
a unique continuous process (M. N) in . -/,, vanishing at zero and such that
MN — (M. N) is a local martingale. Moreover, for any t and any sequence {Л„)
of subdivisions o/[0, t] such that \A,,\ —>¦ 0,
P-1imsup ff" - (A/, /VM =0,
where ff- = Е,,ел„ {Mfi+] - Ml) (*,*+| - tf,*).
Proof. The uniqueness follows again from Proposition A.2) after suitable stop-
stoppings. Moreover the process
(M. N) = -[(M + N.M + N)-(M-N.M- /V>]
4
is easily seen to have the desired properties.
A.10) Definition. The process {M, N) is called the bracket of M and N, the pro-
process {M, M) the increasing process associated with M or simply the increasing
process of M.
In the following sections, we will give general examples of computation of
brackets; the reader can already look at Exercises A.36) and A.44) in this section.
In particular, if M and N are independent, the product MN is a local martingale
hence (A/, N) = 0 (see Exercise A.27)).
A.11) Proposition. IfT is a stopping time,
{MT. NT) = (A/, NT) = (A/. ,/V)r.
Proof. This is an obvious consequence of the last part of Theorem A.9). As an
exercise, the reader may also observe that M1NT — (A/, N)T and MT(N - NT)
are local martingales, hence by difference, so is MTN — (M. N)T. D
The properties of the bracket operation are reminiscent of those of a scalar
product. The map (A/. /V) —> (A/. N) is bilinear, symmetric and (M, M) > 0; it
is also non-degenerate as is shown by the following
A.12) Proposition. (A/, M) = 0 if and only if M is constant, that is M, = A/o
a.s. for every t.
Proof. By Proposition A.11), it is enough to consider the case of a bounded A/
and then by Theorem A.3), E[(M, - Mof] = E[(M, M),]; the result follows
immediately.
This property may be extended in the following way.
A.13) Proposition. The intervals of constancy are the same for M and for (M, M),
that is to say, for almost all w's, M,(a>) = Ми(ш) for a < t < b if and only if
(A/. M)h(co) = (М.М)и(ш).

Proof. We first observe that if M is constant on [a, b], its quadratic variation is
obviously constant on [a, b]. Conversely, for a rational number q, the process
N, = M,+q — Mq is a (.^+9)-local martingale with increasing process (N, N), =
(Л/, M)t+4 — (M, ML. The random variable
T4 = inf {s > 0 : (/V, N)s > 0}
is a (.^"+(/)-stopping time, and for the stopped local martingale N1'1, we have
(NT\ NT") = {N, N)T* = (A/. M)q+Tii - (M. ML = 0.
By Proposition A.12), M is a.s. constant on the interval [q,q + Tq], hence is
a.s. constant on all the intervals [q, q + T4] where q runs through Q+. Since any
interval of constancy of (A/. M) is the closure of a countable union of intervals
[q. q + Тц], the proof is complete.
The following inequality will be very useful in defining stochastic integrals. It
shows in particular that d(M, N) is absolutely continuous with respect to d{M, M).
A.14) Definition. A real-valued process H is said to be measurable if the map
(со, t) ->¦ H,(w) is .Kk <S) ./?(R+)-measurable.
The class of measurable processes is obviously larger than the class of pro-
progressively measurable processes.
A.15) Proposition. For any two continuous local martingales M and N and mea-
measurable processes H and K, the inequality
f
\Ht\\Ks\\d{M. N)U < (j H*d(M, А/)Л (j K]d(N. N)\
f
holds as. for t < oo.
Proof. By taking increasing limits, it is enough to prove the inequality for t < oo
and for bounded H and K. Moreover, it is enough to prove the inequality where
the left-hand side has been replaced by
HsK,d(M.N)s
indeed, if Js is a density of d(M, N)s/ \d(M, N)\, with values in {-1. 1} and we
replace H by HJ sgn (HK) in this expression, we get the left-hand side of the
statement.
By a density argument, it is enough to prove that for those A"s which may be
written
К = Ко 1(oi + К] 1]о.,,] + ... + К„ 1 ],„_,.,„]
for a finite subdivision {t0 — 0 < t\ <...<?„=?} of [0, /] and bounded mea-
measurable r.v. A",'s. By another density argument, we can also take H of the same
form and with the same subdivision.

a.s.
If we now define (M, N)[ = (M, N), - (M, N)s, we have
\(M, N)'s\ < ((M, M)[)W2 {(N, N)'s)]/2
Indeed, almost surely, the quantity
N)'s + r2(N,N)[ = (M +rN,M + rN)'s
is non-negative for every r e Q, hence by continuity for every r e K, and our
claim follows from the usual quadratic form reasoning.
As a result
/'
Jo
.1/2
and using the Cauchy-Schwarz inequality for the summation over /, this is still
less than
1/2 / ч 1/2
ч 1/2
(fd(N.N)A
/ wo /
which completes the proof.
A.16) Corollary (Kunita-Watanabe inequality). For every p > 1 and p~x +
\Hs\\K,\\d(M,N)\s
, M)s
1/2
'^d(N,
1/2
Proof. Straightforward application of Holder's inequality.
We now introduce a fundamental class of processes of finite quadratic variation.
A.17) Definition. A continuous {.7i. P)-semimartingale is a continuous process
X which can be written X = M + A where M is a continuous {¦/], P)-local
martingale and A a continuous adapted process of finite variation.
As usual, we will often drop (.^", P) and we will use the abbreviation cont.
semi. mart. The decomposition into a local martingale and a finite variation process
is unique as follows readily from Proposition A.2); however, if a process X
is a continuous semimartingale in two different filtrations (.^") and (•'?,), the
decompositions may be different even if -К С •'?, for each t (see Exercise C.18)).

More generally one can define semimartingales as the sums of local martin-
martingales and finite variation processes. It can be proved, but this is outside the scope
of this book, that a semimartingale which is a continuous process is a continuous
semimartingale in the sense of Definition A.17), namely that there is a decom-
decomposition, necessarily unique, into the sum of a continuous local martingale and a
continuous finite variation process.
We shall see many reasons for the introduction of this class of processes, but
we may already observe that their definition recalls the decomposition of many
physical systems into a signal (the f.v. process) and a noise (the local martingale).
A.18) Proposition. A continuous semimartingale X — M + A has a finite quad-
quadratic variation and (X, X) = (M, M).
Proof. If Л is a subdivision of [0, /],
(sup\Mu+l-M,,\\
< (sup\Mu+l-M,,\\ Varr(A)
where Var,(A) is the variation of A on [0, t], and this converges to zero when \A\
tends to zero because of the continuity of M. Likewise
A.19) Fundamental remark. Since the process (X, X) is the limit in probability
of the sums TA"(X), it does not change if we replace (.%) by another filtration for
which X is still a semimartingale and likewise if we change P for a probability
measure Q such that Q <K P and X is still a Q-semimartingale (see Sect. 1 Chap.
VIII).
A.20) Definition. IfX = M + A and Y — N + В are two continuous semimartin-
semimartingales, we define the bracket of X and Y by
(X, Y) = (M, N) = - [(X .
-{X -Y.X-Y)].
Obviously, (X, Y), is the limit in probability of ?, (X,Hl - Xt) (Y,i+I - Yti), and
more generally, if H is left-continuous and adapted,
P- lim sup
f H«
the proof of which is left to the reader as an exercise (see also Exercise A.33)).
Finally, between the class of local martingales and that of bounded martingales,
there are several interesting classes of processes among which the following ones
will be particularly important in the next section. We will indulge in the usual
confusion between processes and classes of indistinguishable processes in order
to get norms and not merely semi-norms in the discussion below.

A.21) Definition. We denote by H2 the space of L1-bounded martingales, i.e. the
space of (-У,, P)-martingales M such that
sup E[M2] < +00.
We denote by H2 the subset of L2-bounded continuous martingales, and H^ the
subset of elements of H2 vanishing at zero.
An (.>^)-Brownian motion is not in H2, but it is when suitably stopped, for
instance at a constant time. Bounded martingales are in H2. Moreover, by Doob's
inequality (Sect. 1 in Chap. II), M?_ = sup, \M,\ is in I? if M e H2; hence
M is u.i. and M, = E \_MX | .^"] with M^ e L2. This sets up a one to one
correspondence between H2 and L2{Q..WX, P), and we have the
A.22) Proposition. The space Ш2 is a Hilbert space for the norm
and the set H2 is closed in H2.
Proof The first statement is obvious; to prove the second, we consider a sequence
{M"\ in H2 converging to M in H2. By Doob's inequality,
as a result, one can extract a subsequence for which sup, \м"к - М,\ converges
to zero a.s. which proves that M e H2. D
The mapping M —> ||М?.||2 = E (sup, \M,|) is also a norm on H2; it is
equivalent to || \\щ? since obviously ЦЛ/Ц^г < ||Л?^||2 and by Doob's inequality
ll^^lb < 2||М||я^, but it is no longer a Hilbert space norm.
We now study the quadratic variation of the elements of H2.
A.23) Proposition. A continuous local martingale M is in H2 if and only if the
following two conditions hold
i) Mo e I2;
ii) {M, M) is integrable i.e. E [(M. M)^] < 00.
In that case, M2 — (M, M) is uniformly integrable and for any pair S < T of
stopping times
E [M2T - M2 \Щ = Е [(MT - MsJ \.Ps] = E [(M. M)TS \ .7S].
Proof. Let {7",,} be a sequence of stopping times increasing to +00 and such that
Мт" 1[г„>0] is bounded; we have
\[т„>о]] = E[M2\[Tii>0]].

If M is in H2 then obviously i) holds and, since M^ e L2, we may also pass
to the limit in the above equality to get
E[Ml] - E[(M, M)oo] = E[M2]
which proves that ii) holds.
If, conversely, i) and ii) hold, the same equality yields
? [<л,'[т,>0]] < E [(M, M)X] + E [M2] = К < со
and by Fatou's lemma
which proves that the family of r.v.'s M, is bounded in L2. Furthermore, the
same inequality shows that the set of r.v.'s МТпЫ 1[7-„>о| is bounded in L2, hence
uniformly integrable, which allows to pass to the limit in the equality
to get E [M, I ,?Z\ = Ms. The process M is a L2-bounded martingale.
To prove that M2 — {M, M) is u.i., we observe that
sup \M2 — {M, M),\ < (M^J + (M, Л/)оо
which is an integrable r.v. The last equalities derive immediately from the optional
stopping theorem.
A.24) Corollary. IfMe H2,
\\M\\w = \\(M'M)lJ<}\\2 = E[(M, M)^12 .
Proof If Mo — 0, we have E [M2^] = E [(M, M)^] as is seen in the last proof.
Remark. The more general comparison between the Lp-norms of M^ and
(M, M}^2 will be taken up in Sect. 4.
We could have worked in exactly the same way on [0. t] instead of [0, oo] to
get the
A.25) Corollary. If M is a continuous local martingale, the following two condi-
conditions are equivalent
i) Moe L2 and E[(M,M},]< oo;
ii) {Ms,s < t) is an L2-bounded martingale.
Remark. It is not true (see Exercise B.13) Chap. V) that L2-bounded local mar-
martingales are always martingales. Likewise, E[(M, M}ж] may be infinite for an
L2-bounded cont. loc. mart.

We notice that for M e H2, simultaneously (M, М)ж is in L1 and lim,_oo M,
exists a.s. This is generalized in the following
A.26) Proposition. A continuous local martingale M converges a.s. as t goes to
infinity, on the set {{A/, M)x < oo}.
Proof. Without loss of generality, we may assume Mo = 0. Then, if Tn =
\nf{t : (M, M), > n), the local martingale MT" is bounded in L2 as follows from
Proposition A.23). As a result, Ит,^^ M," exists a.s. But on ({M, M}^ < oo}
the stopping times Т„ are a.s. infinite from some n on, which completes the proof.
Remark. The converse statement that (M, M)x < oo on the set where M, con-
converges a.s. will be shown in Chap. V Sect. 1. The reader may also look at Exercise
A.42) in this section.
#L A.27) Exercise. 1 °) If M and N are two independent continuous local martingales
(i.e. the a -fields a(Ms,s > 0) and cr(Ns,s > 0) are independent), show that
{M, N) = 0. In particular, if В = (в\ ..., Bd) is a BMd, prove that <S\ BJ), =
Slit. This can also be proved by observing that (S' + Bj)/y/l and (В' - В])/-/2
are linear BM's and applying the polarization formula.
2°) If В is a linear BM and T a stopping time, by considering BT and В — BT
prove that the converse to the result in 1°) is false.
[Hint: T is measurable with respect to the cr-fields generated by both BT
(observe that (Sr, BT), = t л Т) and В — BT which thus are not independent if
T is not constant a.s.].
If (X, Y) is a BM2 and T a stopping time, XT and YT provide another example.
# A.28) Exercise. If X is a continuous semimartingale, and T a stopping time, then
X, = XT+, is a (.^+r)-semimartingale. Compute (X, X) in terms of (X, X).
A.29) Exercise. If X = (X1 Xd) is a vector continuous local martingale,
there is a unique process A e . 6 such that lA"!2 — A is a continuous local martin-
martingale.
# A.30) Exercise. 1°) If S and T are two stopping times which reduce M, then
S v T reduces M.
2°) If (Т„) is a sequence of stopping times increasing a.s. to +oo and if MT" is
a continuous local martingale for every n, then M is a continuous local martingale.
This result may be stated by saying that a process which is locally a cont. loc.
mart, is a cont. loc. mart.
3°) If M is a (J^)-cont. loc. mart., the stopping times 5* = inf{r :\M,\ > k]
reduce M. Use them to prove that M is also a cont. loc. mart, with respect to
{¦VIм) and even a(Ms. s < t). The same result for semimarts. (Strieker's theorem)
is more difficult to prove and much more so in the discontinuous case.
4C) In the setting of Exercise B.17) Chap. II, prove that if X is a (.j^>cont.
semimart., then one can find (.>^0)-adapted processes M', A', B' such that
i) M' is a (.j^"°)-cont. loc. mart.;

ii) A' and B' are continuous, A' is of finite variation, B' is increasing;
iii) X is indistinguishable from X' = M' + A';
iv) B' is indistinguishable from {X, X) and M'2 — B' is a (.^"°)-cont. loc. mart..
We will write (A", A") for В'.
5°) Prove that (A", A") is the quadratic variation of A" and that the fundamental
Remark A.19) is still valid. Extend these remarks to the bracket of two cont.
semimarts. X and Y.
A.31) Exercise. If Z is a bounded r.v. and A is a bounded continuous increasing
process vanishing at 0, prove that
A.32) Exercise. (Another proof of Proposition A.2)). If M is a bounded con-
continuous martingale of finite variation, prove that
M2 — M\ + 2 / MsdMs (the integral has then a path by path meaning)
Jo
is a martingale. Using the strict convexity of the square function, give another
proof of Proposition A.2).
* A.33) Exercise. All the processes considered below are bounded. For a subdivi-
subdivision Л — (и) of [0, t] and к 6 [0, 1], we set tf = tt + к (r, + t - ?,).
1°) If A* = M + A, Y — N + В are two continuous semimartingales and H a
continuous adapted process, set
Prove that
lim sup?[(A^J] =0.
2°) If A. = I or if d(X, Y) is absolutely continuous with respect to the Lebesgue
measure, then
lim Y H>, (*> - x< Wf - Y',) = *¦ f H<d(x- r>'
\л^° i ' ' Jo
in the L2-sense. In the second case, the convergence is uniform in X.
3°) If F is а С'-function on R, prove that
P- lim ?(F(A-,i+1)-F(A",,)J= f F'(XsJd(X.X),.
A.34) Exercise. If В is the standard linear BM and 5 and T are two integrable
stopping times with 5 < 7", show that
E [(BT - BsJ] = E\B\- B2s] = E[7" - S].

A.35) Exercise (Gaussian martingales). If Л/ is a continuous martingale and a
Gaussian process, prove that (M, M) is deterministic i.e. there is a function / on
R+ such that (M, M), = f{t) a.s. The converse will be proved in Sect. 1 Chap. V.
A.36) Exercise. 1°) If M is a continuous local martingale, prove that M2 is of
finite quadratic variation and that (M2, M2), = 4/0' M?d{My M)s.
[Hint: Use 2°) in Exercise A.33). In the following sections, we will see the
profound reasons for which M2 is of finite quadratic variation as well as a simple
way of computing (A/2, M2), but this can already be done here by brute force.]
2°) Let R',i = 1,2, ...,r be the squares of the moduli of r independent
d,-dimensional BM's and A., be r distinct, non zero real numbers. We set
?д;, a = 1,2
щ Prove that each Xlk) is of finite quadratic variation and that
(Xu\X{k)),=4 f X?
Jo
Prove that each R' is adapted to (.^X"
A.37) Exercise. Let W = C(R+,M), X be the coordinate process and .V* =
o{Xs.s < t). Prove that the set of probability measures on (W, •>r°) for which
X is a (./^°)-local martingale, is a convex set. Is this still true with the space D
of cadlag functions in lieu of W?
A.38) Exercise. If Х,(ш) = f{t) for every со, then X is a continuous semi-
martingale if and only if / is continuous and of bounded variation on every
interval.
[Hint: Write /(/) = /@) + M, + A, for the decomposition of the semimart.
/. Pick a > 0. Then choose с > 0 sufficiently large so that the stopping time
f
:|M,|+ / \dA\s >c
Jo
satisfies P[T > a] > 0 and |/| is bounded by с on [0, a]. For a finite subdivision
A = (tj) of [0, a], by comparing SA{f) with the sum
prove that SA(f) is less than 3c/ P[T > a]. Another proof is hinted at in Exercise
B.21) and yet another can be based on Strieker's theorem (see the Notes and
Comments).]
*# A.39) Exercise. Let (п.-Т,, Р), t e [0. 1] be a filtered probability space. For a
right-continuous adapted process X and a subdivision A of [0, 1], we set

°) If A' is a refinement of A prove that V&>{X) > V&(X). If
+00,
we say that X is a quasimartingale. If M € H2 prove that Л/2 is a quasimartingale.
2°) If Б is the linear BM, define (#,) as (.3^v cr(S1))+ where .j^ is the
Brownian filtration of Chap. III. Prove that for 0 < s < t < 1,
Prove that B,, t e [0, 1] is no longer a (-f?,)-martingale but that it is a quasimartin-
quasimartingale and compute V{B).
This exercise continues in Exercise C.18).
** A.40) Exercise. (Continuation of Exercise C.16) of Chap. II on BMO). 1°) If
M e BMO, prove that || Л/||Вмо2 is the smallest constant С such that
E[(M, M)OO-{M, M)T \Щ<Сг
for every stopping time T.
2°) Prove that, for every T and n
E [((M, M)x - (M, M)T)" I .Щ] < n\ }\M\\%M02.
[Hint: One can use Exercise A.13) Chap. V.]
3°) If II Af IIbmo, < Uthen
E [exp ((Af, Af >«, - (M, М)т) \.Щ]<{\- II Af |||M02)~' .
* A.41) Exercise (A weak definition of BMO). 1°) Let A be a continuous increas-
increasing process for which there exist constants a,b,ayfi such that for any stopping
time T
Р[Аов-Ат>а\.Щ]<а, P [Ax - AT > b \ .Щ < p. •
Prove that
P[Ax-AT>a + b \,?r\ <afi.
[Hint: Use the stopping time U = infjf : A, - AT > b] and prove that the
event {Ax — A7- > b) is in -Щ.]
2°) A continuous local martingale is in BMO if and only if one can find two
constants a and s > 0 such that
P [(Л/, M)x - (M, M)r > a I -VT\ < 1 - e
for every stopping time T. When this is the case, prove that for a sufficiently
small
E [exp (a ((Af, A/)» - (M, M)T)) I .Щ\
is bounded.

A.42) Exercise. Let (Q, .? , P) be a probability space and (.3^) a complete and
right-continuous filtration. A local martingale on ]0, oo[ is an adapted process M,,
t e ]0, oo[ such that for every s > 0, Me+l, t > 0, is a continuous local martingale
with respect to (.5^+,). The set of these processes will be denoted by .Ж.
1°) If A e .7q and M e .//6, then let 1АЛ/ е .'/6. If 7 is a (.J^")-stopping
time and M e ./№>, then M7 1[г>о] е .-*&.
2°) For any Л/ e .^5?, there is a unique random measure (M)(co, •) on ]0. oo[
such that for every e > 0
M?+e - (M) (• ,]e,t+ ?])
is a (.^"+?)-continuous local martingale.
3°) For any M e .--/&, prove that the two sets A — {a>: lim,^ M,(a>) exists in
Щ and В = {w : {M)(w, ]0, I]) < oo} are a.s. equal and furthermore, that \AM
is a continuous local martingale in the usual sense.
[Hint: Using Iе) , reduce to the case where M is bounded, then use the con-
continuous time version of Theorem B.3) of Chap. II.]
This exercise continues in Exercise C.26).
A.43) Exercise. (Continuation of Exercise C.15) Chap. II). Assume that / e
L2([0, I]) and set
[X, X],(co) = (Hf(co) -
Prove that X] - [X, X], is a uniformly integrable martingale.
[Hint: Compute H((HfJ -2f(Hf)).]
The reader will notice that albeit we deal with a discontinuous martingale X,
the process [X, X] is the quadratic variation process of X and plays a role similar
to that of (X, X) in the continuous case.
A.44) Exercise. Let A" be a positive r.v. independent of a linear Brownian motion
B. Let M, = B,x, t > 0, and (¦7^M) be the smallest right-continuous and complete
filtration with respect to which M is adapted.
P) Prove that M is a {.Yt M)-local martingale and that it is a martingale if and
only if E[XX'2] < oo.
2C) Find the process (M, M).
3°) Generalize the preceding results to M, = BAi where A is an increasing
continuous process, vanishing at 0 and independent of B.
A.45) Exercise. Let M and N be two continuous local martingales vanishing
at 0 such that (M, NJ = (M, M)(N, N). If R = M{s : (M, M)s > 0}, S =
inf{s : (N, N)s > Of prove that a.s. either R v S — oo or R = S and there is a
.Fit П .immeasurable r.v. у vanishing on {R v S = oo} and such that M — yN.
A.46) Exercise. Prove that a local martingale X such that for every integer N,
the process (X')N is of class (D) is a supermartingale. In particular, a positive
local martingale is a supermartingale.

# A.47) Exercise. Let M and N be two continuous local martingales and T a finite
stopping time. Prove that
[Hint: The map (M, N) —*¦ {M, N) has the properties of a scalar product and
this follows from the corresponding "Minkowski" inequality.]
* A.48) Exercise. (Continuous local martingales on a stochastic interval). Let
Г be a stopping time. We define a continuous local martingale M on [0, T[ as
a process on [0, T[ for which there exists a sequence (Г„) of stopping times
increasing to T and a sequence of continuous martingales M" such that M, = M't'
on {/ < Tn]. In the sequel, we will always assume 7" > 0 a.s.
1°) Prove that there exists a unique continuous increasing process (M, M) on
[0, T[ such that M2 — (M, M) is a continuous local martingale on [0, T[. Prove
that M and (M, M) have a.s. the same intervals of constancy.
2°) If ? [supr<7- \M, |] < oo, then lim,^ M, = MT- exists and is finite a.s. and
if we set M, = Mr- for / > T, the process M,, t e K+ is a uniformly integrable
martingale.
3°) If Mo = 0 and E[(M, M)T] < oo, prove that M may be continued in a
continuous uniformly integrable martingale.
A.49) Exercise. (Krickeberg decomposition for loc. marts. Continuation of
Exercise B.18) Chap. II). Let A" be a loc. mart, and set
where T ranges through the family of finite stopping times.
1°) If (Т„) is a sequence of stopping times reducing X and such that
Iim7,, = ex a.s., then N\(X) = sup,, ? [|Xrj]. A loc. mart. X is bounded in
L] iff Ni(X) < oo.
** 2°) If N\(X) < oo prove that there is a unique pair (X"(+), X"(~') of positive
loc. marts, such that
i) X =
[Hint: See Exercise B.18) Chap. II and remember that a positive loc. mart, is
a supermartingale.]
3°) If N\{X) < oo, then (X,) converges a.s. as / tends to infinity, to an
integrable r.v.

§2. Stochastic Integrals
For several reasons, one of which is described at length at the end of Sect. 1 of
Chap. VII, it is necessary to define an integral with respect to the paths of BM.
The natural idea is to consider the "Riemann sums"
where К is the process to integrate and u, is a point in [?,, ti+i]. But it is known
from integration theory that these sums do not converge pathwise because the paths
of В are a.s. not of bounded variation (see Exercise B.21)). We will prove that
the convergence holds in probability, but in a first stage we use L2-convergence
and define integration with respect to the elements of H2. The class of integrands
is the object of the following
B.1) Definition. IfM e H2, we call ^2(M) the space of progressively measur-
measurable processes К such that
If, for any Г е ./7{Ж+) ® .7~x, we set
\r(s,w)d{M,M)s{w)\
we define a bounded measure Рм on .>?(R+) ® .Уж and the space %2{M) is
nothing else than the space of /^-square integrable, progressively measurable,
functions. As usual, L2(M) will denote the space of equivalence classes of ele-
elements of 'A2(M); it is of course a Hilbert space for the norm || • \\M.
Since those are the processes we are going to integrate, it is worth recalling
that they include all the bounded and left (or right)-continuous adapted processes
and, in particular, the bounded continuous adapted processes.
B.2) Theorem. Let M e H2; for each К е L2(M), there is a unique element of
H{2, denoted by К ¦ M, such that
(K-M.N) = K-{M,N)
for every N e H2. The map К —>¦ K-M is an isometry from L2(M) into H^.
Proof, a) Uniqueness. If L and L' are two martingales of H^ such that (L, N) =
{L\ N) for every ./V 6 Я2, then in particular (L - L\ L - L') = 0 which by
Proposition A.12) implies that L - L' is constant, hence L — L'.
b) Existence. Suppose first that M is in #02. By the Kunita-Watanabe inequality
(Corollary A.16)) and Corollary A.24), for every N in Щ we have
¦\ f Ksd(M.N)A

the map N -*¦ E[{K-(M, #))<»] is thus a linear and continuous form on the
Hilbert space Hq and, consequently, there is an element К ¦ M in #02 such that
B.1)
for every N e //02. Let T be a stopping time; the martingales of H2 being u.i.,
we may write
E[(K-M)TNT] =
м, n))t]
which proves, by Proposition C.5) Chap. II, that (K ¦ M)N - К -{М, N) is a
martingale. Furthermore, by eq. B.1),
\\K-M\\2W = E [(K-MJ^} = E[(K2-{M, M))x] = \\K\\2M
which proves that the map К —>¦ К ¦ M is an isometry.
ifNeH2 instead of tf02, then we still have (KM, N) = K-(M, N) because
the bracket of a martingale with a constant martingale is zero.
Finally, if M e H2 we set K'M = К-Ш - Mo) and it is easily checked that
the properties of the statement carry over to that case.
B.3) Definition. The martingale KM is called the stochastic integral of К with
respect to M and is also denoted by
/ KsdMs.
Jo
It is also called the ltd integral to distinguish it from other integrals defined in
Exercise B.18). The Ito integral is the only one among them for which the resulting
process is a martingale.
We stress the fact that the stochastic integral К ¦ M vanishes at 0. Moreover,
as a function off, the process (K • M), may also be seen as an antiderivative (cf.
Exercise B.17)).
The reasons for calling К ¦ M a stochastic integral will become clearer in
the sequel; here is one of them. We shall denote by К the space of elementary
processes that is the processes which can be written
к = лг-Ию) + 2^ JO']'..'.+11
where 0 = /0 < h < h < ..., lim t; = +oo, and the r.v.'s K, are .^-measurable
and uniformly bounded and K-\ 6 .3^. The space У/ is contained in L2(M). For
К е К, we define the so-called elementary stochastic integral К ¦ M by
n-l
{K-M), - J2 K' (M>,+> ~M>,) + K» iM> - Мь)

whenever tn < t < tn+\. It is easily seen that KM e #02; moreover, consider-
considering subdivisions Л including the ?,'s, it can be proved using the definition of the
brackets, that for any N e H2, we have (K-M, N) = K-(M, N).
As a result, the elementary stochastic integral coincides with the stochastic
integral constructed in Theorem B.2). This will be important later to prove a
property of convergence of Riemann sums which will lead to explicit computations
of stochastic integrals.
We now review some properties of the stochastic integral. The first is known
as the property of associativity.
B.4) Proposition. If К e L2(M) and H e L2{K-M) then HK e L2(M) and
(HK)M = H-(K-M).
Proof Since (K-M, K-M) = K2-(M. M), it is clear that HK belongs to L2(M).
For N e H2, we further have
({HK)-M,N) = HK-(M,N) = H-{K-(M,N))
because of the obvious associativity of Stieltjes integrals, and this is equal to
H-(K-M,N) = (H-{K-M),N)\
the uniqueness in Theorem B.2) ends the proof. D
The next result shows how stochastic integration behaves with respect to op-
optional stopping; this will be all important to enlarge the scope of its definition to
local martingales.
B.5) Proposition. IfT is a stopping time,
KMT = К \[0,туМ = (KM)T.
Proof. Let us first observe that MT — l[o.r]-M; indeed, for N e H2,
(MT, N) - (M, N)T = l[0.rr<M, N) = (l[0.r]-M, N).
Thus, by the preceding proposition, we have on the one hand
K-MT = К-(\[0.туМ) = К 1[0.г]-М,
and on the other hand
(K-M)T = ll0.T](KM) = \[0S]K-M
which completes the proof.
Since the Brownian motion stopped at a fixed time t is in H2, if К is a process
which satisfies
< oo, for all t,

we can define /J KsdBs for each t hence on the whole positive half-line and the
resulting process is a martingale although not an element of H2. This idea can of
course be used for all continuous local martingales.
B.6) Definition. IfM is a continuous local martingale, we call ЦЖ(М) the space
of classes of progressively measurable processes К for which there exists a se-
sequence (Tn) of stopping times increasing to infinity and such that
UT" 1
K2d(M, M)s < +00.
J
Observe that ЦЖ(М) consists of all the progressive processes К such that
K2d(M, M)s < 00 for every t.
f
Jo
/0
B.7) Proposition. For any К е Ь2Ж(М), there exists a unique continuous local
martingale vanishing at 0 denoted K-M such that for any continuous local mar-
martingale N
(K-M,N) = K-(M,N).
Proof. One can choose stopping times Tn increasing to infinity and such that MT"
is in H2 and KT" e L2(MT"). Thus, for each n, we can define the stochastic
integral X = KT«-MT". But, by Proposition B.5), X("+1) coincides with Xм
on [0, Tn\, therefore, one can define unambiguously a process KM by stipulating
that it is equal to X(n) on [0, Т„]. This process is obviously a continuous local
martingale and, by localization, it is easily seen that (K-M, N) = K(M, N) for
every local martingale N.
Remark. To prove that a continuous local martingale L is equal to K-M, it is
enough to check the equality (L. N) — K-(M, N) for all bounded Ws.
Again, К М is called the stochastic integral of К with respect to M and is
alternatively written
/ KsdMs.
Plainly, Propositions B.4) and B.5) carry over to the general case after the ob-
obvious changes. Also again if К 6 fi this stochastic integral will coincide with
the elementary stochastic integral. Stieltjes pathwise integrals having been previ-
previously mentioned, it is now easy to extend the definition of stochastic integrals to
semimartingales.
B.8) Definition. A progressively measurable process К и locally bounded if there
exists a sequence (Т„) of stopping times increasing to infinity and constants Cn such
that\KT»\ <Cn.
All continuous adapted processes К are seen to be locally bounded by tak-
taking Tn — inf{? : \K,I > n). Locally bounded processes are in ЦОС(М) for every
continuous local martingale M.

B.9) Definition. If К is locally bounded and X = M + A is a continuous semi-
semimartingale, the stochastic integral of К with respect to X is the continuous semi-
semimartingale
K-X = KM + KA
where KM is the integral of Proposition B.7) and KA is the pathwise Stieltjes
integral with respect to dA. The semimartingale K-X is also written
f K,dXs.
Jo
B.10) Proposition. The map К —» K-X enjoys the following properties:
i) H -(K-X) — (HK)-X for any pair H, К of locally bounded processes;
ii) (K-X)T = (Kl[0.T])-X = KXT for every stopping time T;
Hi) if X is a local martingale or a process of finite variation, so is K-X
4v) if К e '6, then iftn < t < tn+\
n
(K-X), = J2 Ъ (*i,+1 - Xi.) + Kn {X, - X,.).
;=o
Proof. Straightforward.
At this juncture, several important remarks are in order. Although we have used
Doob's inequality and L2-convergence in the construction of K-M for M e H2,
the stochastic integral depends on P only through its equivalence class. This is
clear from Proposition B.7) and the fundamental remark A.19). It is actually true,
and we will see a partial result in this direction in Chap. VIII, that if X is a P-
semimartingale and Q <JC P, then X is also a <2-semimartingale. Since a sequence
converging in probability for P converges also in probability for Q, the stochastic
integral for P of, say, a bounded process is g-indistinguishable from its stochastic
integral for Q.
Likewise, if we replace the filtration by another one for which X is still a
semimartingale, the stochastic integrals of processes which are progressively mea-
measurable for both filtrations are the same.
Finally, although we have constructed the stochastic integral by a global pro-
procedure, its nature is still somewhat local as is suggested by the following
B.11) Proposition. For almost every со, the function (K-X) (to) is constant on any
interval [a. b] on which either K.(a>) — 0 or X.(w) — Xa(co).
Proof. Only the case where X is a local martingale has to be proved and it is then
an immediate consequence of Proposition A.13) since K2-(X, X) hence KX are
then constant on these intervals.
As a result, for К and К locally bounded and predictable processes and X
and X semimartingales we have (K-X), - (KX)a = (K-X), — (K-X)a a.s. on
any interval [a, b] on which К = К and X. — Xa — X. - Xa; this follows from
the equality
K-X - K-X = K(X - X) + (K - K)X.

Remark. That stochastic integrals have a path by path significance is also seen in
3°) of Exercise B.18).
We now turn to a very important property of stochastic integrals, namely the
counterpart of the Lebesgue dominated convergence theorem.
B.12) Theorem. Let X be a continuous semimartingale. If{K") is a sequence of
locally bounded processes converging to zero pointwise and if there exists a locally
bounded process К such that \Kn\ < К for every n, then (K" -X) converges to
zero in probability, uniformly on every compact interval.
Proof. The convergence property which can be stated
P- lim sup|(K"-X)J =0
is clear if X is a process of finite variation. If X is a local martingale and if T
reduces X, then {K"f converges to zero in L2(XT) and by Theorem B.2), (Kn-
X)T converges to zero in H2. The desired convergence is then easily established
by the same argument as in Theorem A.8). D
The next result on "Riemann sums" is crucial in the following section.
B.13) Proposition. If К is left-continuous and (A") is a sequence of subdivisions
of[0, t] such that \A"\ -* 0, then
f KsdXs = P- lim T Kh (X,,+l - X,) .
Proof. If К is bounded, the right-hand side sums are the stochastic integrals of
the elementary processes 51 ^л']/(.'/+il which converge pointwise to К and are
bounded by ||Af Hoo! therefore, the result follows from the preceding theorem. The
general case is obtained by the use of localization. " a
B.14) Exercise. Let X be a continuous semimartingale and Ъ.Щ the algebra of
bounded .^-measurable random variables. Prove the b.^-linearity of the map
H —> H-X namely, for a and b in b.^j,
I (aH + bK)dX = a I H dX + b f К dX.
Jo Jo Jo
B.15) Exercise. Let B\ and Bj be two independent linear BM's. For i — 1, ..., 4,
define the following operations on continuous progressively measurable processes
K:
Oi(K)t= f KsdBt(s), i = 1,2.
Jo
O3(K),= I Ksds,
Jo

Let '6 be the class of processes obtained from {Sb B2) by a finite number of
operations Or, we define by induction a real-valued mapping d on К by
d(B,)=</(B2) = 1/2,
= d(K) + 1/2, d(O3(AT)) = d(K) + 1
Prove that for every К e CK, there is a constant CK, which may be zero, such that
E[Kt] = CKtd{K\
B.16) Exercise. Let / be a locally bounded Borel function on R+ and В be a
BM. Prove that the process
Z, = / f(s)dBs
Jo
is Gaussian and compute its covariance F(s, t). Prove that exp JZ, — ^F(t, r)}
is a martingale. This generalizes the example given in Proposition A.2) iii) of
Chap. II.
* B.17) Exercise. 1°) Let В be a BM and H an adapted right-continuous bounded
process. Prove that for a fixed /
/t+h
HsdBs = H, in probability.
The result is also true for H unbounded if it is continuous.
[Hint: One may apply Schwarz's inequality to
E\\— / (Hu-H0)dBu
1/4
¦]
2°) Let В = (В1 Bd) be a ^-dimensional BM and for each j, Hj be a
bounded right-continuous adapted process. Prove that for a fixed t,
/i+h
H>dB>
converges in law as h -> 0, to /// + Y?j=i H! {Nj/Nl) where (W, ..., Nd) is
a centered Gaussian r.v. with covariance Id, independent of (//,',..., Hf).
* B.18) Exercise. Let X and Y be two continuous semimartingales. For a subdivi-
subdivision Л of [0, t], a function / e C'(K) and a probability measure \i on [0, 1], we
define
,M-yl.) f f{X,,+s(Xll+t -X,,)
Jo

1°) Prove that
lira S»= [ f(Xs)dYi+p. f f'(Xs)d(X. Y),
i^l—0 Jo Jo
in probability, with Д = /o's d)i(s). For /i = 5l/2 and f(x) = x, this limit
is called the Stratonovich integral of X against Y. For \i = 80, we get the Ito
stochastic integral of /(X) against Y. Observe that this is the only integral for
which the resulting process is a local martingale when Y is a local martingale.
[Hint: Use Exercise A.33).]
For ix = S[, the limit is called the backward integral.
2C) If we set
n+, - Уч) f /(x,1+,,,ltl-,1,
and if d{X, Y) is absolutely continuous with respect to the Lebesgue measure,
then S^ has the same limit as 5^ in probability whenever |zi | —> 0.
3°) If w — Y^., fi (x\,... ,x<i)dxi is a closed differential form of class C1 on
an open set U of Mf1 and X = (X1 Xd) a vector semimartingale with values
in U (i.e. P[3t :Х,$и] = 0) then
)dX\ + \ У f' ^-(Xs
' 2 i~f J° J
fj^\ \ f ^\ X>\
X@.i) J ~t ' 2 f J
where X@, t) is the continuous path (Xs(cd), 0 < s < t). We recall that the integral
of a closed form w along a continuous but not necessarily differentiable path
Y '¦ [0, t] -*¦ Rd is defined as n(y(t)) - тг(/@)) where л- is a function such that
drc = w in a string of balls which covers /.
4°) If В is the planar BM, express as a stochastic integral the area swept by
the line segment joining 0 to the point Bs, as s varies from 0 to t.
* B.19) Exercise (Fubini-Wiener identity in law). 1°) Let В and С be two inde-
independent standard BM's. If ф e L2([0, I]2, ds dt), by using suitable filiations, give
a meaning to the integrals
/ dBu / dCs<t>(u,s) and / dCs / dBu<p(u,s)
Jo Jo Jo Jo
and prove that they are almost-surely equal.
2°) Conclude from Iе) that
f' / f^ \^ f' / /"' \^
I dul dB,<t>(u,s)\ = / dull dB^(s,u)) .
Jo \Jo ) Jo \Jo )
3°) If / is а С'-function on [0, 1] and /A) = 1, prove that
s- f f'{t)B,dt] (=' f ds(Bs-f(s)BlJ.
Jo } Jo

If, in particular, В is a Brownian bridge,
f ds(B,- f B,dt\ Ш f dsBl
Jo \ Jo / Jo
[Hint: Take </>(?, u) = 1(M<V) + f(u) - 1.]
B.20) Exercise. Prove that in Theorem B.12) the pointwise convergence of (К„)
may be replaced by suitable convergences in probability.
# B.21) Exercise. Let X be a real-valued function on [0, 1]. For a finite subdivision
Л of [0, 1] and h e W = C([0, 1], Ж), we set
1°) Prove that the map h -*¦ S^(h) is a continuous linear form on W with
norm
!,€A
2°) If Eд,(й)} converges to a finite limit for every /ie\V and any sequence
{Л„} such that \Л„\ tends to 0, prove that X is of bounded variation. This shows
why the stochastic integral with respect to a cont. loc. mart, cannot be defined in
the ordinary way.
[Hint: Apply the Banach-Steinhaus theorem.]
3°) Use the same ideas to solve Exercise A.38).
B.22) Exercise (Orthogonal martingales). Г) Two martingales M and ./V of
H2, vanishing at 0, are said to be weakly orthogonal if E[MsNt] = 0 for ev-
every s and t > 0. Prove that the following four properties are equivalent:
i) M and ,/V are weakly orthogonal,
ii) E[MSNS] = 0 for every s > 0,
iii) E[(M, N)s] — 0 for every s > 0,
iv) E[MtNs] = 0 for every s > 0 and every stopping time T > s.
2°) The two martingales are said to be orthogonal (see also Exercise E.11))
if MN is a martingale. Prove that M and ,/V are orthogonal iff E[MtNs] = 0 for
every s > 0 and every stopping time T < s.
Prove also that M and ./V are orthogonal iff HM and N are weakly orthogonal,
for every bounded predictable process H.
3°) Give examples of weakly orthogonal martingales which are not orthogonal.
[Hint: One can use stochastic integrals with respect to BM.]
4°) If the two-dimensional process (M, N) is gaussian, and if M and N are
weakly orthogonal, prove that they are orthogonal.
Further results relating orthogonality and independence for martingales may
be found in Exercise D.25) Chap. V.

§3. Itd's Formula and First Applications
This section is fundamental. It is devoted to a "change of variables" formula for
stochastic integrals which makes them easy to handle and thus leads to explicit
computations.
Another way of viewing this formula is to say that we are looking for functions
which operate on the class of continuous semimartingales, that is, functions F such
that F(Xi) is a continuous semimartingale whatever the continuous semimartingale
X is. We begin with the special case F{x) = x2.
C.1) Proposition (Integration by parts formula). If X and Y are two continu-
continuous semimartingales, then
f XsdYs + f YsdXs
Jo Jo
X,Y, = XqYq + f XsdYs + f YsdXs + (X, Y),;
J J
In particular,
f
Jq
{X,X)t.
Proof. It is enough to prove the particular case which implies the general one by
polarization. If A is a subdivision of [0, t], we have
letting \A\ tend to zero and using, on one hand the definition of (X, X), on the
other hand Proposition B.13), we get the desired result. D
If X and Y are of finite variation, this formula boils down to the ordinary
integration by parts formula for Stieltjes integrals. The same will be true for the
following change of variables formula. Let us also observe that if M is a local
martingale, we have, as a result of the above formula,
] - (M, M), = Ml + 2 /
Jo
MsdMs;
о
we already knew that M2 - (M, M) is a local martingale but the above formula
gives us an explicit expression of this local martingale. In the case of BM, we
have
B2 -t=2 ( B,dBs,
Jo
which can also be seen as giving us an explicit value for the stochastic integral
in the right member. The reader will observe the difference with the ordinary
integrals in the appearance of the term t. This is due to the quadratic varia-
variation.
All this is generalized in the following theorem. We first lay down the

C.2) Definition and notation. A d-dimensional vector local martingale (resp.
vector continuous semimartingale,) is a W-valued process X = (X',...,Xrf)
such that each X' is a local martingale (resp. cont. semimart.). A complex lo-
local martingale (resp. complex cont. semimart.) is a ^-valued process whose real
and imaginary parts are local martingales (resp. cont. semimarts.).
C.3) Theorem (Ito's formula). Let X = (X1 , Xd) be a continuous vector
semimartingale and F e C2(M.d, Щ; then, F(X) is a continuous semimartingale
and
F(Xt) = F(X0) + V f
Proof. If F is a function for which the result is true, then for any /, the result is
true for G(x\,...,Xd) = XjF(x\,... ,Xd); this is a straightforward consequence
of the integration by parts formula. The result is thus true for polynomial func-
functions. By stopping, it is enough to prove the result when X takes its values in a
compact set К of Rd. But on K, any F in С2(М^,1К) is the limit in C2(K,R)
of polynomial functions. By the ordinary and stochastic dominated convergence
theorems (Theorem B.12)), the theorem is established.
Remarks. 1°) The differentiability properties of F may be somewhat relaxed. For
instance, if some of the X"s are of finite variation, F needs only be of class C1 in
the corresponding coordinates; the proof goes through just the same. In particular,
if X is a continuous semimartingale and A e . ¦?, and ifd2F/dx2 and dF/ду exist
and are continuous, then
F(X,,A,) - F(Xo,Ao)+ f ^(Xs,AJdXs+ [
J Э Jo dy
f ^(Xs,AJdXs
Jo Эх
- l -—г{Х„А,тх,Х),.
2 Jo dx2
2°) One gets another obvious extension when F is defined only on an open
set but X takes a.s. its values in this set. We leave the details to the reader as an
exercise.
3°) Ito's formula may be written in "differential" form
dF{X,) = У ^(Xt)dXi + \Т AJL{X,)d{X', X')t.
t--1 dxt 2 y-i dXjdxj
More generally, if X is a vector semimartingale, dY, — J], H'tdX\ will mean
H'sdX\.
In this setting, Ito's formula may be read as "the chain rule for stochastic differ-
differentials".

4°) Ito's formula shows precisely that the class of semimartingales is invariant
under composition with C2-functions, which gives another reason for the introduc-
introduction of semimartingales. If M is a local martingale, or even a martingale, F(M)
is usually not a local martingale but only a semimartingale.
5°) Let ф be а С'-function with compact support in ]0, 1[. It is of finite
variation, hence may be looked upon as a semimartingale and the integration by
parts formula yields
[ фШХ, + [ Xt<f>\s
Jo Jo
i + / ф^Х, + / Х;ф (s)ds + (X, ф}]
Jo Jo
which reduces to
/ фШХ* = - I
Jo Jo
Stochastic integration thus appears as a random Schwartz distribution, namely
the derivative of the continuous function / —> Х,(ш) in the sense of distributions.
The above formula, a special case of the integration by parts formula was taken as
the definition of stochastic integrals in the earliest stage of the theory of stochastic
integration and is useful in some cases (cf. Chap. VIII, Exercise B.14)). Of course,
the modern theory is more powerful in that it deals with much more general
integrands.
To some extent, the whole sequel of this book is but an unending series of
applications of Ito's formula. That is to say that Ito's formula has revolutionized
the study of BM and other important classes of processes. We begin here with a
few remarks and a fundamental result.
In the following proposition, we introduce the class of exponential local mar-
martingales which turns out to be very important; they are used in many proofs and
play a fundamental role in Chap. VIII. For the time being, they provide us with
many new examples of local martingales.
C.4) Proposition. If f is a complex valued function,'defined опЖх R+, and such
that -^ and j- exist, are continuous and satisfy ^ + \ ^4 = 0, then for any cont.
local mart. M, the process f (А/,, (M, М),) is a local martingale. In particular for
any X e С the process
= expUA/, (M,M),
is a local martingale.
For к — 1, we write simply l6 (M) and speak of the exponential of M.
Proof. This follows at once by making A — (A/, M) in Remark 1 below Theorem
C.3).
Remarks. 1°) A converse to Proposition C.4) will be found in Exercise C.14).
2°) For BM, we already knew that exp{XB, — y/} is a martingale. Let us
further observe that, for / e ifoc(R+), the exponential

«/ = exp lj f(u)dBu - X- f f2(u)du J
is a martingale; this follows easily from the fact that /' f(u)dBu is a centered
Gaussian r.v. with variance f's f2(u)du and is independent of .'71. Likewise for
BMrf and a rf-uple / = (/,, /rf) of functions in L2X(R+), and for the same
reason,
K/ = exp
is a martingale. These martingales will be used in the following chapter.
3°) Following the same train of thought as in the preceding remark, one can ask
more generally for the circumstances under which exponentials are true martingales
(not merely local martingales). This will be studied in great detail in connection
with Girsanov's theorem in Chap. VIII. We can already observe the following
facts:
a) As already mentioned in Sect. 1, a local martingale which is > 0 is a
supermartingale; this can be seen by applying Fatou's lemma in passing to the
limit in the equalities
Е[М,лТп \.Щ = М,лг„.
To obtain a martingale, one would have to be able to use Lebesgue's theorem. This
will be the case if M is bounded, hence the exponential of a bounded martingale
is a martingale.
b) If Mo = 0then KX(M) is a martingale if and only if E[Kx(M)t] = 1. The
necessity is clear and the sufficiency comes from the fact that a supermartingale
with constant expectation is a martingale.
4°) For a cont. semimart. X, we can equally define Xk(X), — exp{AX,—
у (X, X),} and we still have
Xk(X), = Zl(XH+ I MHX)sdXs.
Jo
This can be stated as: <CX(X) is a solution to the stochastic differential equation
dY, =XY,dXt.
When Xo = 0, it is in fact the unique solution such that Yo = 1 (see Exercise
C.10)).
In the same spirit as in the above result we may also state the
C.5) Proposition. If В is a d-dimensional BM and f e C2(R+ x W1). then
Ml = f(i, B,) - jf {^-Af + ^Q (.v, Bs)ds
is a local martingale. In particular if f is harmonic in Rd then f(B) is a local
martingale.

Proof. Because of their independence, the components B' of В satisfy (B\ BJ), =
Sijt (see Exercise A.27)). Thus, our claim is a straightforward consequence of Ito's
formula. ?
The foregoing proposition will be generalized to a large class of Markov pro-
processes and will be the starting point of the fundamental method of Martingales
problems (see Chap. VII). Roughly speaking, the idea is that Markov processes
may be characterized by a set of local martingales of the above type. Actually
much less is needed in the case of BM where it is enough to consider Mf for
f(x) — x' and f(x) — x'x1. This is the content of the fundamental
C.6) Theorem (P. Levy's characterization theorem). For a {-^)-adap(ed con-
continuous d-dimensional process X vanishing at 0, the following three conditions are
equivalent:
i) X is an .J^-Brownian motion;
ii) X is a continuous local martingale and (X', X*)t = <5,;/ for every 1 < ;',
j < d;
Hi) X is a continuous local martingale and for every d-uple f = {f\, ..., fd) of
functions in L2(K+), the process
is a complex martingale.
Proof That i) implies ii) has already been seen (Exercise A.27)).
Furthermore if ii) holds, Proposition C.4) applied with X = i and M, =
/L/t /o fk(s)dXks implies that f^'f is a local martingale; since it is bounded, it is a
complex martingale.
Let us finally assume that iii) holds. Then, if / = | l[o.r] for an arbitrary f
in Rd and T > 0, the process
?,'7 = exp j/(?, Х,лг) + ^$|2(f л 7)
is a martingale. For A € .^, s < t < T, we get
Е[1Аехр[Ц1;,Х, - Xs)}] = Р(А)ац>(-Ц-« -s)\.
(Here, and below, we use the notation (x, y) for the euclidean scalar product of
x and у in Rd, and \x\2 = (x,x).)
Since this is true for any % e Rd, the increment X, - Xs is independent of.%
and has a Gaussian distribution with variance (t — s); hence i) holds. ?
C.7) Corollary. The linear BM is the only continuous local martingale with t as
increasing process.

Proof. Stated for d = 1, the above result says that X is a linear BM if and only
if X, and X2 — t are continuous local martingales.
Remark. The word continuous is essential; for example if N is the Poisson process
with parameter с = \, N, —t and (N, — tJ — t are also martingales. (See Exercise
A.14) Chap. II).
The same principle as in Proposition C.4) allows in fact to associate many other
martingales with a given local martingale M. We begin with a few prerequisites.
The Hermite polynomials hn are defined by the identity
~~ hn(x) = exp | их >, иде!,
whence it is deduced that
z exnfr2/2U-n'
dx
hn(x) = exP(x2/2)(-l)"— (exP(-x2/2)).
For a > 0, we also have
if we set H,,(x, a) = an/2h,,(x/^/a); we also set Н„(х, 0) = x".
C.8) Proposition. If M is a local martingale and Л/о = О, the process
L™ = Hn{Mt,(M,M)t)
is, for every n, a local martingale and moreover
dMSl / dMSl... / dMSn.
Jo Jo
Proof. It is easily checked that (\д2/дх2 + д/да) Н„(х. а) = 0 and that
дН„/дх — пН„-\; thus lto's formula implies that
L)n)=n / L\n-{)dMs,
Jo
which entails that L{n) is a loc. mart, and its representation as a multiple stochastic
integral is obtained by induction. ?
Remark. The reader is invited to compute explicitly L{n) for small n. For n =
0, 1, 2, one finds the constant 1, M and M2 — (M, M), but from n = 3 on, new
examples of local martingales are found.

C.9) Exercise. Prove the following extension of the integration by parts formula.
If f(t) is a right-continuous function of bounded variation on any compact interval
and X is a continuous semimartingale
f(t)X, = /@)Xo + / f(s)dXs + f Xsdf(s).
Jo Jo
# C.10) Exercise. 1°) If X is a semimartingale and Xo = 0, prove that %X(X), is
the unique solution of dZ, = XZ,dX, such that Zo — 1.
[Hint: If Y is another solution, compute Y rCl(X)~l using the integration by
parts formula and Remark 2 below Ito's formula.]
2°) Let X and Y be two-continuous semimartingales. Compute % (X + Y)
and compare it to K(X)K(Y). When does the equality occur? This exercise is
generalized in Exercise B.9) of Chap. IX.
C.11) Exercise. 1°) If X — M + A and Y = N + В are two cont. semimarts.,
prove that XY - {X, Y) is a cont. loc. mart, iff X В + Y A — 0. In particular,
X2 - (X, X) is a cont. loc. mart, iff P-a.s. X, = 0 c/A,-a.e.
2°) If the last condition in 1°) is satisfied, prove that for every C2-function /,
f(X,) - /(Xo) - f'@)A, - A/2) / f"(Xs)d(X, X),
J
Jo
is a cont. loc. mart.
The class E of semimartingales X which satisfy the last condition in P) is
considered again in Definition 4.4, Chap. VI.
# C.12) Exercise. (Another proof and an extension of Ito's formula). Let X be
a continuous semimart. and g : К x (Q x K+) -> К a function such that
i) {x, u) —> g(x, со, и) is continuous for every со;
ii) x —> g(x, со, u) is C2 for every (со, и);
iii) (со, и) —> g(x, со, и) is adapted for every x.
1°) Prove that, in the notation of Proposition B.13),
[ dg/dx(Xu,u)dXu+(\/2) [ d2g/dx2(Xu,u)d(X,X)u.
Jo Jo
2°) Prove that if in addition g@, со, и) = 0, then
Г;€Д,
/ dg/dx@,u)dXu + (l/2) f d2g/dx2@.u)d{X,X)u.
Jo Jo
3°) Resume the situation of 1°) and assume moreover that g satisfies

iv) for every (x, со), the map и —>¦ g(x, со, и) is of class C1 and the derivative is
continuous in the variable x,
then prove the following extension of Ito's formula:
g(Xt, t) = g(X0, 0) + f (dg/dx)(Xu, u)dXu + f (dg/du)(Xu. u)du
Jo Jo
+ A/2) / (d2g/dx2)(Xu,u)d{X,X)u.
Jo
4°) Extend these results to a vector-valued cont. semimart. X.
C.13) Exercise (Yet another proof of Ito's formula). 1 ) Let x be a continuous
function which is of finite quadratic variation on [0. /] in the following sense:
there exists a sequence (A,,) of subdivisions of [0, t] such that \Л„ | —> 0 and the
» measures
J2 (x,l+l-x,,) «'¦
converge vaguely to a bounded measure whose distribution function denoted by
(x, x) is continuous. Prove that for a C2-function F,
F(x,) = F(xo)+ ( F'(xs)dx5 + - ( F"(xs)d{x,x)s
Jo *¦ Jo
where /0' F\xs)dx, = lim,,^^ Х),.б4л F' (x,t) (x,^, -xti).
[Hint: Write Taylor's formula up to order two with a remainder r such that
r(a, b) < ф(\а - b\)(b - aJ with ф increasing and linv-o ф(с) — 0.]
2°) Apply the result in 1°) to prove Ito's formula for continuous semimartin-
gales.
# C.14) Exercise. If M is an adapted continuous process, A is an adapted continu-
continuous process of finite variation and if, for every X, the process exp JAM, — уЛ, J
is a local martingale, then M is a local martingale and (A/, M) — A.
[Hint: Take derivatives with respect to X at X = 0.]
C.15) Exercise. If X and Y are two continuous semimartingales, denote by
/ XsodYs
o
the Stratonovich integral defined in Exercise B.18). Prove that if F e C3(IRrf, R)
and X — (X1, Xd) is a vector semimartingale, then
/' r) F
— {Xs)odX\.
dXj
# C.16) Exercise. (Exponential inequality, also called Bernstein's inequality). If
M is a continuous local martingale vanishing at 0, prove that
P [M^ > x, (M, M)^ < y] < exp(-x2/2y).

Derive therefrom that, if there is a constant с such that (M, M), < ct for all t,
then
Г 
exp(-a2t/2c).
1 sup Ms > at\
[Hint: Use the maximal inequality for positive supermartingales to carry
through the same proof as for Proposition A.8) in Chap. П.]
* C.17) Exercise. Let ц be a positive measure on M+ such that the function
Г°° ( у2 \
/„(*, 0 = / exp yx - ~t dn(\)
Jo \ l )
is not everywhere infinite. For s > 0, define
>e\
and suppose that this is a continuous function of /.
1°) Prove that for any stopping time T of the linear BM,
p\sup(B,-A(t,?))>0\.^\ =
lt>T J
[Hint: Use the result in Exercise C.12) Chap. II. To prove the necessary con-
convergence to zero, look at times when В vanishes.]
2°) By suitably altering ji, prove that for h > 0 and b € K,
3°) If ji is a probability measure, prove that
sup(fi,-A(/,l))(=
where e is an exponential r.v. with parameter 1, У is a r.v. with law /u, and e and
Y are independent.
4°) If м({0|) = 0 and if there is an N > 0 such that f exp(-Ny)fi(dy) < oo,
then, for every n, the following assertions are equivalent
i) ?[(sup,>0(S, -A(>, ?))+)"] <<x>;
*# C.18) Exercise (Brownian Bridges). 1°) Retain the situation and notation of Ex-
Exercise A.39) 2°), and prove that
/¦(Л1
= B< ~ L
is a .^-Brownian motion, independent of B\. In particular, В is a '^,-semi-
martingale.

2°) If Xх = xt + B, - tBx is the Brownian Bridge of Sect. 3 Chap. I then
/' x — Xх
1 - s
The same equality obtains directly from 1°) by defining Xх as the BM conditioned
to be equal to x at time 1.
The following questions, which are independent of 2°) are designed to give
another (see Exercise C.15) Chap. II) probabilistic proof of Hardy's L2-inequality,
namely, if for / e L2([0, 1]) one sets
Hf{x) = - / f(x)dy,
x Jo
then Hf e L2([0, 1]) and \\Hf\\2 < 2||/||2.
3°) Prove that if / is in L2([0, 1]), there exists a Borel function F on [0, 1[
such that for any / < 1,
Jo
—. —du= ( F(vAt)dBv.
1 - и Jo
Then, observe that
4°) Prove that
/ f{u)dfa= [ (f(u)~F(u))dBu.
o Jo
/ F(vJdv <4 f2(u)du,
Jo Jo
Jo Jo
then, prove by elementary transformations on the integrals, that this inequality is
equivalent to Hardy's L2-inequality.
* C.19) Exercise. 1°) Let if be a Borel function on ]0, 1] such that for every s > 0
/"'
I \\lf(u)\du < oo,
Je
and define ф(и) = fu \j/(s)ds. If В is the standard linear BM, prove that the limit
lim / Tfr(s)B,ds
exists in probability if and only if
/ 4>2(u)du < oo and lim \[ёф(е) = О.
Jo e^°
Compare with Exercise B.31) of Chap. 111.
[Hint: For Gaussian r.v.'s, convergence in probability implies convergence
in /A]

2°) Deduce from Exercise C.18) 3°) that for / e L2([0, 1])
lim I f(u)u]Budu
exists a.s. If //* is the adjoint of the Hardy operator H (see Exercise C.18)),
prove that for every / e L2([0, 1]),
lim *JeH*f(e) -Q.
3°) Admit the equivalence between iii) and iv) stated in Exercise B.31)
Chap. III. Show that there exists a positive function / in L2([0. 1]) such that
lim/ f(u)u~lBudu exists a.s. and / /(m)m
'->°Л Jo
-1 JSH|<iM = oc a.s.
[Hint: Use/(ii) = \[u<]/2]/u]'2(-logu)a, 1/2 <«< 1.]
4°) Let X be a stationary OU process. Prove that
lim f
g(s)Xsds
exists a.s. and in L2 for every g e L2([0, oo[).
[Hint: Use the representation of X given in Exercise C.8) of Chap. I and the
fact that for p > 0, the map g -> BPu)~1/2g (B/3) log(l/M)) is an isomorphism
from L2([0, oo[) onto L2([0, 1[).]
Moreover using the same equivalence as in 3°) prove that if g is a positive
function of Z-i'oc([0, oof) then
/ g(s)\Xs\ds < oo a.s. iff / g(s)ds < oo.
Jo Jo
5°) For д е М, /n ^ 0, and g locally integrable prove that
for a suitable stationary OU process X. Conclude that limbec /0' g{s)e'^Bsds exists
in L2 whenever g is in L2([0, oo[). Show that the a.s. convergence also holds.
* C.20) Exercise. Let A be a d x(/-matrix and S a BM^(O). Prove that the processes
(ABt, Bt) (where ( , ) is the scalar product in Ш1) and /„' ((A + А')В„ dBs) have
the same filtration.
# C.21) Exercise. Prove the strong Markov property of BM by means of P. Levy's
characterization theorem. More precisely, if В is an (.>^")-BM, prove that for any
starting measure v and any (..^-stopping time T, the process (Вт+, — Bj) \(т<х)
is a standard BM for the conditional probability Pv(-1 T < oo) and is independent

C.22) Exercise. Let В be a BMd and O(t) = (oj(t)) be a progressively mea-
measurable process taking its values in the set of d x rf-orthogonal matrices.
Remark that /„' ||0(.s)||2rf.? < oo for every t. Prove that the process X defined
by d
X\ =Y\ I OUs)dBJ(s)
is a BM^.
C.23) Exercise. Let (X, Y) be a standard BM2 and for 0 < p < 1 put
Z, = PXt + y/\ - p2Y,.
1°) Prove that Z is a standard linear BM1 and compute (X, Z) and (У, Z).
2°) Prove that X\ and cr(Zs, s > 0) are conditionally independent with respect
toZi.
C.24) Exercise. 1°) If M is a continuous process and A an increasing process,
then M is a local martingale with increasing process A if and only if, for every
f(Mt) - f(M0) - \ f f"{Ms)dAb
is a local martingale.
[Hint: For the sufficiency, use the functions x and x2 suitably truncated.]
2°) If M and N are two continuous local martingales, the process {M, N) is
equal to the process of bounded variation С if and only if for every / e C^(K2, E),
the process
f(M,. N,) - /(Л/о, N0)-l-J ftW,, Ns)d{M, M)s
~\ I f?Ws, N,)d(N, N)s - J /;V(MS, Ns)dC,
is a local martingale.
C.25) Exercise. If M is a continuous local martingale such that Mo = 0, prove
that
{^(M 0} {(M Mj^^oc) a.s.
[Hint: K(M) = ^(|Л/Jехр(-|(М, М)).}
** C.26) Exercise (Continuation of Exercise A.42)). 1°) Let X - Xo + M + A be
a positive semimartingale such that A is increasing and X, < с a.s. for every /.
Prove that ?[Aoo] < с
2°) Let M 6 . //> and suppose that M, < к for every / > 0. Prove that
V, = (*+]- Af,) - f (k+\- M,2
Jo
is in . /^ and that (V>(-, ]0. 1]) is a.s. finite.

3°) Let M 6 ,J'/j and suppose that \\mt^Mt < oo a.s. Prove that lim,i0 A/,
exists a.s.
4°) Prove that for M e . /ei and for a.e. со one of the following three properties
holds
i) lim,;o A/,(a>) exists in R;
ii) limao I A/, (ft>) | = +oo;
iii) iirn,|0A/,(w) = —oo and lim,;oA/, (ft>) — +oo.
C.27) Exercise. Let An = @ — г0 < 'l < • ¦ ¦ < tPll = T) be a sequence of sub-
subdivisions of [О, Г„] such that Д,+1 is a refinement of 4„ and \A,,\ tends to zero.
1°) If F is a continuous function on [0, 7], prove that the sequence of measures
converges to F' in the sense of Schwartz distributions.
2°) Let В be the standard linear BM. Prove that, for / e L2([0, T]), the
sequence of random variables
?
converges a.s. and in L2 to a limit which the reader will identify.
3°) Prove that nonetheless the sequence of measures defined by making F —
В {со) in 1°) does not converge vaguely to a measure on [0, T].
4°) Prove that if / is a function of bounded variation and 9p is a point of the
interval \tp, tp+\\, then
X -f (/3 \ ( Z? Z? ^
7 I \"n) \ Of I — Or I
converges a.s.
* C.28) Exercise (Continuation of Exercise A.48)). P) If M is a continuous local
martingale on [0, T[, vanishing at 0 and with increasing process t Л T and if В is
a BM independent of Л/, then M, + B, — В1Лт is a BM. We will say that M is a
BM on [0, T[.
2°) State and prove an Ito formula for continuous local martingales on [0, T[.
(The two questions are independent).
** C.29) Exercise. (Extension of P. Levy's characterization theorem to signed
measures). Let (J2, .Wt, P) be a filtered probability space and Q a bounded signed
measure on .7^ such that Q <$c P. We suppose that (.^) is right-continuous
and P-complete and call M the cadlag martingale such that for each t, M, =
{dQ/dP)\.yr a.s. A continuous process X is called a (Q, P)-local martingale if
i) X is a P-semimartingale;
ii) XM is a P-local martingale (see Sect. 1 Chap. VIII for the significance of this
condition.

1°) If Я is a locally bounded (.J^")-predictable process and HX is the stochastic
integral computed under P, then if X is a (Q, P)-local martingale, so is H X.
2°) We assume henceforth that X, and Xf - t are {Q, P)-local martingales,
and Xq = 0, P-a.s. Prove that for any real и the process
u2 f
Y, = exp(iuXt) -1-1 / exp(iuXs)ds
2 Jo
is a (<2, /3)-local martingale and conclude that f Y,dQ = 0.
3°) Prove that
exp(iuX,)dQ = Q(\)exp(-tu2/2)
and, more generally, that for 0 < t\ < ... < tn and real numbers uk,
ехр/(м,Х„ +u2(X,2 - X,,) + ... +м„(Х,„ - Xtn_t))dQ
= Q(l)exp (-- (nil] + (t2-t0u22 + ... + (tn- tn-\)u
/¦
Conclude that
- if Q(\) = 0, then Q = 0 on a {Xs, s > 0},
- if GO) Ф 0, then QIQ{\) is a probability measure on a {Xs,s > 0} under
which X is a standard BM.
C.30) Exercise (The Kailath-Segal identity). Let M be a cont. loc. mart, such
that Mo = 0, and define the iterated stochastic integrals of M by
h = f /„-.
Jo
/o= I, /„ = / 1n-X(s)dMs.
Prove that for и > 2,
л/л = /л_,Л/-/и_2<Л/,Л/).
Relate this identity to a recurrence formula for Hermite polynomials.
[Hint: See Proposition C.8).]
* C.31) Exercise. (A complement to Exercise B.17)). Let В be а ВМ'@) and
Я and К two (.S^"fl)-progressively measurable bounded processes. We set X, —
/0 HsdBs + /0' Ksds and assume in addition that H is right-continuous at 0.
1 ) If 0 is а С'-function, show that
Г1'
h~l/2 / 4>(Xs)dX,
Jo
converges in law, as h tends to 0, to ф(О)НоВ\.
2°) Assume now that ф@) = 0 and that ф is C2, and prove that
fi,
h~l / 4>(Xs)dX,
Jo
converges in law to ф'@)Н<$(В2 - l)/2.

3°) State and prove an extension of these results when ф is Cp+I with 0@) =
ф'@) = ... =ф(Р~1)@) =0.
[Hint: Use Proposition C.8).]
* C.32) Exercise. Let В and С be two independent BM@)'s. Prove that
/l /.l
(B, + C|.fJdfU / (B2 + (B\-B,J)dt.
Jo
[Hint: The Laplace transform in (A2/2) of the right-hand side is the character-
characteristic function in X of the sum of two stochastic integrals; see Exercise B.19).]
C.33) Exercise. 1°) Let В be a BM1 and H a (.5*J"B)-adapted process such that
/„' Hjds < oo for every t, and /0°° Hjds = oo. Set
t > 0, / H2ds = a2 \
Jo J
where cr2 is a strictly positive constant. Prove that fQ HsdBs — . / @. a2).
2°) (Central-limit theorem for stochastic integrals) Let (Bn) be a sequence
of linear BM's defined on the same probability space and (K") a sequence of
(J^"fl")-adapted processes such that
CT"
' - lim / (K"
Jds = a2.
for some constants (Tn). Prove that the sequence (/0" K" dB"\ converges in law
to. Г@, a2).
[Hint: Apply 1°) to the processes H" - АГ[0.г„] + ст 1]г„.7-„ + 1]]
§4. Burkholder-Davis-Gundy Inequalities
In Sect. 1, we saw that, for an L2-bounded continuous martingale A/ vanishing
at zero, the norms ]|Л/^||2 and ||(A/, M)^\\2 are equivalent. We now use the Ito
formula to generalize this to other Lp-norms. We recall that if M is a continuous
local martingale, we write M* = sups<, \MS\.
The whole section will be devoted^ to proofs of the Burkholder-Davis-Gundy
inequalities which are the content of
D.1) Theorem. For every p e ]0, oo[, there exist two constants cp and Cp such
that, for all continuous local martingales M vanishing at zero.
cpE [{M, M)^2} < E [{M*x)p] < C,,E [(M, M)^2].
It is customary to say that the constants cp and Cp are "universal" because
they can be taken the same for all local martingales on any probability space

whatsoever. If we call Hp the space of continuous local martingales such that
Л/?, is in Lp, Theorem D.1) gives us two equivalent norms on this space. For
p > 1, the elements of Hp are true martingales and, for p > 1, the spaces Hp are
the spaces of continuous martingales bounded in Lp; this is not true for p = 1,
the space Hx is smaller than the space of continuous Lx -bounded martingales and
even of uniformly integrable martingales as was observed in Exercise C.15) of
Chap. II.
Let us also observe that, by stopping, the theorem has the obvious, but nonethe-
nonetheless important
D.2) Corollary. For any stopping time T
cpE [<M, M)f] < E [(M*)»] < CpE [{M. M)f] .
More generally, for any bounded predictable process H
\ j HsdMs
< CPE
[{[ h
The proof of the theorem is broken up into several steps.
D.3) Proposition. For p > 2, there exists a constant Cp such that for any contin-
continuous local martingale M such that Mo = 0,
Proof. By stopping, it is enough to prove the result for bounded M. The function
x -> \x\p being twice differentiable, we may apply lto's formula to the effect that
\Mx\p = [ p\M,\p-1 (sgn Ms)dMs+ ]- f p(p-\)\M,\p~2d{M,M)s.
Jo 2 J
s+ ]- f
2 Jo
Consequently,
E[\MX\"] =
On the other hand, by Doob's inequality, we have ||Л/^||р < (p/p -
and the result follows from straightforward calculations.
D.4) Proposition. For p > 4, there exists a constant cp such that

Proof. By stopping, it is enough to prove the result in the case where (M, M) is
bounded. In what follows, ap will always designate a universal constant, but this
constant may vary from line to line. For instance, for two reals x and у
From the equality M2 — 2/0' MsdMs + (M, M)t, it follows that
E [(M, MVj2} < a,,{E [{M*x)p] + 41/" M'dM>\P'2])
and applying the inequality of Proposition D.3) to the local martingale /„' MsdMs,
we get
E[(M,MfJ2] < a
< ap (E [(*?)'] + (E [(ЛС)"] E [(A/,
If we set л- = E [(Л/, M)PJ^ ' and v = E [{М*Ж)Р]Х'2, the above inequality
reads x2 — apxy — apy2 < 0 which entails that x is less than or equal to the
positive root of the equation x2 — apxy — apy2 = 0, which is of the form apy.
This establishes the proposition.
Theorem D.1) is ь "onsequence of the two foregoing propositions and of a
reduction procedure whicli we now describe.
D.5) Definition (Domination relation). A positive, adapted right-continuous pro-
process X is dominated by an increasing process A, if
Е[ХТ\.Щ<Е[АТ\.Щ
for any bounded stopping time T.
D.6) Lemma. IfX is dominated by A and A is continuous, for x and у > 0,
P[X*X >x;A0C < v]< -
where X^ = sup,
Proof. It suffices to prove the inequality in the case where P(Aq < у) > О
and, in fact, even P(A0 < v) = 1, which may be achieved by replacing P by
P' = P{- \ Aq < y) under which the domination relation is still satisfied.
Moreover, by Fatou's lemma, it is enough to prove that
P[X* >x;An <y] < -Е[Ахлу]\
but reasoning on [0, n] amounts to reasoning on [0, 00] and assuming that the
r.v. Xoo exists and the domination relation is true for all stopping times whether

bounded or not. We define R = inf{t : A, > y], S = inf{/ : X, > x), where
in both cases the infimum of the empty set is taken equal to +00. Because A is
continuous, we have {Aoo < y} = {R = 00} and consequently
P [X*^ > x: Ax < y] = P[X*^ > x\ R = 00]
< P[XS > x\ (S < 00) П (R = 00)]
< P[XSAR>x]<-E[XSAR]
x
< -Е[А5лК]<-Е[Аклу].
x x
the last inequality being satisfied since, thanks to the continuity of A, and Ao < y
a.s., we have ASaR < Аж л у. а
D.7) Proposition. Under the hypothesis of Lemma D.6), for any к e ]0, 1[,
Proof. Let F be a continuous increasing function from K+ into M+ with F@) = 0.
By Fubini's theorem and the above lemma
E[F(X'X)] - eJjT l(^>
/ (P[X^ >л-;Аос <
if we set F(x) = 2F(x) + * /^ ^^. Taking F(x) = xk, we obtain the desired
result.
Remark. For к > 1 and /Or) = xk, F is identically +00 and the above reasoning
has no longer any interest. Exercise D.16) shows that it is not possible under the
hypothesis of the proposition to find a universal constant с such that E [X^] <
cE[Aoc]. This actually follows also from the case where X is a positive martingale
which is not in Hx as one can then take A, — Xq for every /.
To finish the proof of Theorem D.1), it is now enough to use the above
result with X = (M*J and A — Сг(Л/, М) for the right-hand side inequality,
X — {M, MJ and A = C4(M*L for the left-hand side inequality. The necessary
domination relations follow from Propositions D.3) and D.4), by stopping as in
Corollary D.2).

Other proofs of the BDG inequalities in more or less special cases will be
found in the exercises. Furthermore, in Sect. 1 of the following chapter, we will
see that a method of time-change permits to derive the BDG inequalities from the
special case of BM. We will close this section by describing another approach to
this special case.
D.8) Definition. Let ф be a positive real function defined on ]0, a], such that
limJC_o0(*) — 0 and /3 a real number > I. An ordered pair (X, Y) of positive
random variables is said to satisfy the "good к inequality" ](ф, /3) if
P [X > 0k; Y < Sk] < ф(8)Р[Х > к]
for every A > 0 and 8 e]0, a]. We will write (X. Y) e 1(ф, /3).
In what follows, F will be a moderate function, that is, an increasing, contin-
continuous function vanishing at 0 and such that
sup F(ctx)/F(x) = у < oo for some a > 1.
д>0
The property then actually holds for every a > 1 with у depending on a. The
function F(x) = x1', 0 < p < oo is such a function.
The key to many inequalities is the following
D.9) Lemma. There is a constant с depending only on ф, ft and у such that if
(X, Y) e 1(ф.Р), then
E[F(X)]<cE[F(Y)].
Proof. It is enough to prove the result for bounded F's because the same у works
for F and F л п. We have
= f P[X>pk]dF(k)
Jo
< / ф(8)Р[Х > k]dF(k) + P[Y > 8k]dF(k)
Jo Jo
By hypothesis, there is а у such that F(x) < у F(x/fi) for every x. Pick 8 e]0. a[
such that уф(8) < 1; then, we can choose y' such that F(x/S) < y'F(x) for
every x, and it follows that
E[F(X)] < y'E[F(Y)]/(\ - уф(8)).
The foregoing lemma may be put to use to prove Theorem D.1) (see Exercise
D.25)). We will presently use it for a result on BM. We consider the canonical BM
with the probability measures Px, x € Ж, and translation operators в,, / > 0. We
denote by (.Л") the Brownian filtration of Chap. III. Then, we have the following

D.10) Theorem. Let A,, t > 0, be an (-F^)-adapted, continuous, increasing pro-
process such that
(i) Wmh^x supv x PX[AX2 > bk] = 0,
(ii) there is a constant К such that for every s and t
A,+s - As < KA, o6s.
Then, there exists a constant Cf such that for any stopping time T,
E0[F(AT)]<cFE0[F(T1'2)].
Proof. It is enough to prove the result for finite T and then it is enough to prove
that there exist ф and fS such that (Ат, Tx/2) е 1(ф,р) for every finite T. Pick
any /3 > 1 and set S — inf (/ : A, > X). Using the strong Markov property of BM
at time S, we have
' Po[AT >pk,Tl/2 <Sk] = Po[AT-As>(p-l)X,T<82X2.S<T]
< Po [Aj+W - As > (/3 - 1 )X, S < T]
< P0[AS2lioes>(l3-l)XK-l.S<T]
< E0[EBs[A62Xi > (p-\)XK-x],S < T]
< sup Px [AS2k2 >(fi~ \)XK~X] ¦ P0[S < T]
X
< sup Px \a,2 > ^-^л] • P0[AT > Л]
x.X L Л0 J
which ends the proof. ?
We may likewise obtain the reverse inequality.
D.11) Theorem. If A,, t > 0, is an {.Ц')-adapted, continuous, increasing process
such that
(i) lim/,^0 supv x PX [Ak2 < bX] = 0,
(ii) there is a constant К such that for every s < t
A,-.so6>v < К A,.
Then, there is a constant CF such that for any stopping time T,
E0[F(Tl/2)]<CFE0[F(AT)].
Proof. It follows the same pattern as above. Pick f}>l,S<l;we have
Pq[T1/2 >PX.AT <SX] < P0[T > р2Х2,Ат_к2овХ2 < KSX]
< P0[T>X2, AP212_12O0X: <KSX]
= Eo [EB/1 [Apv.k: < KSX], T > Л2]
< sup Px [A{P2_UX2 < KSX] ¦ Po [T1'2 > X]
.v .A
which ends the proof.

The reader will check that these results apply to A, = sup55, \BS — Bo\, thus
yielding the BDG inequalities for B, from which by time-change (see Sect. 1
Chap. V), one gets the general BDG inequalities. This method is actually extremely
powerful and, to our knowledge, can be used to prove all the BDG-type inequalities
for continuous processes.
D.12) Exercise. Let В and B' be two independent standard linear BM's. Prove
that for every p, there exist two constants cp and Cp such that for any locally
bounded (.^"B)-progressively measurable process H,
[{{H-B>yj} < E [((H.*?/] < CpE
D.13) Exercise. For a continuous semimartingale X — M + V vanishing at 0, we
set
p = \{M, M)l'2+ / \dV\A .
II Jo IIV
1°) Check that the set of A"s such that ЦХЦ./я < oo is a vector-space denoted
by У and that X -> ||X||.yi. is a semi-norm on У р.
2D) Prove that if X* = sup, |Х|„ then ||X*||p < ср\\Х\\у„ for some universal
constant cp. Is there a constant c'p such that ЦХЦ.у/- < c'p\\X*\lp?
3°) For p > 1, the quotient of Ур by the subspace of processes indistinguish-
indistinguishable from the zero process is a Banach space and contains the space Hp.
# D.14) Exercise. 1°) If M is a continuous local martingale, deduce from the BDG
inequalities that [M^ < oo} = {(M, M)^ < oo} a.s. (A stronger result is proved
in Proposition A.8) Chap. V).
2°) If M" is a sequence of continuous local martingales, prove that {M")*x
converges in probability to zero if and only if (Л/", M")^ does likewise.
[Hint: Observe that it is enough to prove the results when the A/"'s are uni-
uniformly bounded, then apply Lemma D.6).]
* D.15) Exercise. (A Fourier transform proof of the existence of occupation
densities). 1°) Let M be a continuous local martingale such that E [(M, M)j] <
oo; let pi, be the measure on E defined by
Д,(/)= / f(Ms)d(M,M)s
Jo
and Д, its Fourier transform. Prove that
\/l,(?;)\2di; < 00,
and conclude that n,(dx) <$C dx a.s.
2°) Prove that for fixed /, there is a family Ц of random variables, ,/?(R)®.^-
measurable, such that for any positive Borel function /

f f(M,)d(M,M),= f X L°f{a)da.
J0 J-cc
This will be taken up much more thoroughly in Chap. VI.
D.16) Exercise. 1°) If В is a BM, prove that \B\ is dominated by 25 where
St = sup Bs.
[Hint: If x and у are two real numbers and у > x+, one has |дг| < (y — x) + y.]
2°) By looking at X, = |В,лг, I where T\ — inf{? : B, = 1}, justify the remark
following Proposition D.7).
D.17) Exercise. Let M be a continuous local martingale with Mo = 0 and define
5'/=supMs, s, — infMs.
Let A be an increasing adapted continuous process with Ao > a > 0.
Г) Remark that
E, - M.s)dSs = 0.
f
2°) Suppose M bounded and prove that
E [Л (Soo - M^J} <e\J Ajld(M, M),] .
3°) Prove that (M*J < 2 ((St - M,J + (M, - s,J) and that
E \[М*ХJ (М, M)-J/2\ < 4e\ f (M,M)Jl/2d(M, M)A
and extend this result to non bounded M's.
[Hint: To prove the last equality, use the time-change method of Sect. 1 in
Chap. V.]
4°) Derive therefrom that
E[M*X] <2V2E[(M,M)]J2].
5°) Using 2°) , prove also that
E[(Sx~sKJ]<4E[Mi].
For another proof of this inequality, see Exercise D.11), Chap. VI.
* D.18) Exercise. 1°) Let M be a continuous local martingale with Mo — 0 and A,
В, С three continuous increasing processes such that Bo = Co = 0, and До > О.
If X = M + В - С is > 0, prove that A~XX is dominated by Y where
Jo
(It is understood that 0/0 is taken equal to 0).
Y, = I A7]dBx.
о

[Hint: Replace Ao by Ao + e, then let s decrease to 0.]
2°) Prove that for p > q > 0, there exists a constant Cpq such that
e [(м*ху (M, m)zs'2] < cpqE
# D.19) Exercise. 1°) Let A be an increasing continuous process and X a r.v. in
L\ such that for any stopping time S
E [Ax - As 1 <5>o)] < E [X 1 ,s<oo)] •
Prove that for every X > 0,
E[(A^~X)\iA^>X)] <E[X 1<^>A)].
[Hint: Consider the stopping time S — inf {f : A, > k}.]
2°) Let F be a convex, increasing function vanishing at zero and call / its
right derivative. Prove that, under the hypothesis of 1°) ,
E[F(Ax)]<E[Xf(Ax)}.
[Hint: Integrate the inequality of P) with respect to df(X).]
3°) If M is a continuous local martingale, show that
for a universal constant с
4°) For an L2-bounded martingale M, define
5(M), - sup(m;,<M,M),1/2),
Prove that ? [5(M)^] < dE [/(A/)^] for a universal constant d.
D.20) Exercise. Iе) For 0 < p < 1 set, in the usual notation,
N, = f {M,M)\p-X)l2dMs
Jo
(to prove that this integral is meaningful, use the time-change method of Sect. 1
Chap. V) and prove that, if E [(M, M)p] < oo, then E [{M, M)P] = pE [N}\
2°) By applying the integration by parts formula to N,{M, M)\x~p)l , prove
that \M,\ < 2N*{M, M)\X~P)I2, and conclude that
e[(m;Jp]<(\6/P)pE[(M,M)p].
D.21) Exercise. Let M = (M1,..., Md) be a vector local martingale and set
A = JZf_,(M(, M'). For e, ц > 0 and two finite stopping times S < T, prove that
P sup \M, - Ms|2 > s \< - + P [Ar - A5 > ?j].

D.22) Exercise. For a continuous local martingale M let (P) be a property of
{M, M)x such that i) if {N, N)oo < {M, М)ж and M satisfies (P) then N satisfies
(P), ii) if M satisfies (P) then M is a uniformly integrable martingale.
\°)lf M satisfies (P), prove that
I П f
sup < ? / HsdMs
1
; Я
J
I
progressively measurable and \H\ < \\ < oo.
[Hint: Use the theorem of Banach-Steinhaus.]
2°) By considering Я = ? 1,1],. r+l] for a subdivision Л = (?,•) prove that (P)
entails that ? \{M, М)Ц2] < +oo. As a result, the property ? {M. M)lJ.2 < oo
is the weakest property for which i) and ii) are satisfied.
int: Prove that sup^ E U? (M,i+l - M,J^' < oo.]
[Hi
:# D.23) Exercise. Let R, be the modulus of the BMd, d > 3, started at x ^ 0.
Г) After having developed log R, by lto's formula, prove that
tJ^
sup? / R;zds/ log t) | < +00
t>2 l\Jo
for every p > 0.
[Hint: One may use the argument which ends the proof of Proposition D.4).]
2°) Prove that (log Rt/ log?) converges in probability as / goes to infinity to a
constant с and conclude that /„' Rj2ds/ logf converges in probability to \/{d — 2).
The limit holds actually a.s. as is proved in Exercise C.20) of Chap. X.
3°) Let now x be 0 and study the asymptotic behavior of/f R~2ds as e tends
to zero.
[Hint: Use time-inversion.]
D.24) Exercise. (The duality between Hl and BMO revisited). 1°) Prove that
111*111//' = e\{X, Х)!?2] is a norm on the space Я1 of Exercise A.17) of Chap. II
which is equivalent to the norm ||X||wi.
2°) (Fefferman's inequality). Let X e Я1 and Y e BMO; using the result in
Exercise A.40), prove that
U <
\d(X,Y)U <2|||X|||h.||J1bmo2.
[Hint: Write /~ \d(X. Y)U = J^(X- ХOи\х, X)\/A\d{X, Y)\, and apply the
Kunita-Watanabe inequality.]
3=) Prove that the dual space of Я1 is BMO and that the canonical bilinear
form on Я1 x BMO is given by
(X.Y)^ E[(X,Y)X].

* D.25) Exercise. Г) Let A and В be two continuous adapted increasing processes
such that Aq = Bo = 0 and
E[(AT ~ As)p] < DWBtW^PIS < T]
for some positive real numbers p and D and every stopping times 5, T with S < T.
Prove that (A,*,, fi№) e 1(ф, ?) for every p> > 1 and 0(*) = D{fi - \)~PxP.
[Hint: Set T = infff : S, = 51}, 5„ = inf{? : A, > X(\ - \/n)} and prove that
the left-hand side in 1(ф, /?) is less than P [AT - ATaS., > tf - \ + \/n)X].]
2°) If M is a continuous local martingale vanishing at 0, prove, using only
the results of Sect. 1, that for A = (M, M)]/2 and В = M* or vice-versa, the
conditions of 1°) are satisfied with p = 2. Conclude that for a moderate function
F there are constants с and С such that
c? [F {{M, M)^2)] <E[F (M*K)] < CE [F ((M, M)'J})}.
3°) Derive another solution to 2°) in Exercise D.14) from the above inequali-
inequalities.
[Hint: The function x/(\ + x) is moderate increasing.]
* D.26) Exercise. If Z is a positive random variable define
o- = supXP[Z>k], I- = lim kP[Z > X].
If (X, Y) € I(ф, /J), prove that there is a constant с depending only on ф and fi
such that
ox < caY, h < cly.
D.27) Exercise. Apply Theorems D.10) and D.11) to
A,- sup (IB.-Brl/ls-
0<r<s<t
where 0 < e < 1/2.
* D.28) Exercise. Let A (resp. B) satisfy the assumptions of Theorem D.10) (resp.
D.11)). For a > 0, prove that there is a constant Cf such that for any stopping
time
[(^)] <cFE[F(BT)\.
* D.29) Exercise (Garsia-Neveu lemma). Retain the situation and notation of Ex-
Exercise D.19) and assume further that s\xpx>uxf{x)/F(x) = p < +oo.
1°) Prove that if U and V are two positive r.v.'s such that
E[UF(U)]<+oo, E[UF(U)]< E[VF(U)],
then
E[F(U)] < E[F(V)].
[Hint: If G is the inverse of F (Sect. 4 Chap. 0) then for u, v > 0,
G(i)c/j, и/(и) = F(u) + / G(iW.9.
./o

2°) Prove that
)] < E [F(pX)] < p>>E[F(X)].
D.30) Exercise (Improved constants in domination). Let Ck be the smallest
constant such that ?Г(Х^)*1 < CkE[Akx] for every X and A satisfying the
condition of Definition D.5).
1°) Prove that Ck <k~k(\ ~k)X < B-*)/(! - k), fork <= ]0, 1[.
Reverse inequalities are stated in Exercise D.22) Chap. VI.
[Hint: Follow the proof of Proposition D.7) using kx instead of x and y/k
instead of у in the inequality of Lemma D.6).]
2°) Prove that, for k, k' <= ]0, 1[, Cky < Q< (Ckf.
§5. Predictable Processes
Apart from the definition and elementary properties of predictable processes, the
notions and results of this section are needed in very few places in the sequel.
They may therefore be skipped until their necessity arises.
In what follows, we deal with a filtration (.^) supposed to be right-continuous
and complete. We shall work with the product space J2xR+ and think of processes
as functions defined on this space. Recall that a ст-field is generated by a set of
functions if it is the coarsest ст-field for which these functions are measurable.
E.1) Proposition. The a-fields generated on Q x K+ by
i) the space <C of elementary processes,
ii) the space of adapted processes which are left-continuous on ]0, oo[,
Hi) the space of adapted continuous processes
are equal.
Proof. Let us call ту, / = 1, 2, 3, the three ст-fields of the statement. Obviously
Тз с Ti, moreover т> С Х\ since a left-continuous process X is the pointwise limit
of the processes
X't'(a>) =
k=0
On the other hand, the function 1]„ „] is the limit of continuous functions /"
with compact support contained in ]u, v + \/n]. If H e .Уи, the process Hf is
continuous and adapted which implies that r, Cij.
E.2) Definition. The unique a-field discussed in the preceding proposition is
called the predictable a-field and is denoted by .У or .-У (.^") (when one wants to
stress the relevant filtration). A process X with values in (U, M) is predictable if
the map (со, t) —> X, (to) from (?2 x K+) to (U, M) is measurable with respect to

Observe that if X is predictable and if Xo is replaced by another .^-measurable
r.v., the altered process is still predictable; predictable processes may be thought
of as defined on ]0, oof. It is easily seen that predictable processes are adapted;
they are actually (.3^-)-adapted.
The importance of predictable processes comes from the fact that all stochas-
stochastic integrals are indistinguishable from the stochastic integrals of predictable pro-
processes. Indeed, if we call L2y,(M) the set of equivalent classes of predictable
processes of lZ2(M), it can be proved that the Hilbert spaces L2(M) and L2^,(M)
are isomorphic, or in other words, that every process of '/;2{M) is equivalent to a
predictable process. We may also observe that, since '6 is an algebra and a lattice,
the monotone class theorem yields that # is dense in L2y,(M). Consequently,
had we constructed the stochastic integral by continuity starting with elementary
stochastic integrals, then L2y,(M) would have been the class of integrable pro-
processes.
We now introduce another important a -field.
E.3) Definition. The a -field generated on Q x!R+ by the adapted cadlag processes
is called the optional ст-field and is denoted by f or C- (.^"). A process which is
measurable with respect to f- is called optional.
It was already noticed in Sect. 4 Chap. I that, if Г is a stopping time, the
process l]o.r[> namely (со, t) -> 1[о<г<г<ш)], is predictable. We can now observe
that Г is a stopping time if and only if l[0 r[ is optional.
Since the continuous processes are cadlag, it is obvious that С- э У and this
inclusion is usually strict (however, see Corollary E.7) below). If we denote by
Prog the progressive ст-field, we see that
.>& <g) .J?(R+) D Prog D Г D .У.
The inclusion Prog D Cj may also be strict (see Exercise E.12)).
We proceed to a few properties of predictable and optional processes.
E.4) Definition. A stopping time T is said to be predictable if there is an increasing
sequence (Т„) of stopping times such that almost-surely
i) lim,, Tn — T,
ii) Tn < T for every n on {T > 0}.
We will now state without proof a result called the section theorem. Let us
recall that the graph [T] of a stopping time is the set {(со, t) e Q xK+ : T(co) = t].
If T is predictable, this set is easily seen to be predictable. Let us further call л
the canonical projection of Q x K+ onto Q.
E.5) Theorem. Let A be an optional (resp. predictable) set. For every e > 0,
there is a stopping time (resp. predictable stopping time) such that
0 [ПСА,
ii) P[T < oo] > P(n(A)) -e.

This will be used to prove the following projection theorem. The a -field .Щ-
is defined in Exercise D.18) of Chap. I. By convention .5fo- = -ЯГ
E.6) Theorem. Let X be a measurable process either positive or bounded. There
exsists a unique (up to indistinguishability) optional process Y (resp. predictable
process Z) such that
E [XTl(T<oo) I -Ft] = Yt I a <oc) a.s. for every stopping time T
(resp. ?[Xrl(r<Oo) \-?f-] = ZT\{T<ao) a.s. for every predictable stopping time
T).
The process Y (resp. Z) is called the optional (predictable) projection of X.
Proof. The uniqueness follows at once from the section theorem. The space of
bounded processes X which admit an optional (predictable) projection is a vector
space. Moreover, let X" be a uniformly bounded increasing sequence of processes
with limit X and suppose that they admit projections Y" and Z". The section
theorem again shows that the sequences (Y") and (Z") are a.s. increasing; it is
easily checked that lim Y" and lim Z" are projections for X.
By the monotone class theorem, it is now enough to prove the statement for a
class of processes closed under pointwise multiplication and generating the a -field
J^o ® .>?(R+). Such a class is provided by the processes
Х,И = 1м(г)Я(и), 0<«<оо, //€L°°(J?>).
Let H, be a cadlag version of E[H \ .Щ (with the convention that Ho = Ho).
The optional stopping theorem (resp. the predictable stopping theorem of Exercise
C.18) Chap. II) proves that
Y, = \[o.u[(t)H, (resp. Z, = 1[о.и[(г)Я,-)
satisfies the condition of the statement. The proof is complete in the bounded case.
For the general case we use the processes X An and pass to the limit.
Remark. The conditions in the theorem might as well have been stated
?'[А'г1<г<оо)] = E[YT\(T«x»~\ for any stopping time T
(resp.
E [Xt l<r<oc)] = E [Zr 1(г«»)] for any predictable stopping time T).
E.7) Corollary. Let ^ be the a-field generated by the processes M, — M,- where
M ranges through the bounded {.^)~martingales; then
In particular, if all (.^)-martingales are continuous, then f- = .У.

Proof. Since every optional process is its own optional projection, it is enough to
prove that every optional projection is measurable with respect to -У v ^, but
this is obvious for the processes \[o,u[H considered in the above proof and an
application of the monotone class theorem completes the proof.
E.8) Exercise. Prove that .-/" is generated by the sets
[5, T] = {(t, со) : S(oj) < t <T{w)}
where 5 is a predictable stopping time and T an arbitrary stopping time larger
than S.
# E.9) Exercise. Prove that if M, is a uniformly integrable cadlag martingale, its
predictable projection is equal to M,-.
E.10) Exercise. For any optional process X, there is a predictable process Y such
that the set {(a>, t) : X,(a>) ф Y,{u>)} is contained in the union of the graphs of
countably many stopping times.
[Hint: It is enough to prove it for a bounded cadlag X and then Y = X_ will
do. For s > 0, define T(s) = infj? > 0 : \X, - X,-\ > e}
Тп{в) = inf{r > Tn-X(e) : \X, - X,-\ > e}
and use \Jpn Tn(sp) with sp j 0.]
# E.11) Exercise. A closed subspace S of H^ is said to be stable if
i) for any M e S and stopping time T, MT is in 5;
ii) for any Г е .Щ, and M e S, \pM is in S.
1°) If 5 is stable and Г е .Щ, prove that 1 r(M - MT) e S for any M e S.
2°) The space S is stable if and only if, for any M e S and any predictable
H e L2(M), the martingale H ¦ M is in 5.
3°) Let S{M) be the smallest stable subspace of Щ containing the martingale
M. Prove that
= {H-M: H e L2{M)}.
4°) Let 5 be stable and SL be its orthogonal subspace in #02 (for the Hilbert
norm). For any M e S and N e S1, prove that (M, /V) = 0. Show that S1 is
stable.
5°) If M and N are in H^, prove that the orthogonal projection of Л' on
S(M) is equal to H ¦ M where H is a version of the Radon-Nikodym derivative
of d{M, N) with respect to d{M, M).
* E.12) Exercise. Retain the situation of Proposition C.12) Chap. Ill and for t > 0
set d, = t + To о в,.
Г) Prove that the map t -> d, is a.s. right-continuous.
2°) Let H = {(a>, t) : B,(co) - 0}. For each со, the set (H(co, •))'" is a countable
union of open intervals and we denote by F the subset of ?2 xR+ such that F(co, •)

is the set of the left-ends of these intervals. Prove that F is progressive with respect
to the Brownian filtration.
[Hint: Prove that H\F = {(со, t) : d,(co) = /}.]
3°) Prove that for any stopping time T, one has Р[[Т] С F~\ = 0.
[Hint: Use the strong Markov property of BM]
4°) Prove that F is not optional, thus providing an example of a progressive
set which is not optional.
[Hint: Compute the optional projection of If.]
E.13) Exercise. Let (.5f) С (¦'?,) be two filiations. If M is a (-r?r)-martingale,
prove that its optional projection w.r.t. (.>O is a (.5f )-martingale.
E.14) Exercise. Let A" be a measurable process such that, for every t,
E\ f \Xs\ds\ <oo;
and set Y, = /„' Xsds. If we denote by H the optional projection of H w.r.t. a
filtration (.7j), prove that Y, — jjj Xsds is a (.^-martingale.
E.15) Exercise. Let (.Vt) be a filtration, В a (.^)-BM and h a bounded optional
process. Let Y, = /„' hsds + B,; if й is the optional projection of A w.r.t. (-^Y),
prove that the process
N, = Y,- f hsds
Jo
is a .^"K-BM. In filtering theory, the process N is called the innovation process.
[Hint: Prove first that E[N}] < oo for every t, then compute E[Nt] for
bounded stopping times Т.]
E.16) Exercise. Let X and X' be two continuous semimartingales on two filtered
probability spaces; let H and H' be two predictable locally bounded processes
such that (H, X) = (H\ X'). Prove that (H ¦ X, H, X) = (H' ¦ X', H\ X').
[Hint: Start with the case of elementary H and #'.]
*# E.17) Exercise (Fubini's theorem for stochastic integrals). 1°) Let (A,. ¦/,) be
a measurable space, (Q,.W, P) a probability space and Х„(а, •) a sequence of
. -/> <S> .F"-measurable r.v.'s which converge in probability for every a. Prove that
there exists a . -d ® .i^-measurable r.v., say X, such that for every a, X(a, •) is
the limit in probability of Х„(а. ¦).
[Hint: Define inductively a sequence пь(а) by n$(a) = 1 and
nk(a) = inf \m >nk-i(a) : sup P [\X,,(a, -)-X4(a, )| > 2~k] < 2-k\;
prove that lim^ Хп,(Я)(а, •) exists a.s. and answers the question.]

2°) In the setting of Sect. 2, if H(a, s, w) is a uniformly bounded , -A <g> -У>-
measurable process and X a continuous semimartingale, there exists a . -# ® ¦^>-
measurable process /Г(a, -, •) such that for each a, K(a, •, •) is indistinguishable
from /0 H(a. s, -)dXs. Moreover, if v is a bounded measure on . /,, then a.s.,
/ K(a.t,)v(da) = f ( f H(a,s,-)v(da))dX,.
Ja Jo \Ja /
[Hint: These properties are easily checked when H(a,s,cv) = h{a)g(s,co)
with suitable h and g. Apply the monotone class theorem.]
E.18) Exercise. 1°) Let M e H2 and К be a measurable, but not necessarily
adapted process, such that
KJd{M, M)A< oo.
Prove that there exists a unique Z e H2 such that for N e H2
2°) Let К be the projection (in the Hilbert space sense) of К on L2y,{M).
Prove that Z = K-M.
# E.19) Exercise. Iе) If К is a continuous adapted process and X a cont. semimart.,
prove that
P-\\m- [ Ks(X,+,-X,)ds= [ KsdXs.
[Hint: Use Fubini's theorem for stochastic integrals.]
2°) Under the same hypothesis, prove that
P-\\m - f Ks (Xs+F - Xsfds = I K,d(X, X),.
f-«f/o Jo
Notes and Comments
Sect. 1. The notion of local martingale appeared in Ito-Watanabe [1] and that of
semimartingale in Doleans-Dade and Meyer [1]. For a detailed study, we direct the
reader to the book of Dellacherie and Meyer ([1] vol 2) from which we borrowed
some of our proofs, and to Metivier [1].
The proof of Theorem A.3) is taken from Kunita [4]. The existence of quadratic
variation processes is usually shown by using Meyer's decomposition theorem
which we wanted to avoid entirely in the present book. Proposition A.13) is due
to Getoor-Sharpe [1]. Let us mention that we do not prove the important result of

Strieker [1] asserting that а (.У, )-semimartingale X is still a semimartingale in its
own filtration (.F^x).
Exercise A.33) is from Yor [2], Exercise A.41) from Emery [1] and Exercise
A.42) from Sharpe [1]. Exercise A.48) is due to Maisonneuve [3] and Exercise
A.49) is taken from Dcllacherie-Meyer [1], Vol. II. The method hinted at in Ex-
Exercise A.38) was given to us by F. Delbaen (private communication).
Sect. 2. Stochastic integration has a long history which we will not attempt to
sketch. It goes back at least to Paley-Wiener-Zygmund [1] in the case of the
integral of deterministic functions with respect to Brownian motion. Our method
of defining stochastic integrals is that of Kunita and Watanabe [1] and is originally
due to Ito [1] in the BM case. Here again, we refer to Dellacherie-Meyer [1] vol.
2 and Metivier [ 1 ] for the general theory of stochastic integration with respect to
(non continuous) semimartingales.
Exercise B.17) is due to Isaacson [1] and Yoeurp [2] and Exercise B.18) is
from Yor [2]. Exercise B.19) is taken from Donati-Martin and Yor [2]; a number
of such identities in law have been recently discussed in Chan et al. [1] and
Dean-Jansons [1]. The important Exercise B.21) is taken from a lecture course of
Meyer.
Sect. 3. Ito's formula in a general context appears in Kunita-Watanabe [1] and
our proof is borrowed from Dellacherie-Meyer [1], Vol. II. Exercise C.12) is but
one of many extensions of Ito's formula; we refer to Kunita [6]. The proof of
Exercise C.13) is due to Follmer [2].
Exponentials of semimartingales were studied by Doleans-Dade [2] and are
important in several contexts, in particular in connection with Girsanov's theorem
of Chap. VIII. We refer to the papers of Kazamaki and Sekiguchi from which
some of our exercises are taken, and also to Kazamaki's Lecture Notes [4].
The proof given here of P. Levy's characterization theorem is that of Kunita-
Watanabe [1]; the extension in Exercise C.29) is due to Ruiz de Chavez [1].
This result plays a central role in Chap. V and its simplicity explains to some
extent why the martingale approach to BM is so successful. Moreover, it contains
en germe the idea that the law of a process may be characterized by a set of
martingale properties, as is explained in Chap. VII. This eventually led to the
powerful martingale problem method of Stroock and Varadhan (see Chap. VII).
Exercise C.17) is from Robbins-Siegmund [1]; Exercise C.19) is from Donati-
Yor [1] and Exercise C.26) from Calais and Genin [1]. Exercise C.30) comes from
Carlen-Kree [1] who obtain BDG type inequalities for multiple stochastic integrals.
Sect. 4. Our proof of the BDG inequalities combines the method of Getoor-Sharpe
[1] with the reduction procedure of Lenglart [2] (see also Lenglart et al. [1]). The
use of "good A. inequalities" is the original method of Burkholder [2]. The method
used in Theorems D.10) and D.11), due to Bass [2] (see also Davis [5]), is the
most efficient to date and works for all inequalities of BDG type known so far.
For further details about the proof of Fefferman's inequality presented in Exer-
Exercise D.24), see e.g. Durrett [2], section 7.2, or Dellacherie-Meyer ([1], Chap. Vll).

The scope of Fefferman's inequality may be extended to martingales Y not
necessarily in BMO; see, e.g., Yor [26] and Chou [1].
Exercise D.30) gives an improvement on the constant B — k)/{\ — k) obtained
in Proposition D.7), but the following natural question is still open.
Question 1: For к e ]0, 1[, find the best constant Ck for the inequality
which follows from the domination relation.
Sect. 5. This section contains only the information on predictable processes which
we need in the sequel. For a full account, we direct the reader to Dellacherie-
Meyer [1].
The reader shall also find some important complements in Chung-Williams [1]
(Sect. 3.3) to our discussion, following Definition E.2), of the various classes of
stochastic integrands.

Chapter V. Representation of Martingales
In this chapter, we take up the study of Brownian motion and, more generally, of
continuous martingales. We will use the stochastic integration of Chap. IV together
with the technique of time changes to be introduced presently.
§1. Continuous Martingales as Time-changed Brownian Motions
It is a natural idea to change the speed at which a process runs through its path;
this is the technique of time changes which was described in Sect. 4 Chap. 0 and
which we now transpose to a stochastic context.
Let (.3^") be a right-continuous filtration. Throughout the first part of this
section, we consider an increasing, right-continuous, adapted process A with which
we associate
Cs — inf{? : A, > s)
where, as usual, inf@) = +oo. Since CS increases with s, the limit CS- =
limM^s Cu exists and
Cs- =inf{r : A, >s).
By convention Co- = 0.
A.1) Proposition. The family (Cs) is an increasing right-continuous family of
stopping times. Moreover for every t, the r.v. A, is an {.7c\. )¦-stopping time.
Proof. Each Cs is a stopping time, by the reasoning of Proposition D.6) Chap. I.
The right-continuity of the map s —* Cs was shown in Lemma D.7) Chap. 0, and it
follows easily (Exercise D.17) Chap. I) that the filtration (-ScJ is right-continuous.
Finally it was also shown in Lemma D.7) Chap. 0, that A, = mf{s : Cs > t) which
proves that A, is an (-^cj-stopping time. ?
It was proved in Chap. 0 that Ac, > t with equality if/ is a point of increase
of C, that is C,+F — C, > 0 for every s > 0. If A is strictly increasing, then X is
continuous; if A is continuous and strictly increasing, then С is also continuous
and strictly increasing and we then have Ac, — С a, = t. While reasoning on this
situation, the reader will find it useful to look at Figure 1 in Sect. 4 Chap. 0. He
will observe that the jumps of A correspond to level stretches of С and vice-versa.
Actually, A and С play symmetric roles as we will see presently.

A.2) Definition. A time-change С is a family Cs, s > 0, of stopping times such
that the maps s —*¦ Cs are a.s. increasing and right-continuous.
Thus, the family С defined in Proposition A.1) is a time-change. Conversely,
given a time change C, we get an increasing right-continuous process by setting
A, = inf {s : Cs > t}.
It may happen that A is infinite from some finite time on; this is the case if
Coc — Нш^-.эс Cs < oo, but otherwise we see that time-changes are not more
general than the inverses of right-continuous increasing adapted processes.
In the sequel, we consider a time-change С and refer to A only when necessary.
We set ,Wt = .Vq,- If X is a (..^-progressive process then X, = Xc, is a (.W,)-
adapted process; the process X will be called the time-changed process of X.
Let us insist once more that, if Л is continuous strictly increasing and Аж = oo,
then С is continuous, strictly increasing, finite and CTO = oo. The processes A
and С then play totally symmetric roles and for any (..^-progressive process X,
we have Хд = X. If Лос < oo, the same holds but X is only defined for t < Аж.
Finally let us observe that by taking C, = t л T where T is a stopping time, one
gets the stopped processes as a special case of the time-changed processes.
An important property of the class of semimartingales is its invariance under
time changes. Since, in this book, we deal only with continuous semimartingales,
we content ourselves with some partial results. We will need the following
A.3) Definition. If С is a time-change, a process X is said to be C-continuous //
X is constant on each interval [С,-, С,].
If X is increasing and right-continuous, so is X; thus, if X is right-continuous
and of finite variation, so is X. The Stieltjes integrals with respect to X and X are
related by the useful
A.4) Proposition. If H is (.Y,) -progressive, then H is {.V,)-pwgressive and if X
is a C-continuous process of finite variation, then HX — HX; in other words
/ HsdXs=\ 1 ,с„ <oo) HCudXc,, = \ HCudXCu.
JCo JO JO
Proof. The first statement is easy to prove. The second is a slight variation on
Proposition D.10) of Chap. 0. ?
In many cases, Q = 0 and C, is finite for every t and the above equality reads
re,
I HsdXs = f HCudXCll.
Jo Jo
We now turn to local martingales. A first problem is that, under time changes,
they remain semimartingales but not always local martingales; if for instance, X
is a BM and At = S,, then X, = t. Another problem for us is that С may have
jumps so that X may be discontinuous even when X is continuous. This is why
again, we will have to assume in the next proposition that X is C-continuous.

A.5) Proposition. Let С be a.s. finite and X be a continuous (.^)-locaJ martin-
martingale.
i) If X is C-continuous, then X is a continuous (-T^)-local martingale and
(X, X) = (O);
ii) If moreover H is (-i^)-progressive and /0 Hfd(X, X)s < oo a.s. for every
t then /0' HJd{X, X)s < oo a.s. for every t and HX = /Tx.
Proof, i) Since X is C-continuous the process X is continuous. Let T be a (.V,)-
stopping time such that XT is bounded; the time T = inf[t; C, > 7"} is a (.V,)-
stopping time because 1 ^ ^ = 1 \т.~о\ is .^"-adapted. It is clear that XT is bounded.
Moreover
XJ = XTAt = XC,T
and because X is C-continuous, X is constant on [7\ Cf] and we have Хс,^ —
Xf ; it follows from the optional stopping theorem that XT is a (.^-martingale.
Finally if (Tn) increases to +oo, so does the corresponding sequence (Т„); thus,
we have proved that X is a cont. loc. mart.
By Proposition A.13) of Chap. IV, the process {X, X) is also C-continuous,
thus, by the result just proved, X2 — (X, X) is a cont. loc. mart, which proves that
<х,х> = <хГх>.
ii) The first part follows from Proposition A.4). To prove the second part, we
need only prove that the increasing process of the local martingale H X — H -X
vanishes identically and this is a simple consequence of i) and Proposition A.4).
?
Thus, we have proved that suitably time-changed Brownian motions are local
martingales. The following converse is the main result of this section.
A.6) Theorem (Dambis, Dubins-Schwarz). If M is a (.i^", P)-cont. loc. mart,
vanishing at 0 and such that (Л/, M)x — oo and if we set
T, = \nf{s : (M, M), > t),
then, B, = Mt, is a (.^j^-Brownian motion and M, = Вщ.м),-
The Brownian motion В will be referred to as the DDS Brownian motion of M.
Proof. The family T — (T,) is a time-change which is a.s. finite because
(M, М)ж = oo and, by Proposition A.13) of Chap. IV, the local martingale M is
obviously Г-continuous. Thus, by the above result, В is a continuous (.>^()-local
martingale and (B. B), = (M, M)Tl = t. By P. Levy's characterization theorem,
В is a (.>^()-Brownian motion.
To prove that В(М.м) = M, observe that S<m.m) = MT,M „, and although
T(M.M), may be > t, it is always true that Mj:mm%i = M, because of the constancy
of M on the level stretches of (Л/, М). a

In the above theorem, the hypothesis (M, M)^ = oo ensures in particular that
the underlying probability space is rich enough to support a BM. This may be also
achieved by enlargement.
We call enlargement of the filtered probability space (,f2, .5^", P) another filtered
probability space (?2,.^, P) together with a map я from ?2 onto ?2, such that
л- '(•^') С .^i for each t and я(Р) — P. A process X defined on ?2 may be
viewed as defined on ?2 by setting X(a>) = X(co) if л{й>) = a).
If (M, М)ж < oo, recall that M^ = lim,^^ M, exists (Proposition A.26)
Chap. IV). Thus we can define a process W by
W, = MTl for / < (M, M)x, W, = Mx if / > (A/, A/)oc.
By Proposition A.26) Chap. IV, this process is continuous and we have the
A.7) Theorem. There exist an enlargement (&,.&,, P) of (?2, .7jr P) and a BM
ft on Q independent of M such that the process
„ I Mr, if t < (M, Af >«,,
' \ Mx + 0,-{U.H)x if t > Ш, M)K
is a standard linear Brownian motion. The process W is а {.Щ-^-ВМ stopped at
{M,M)co.
Proof. Let (??', .^', P') be a probability space supporting a BM ^ and set
?2 = Пх?2', .Wt = .TTl ® .Vt', P=P®P\ p,(co,co') = P,(co').
The process ft is independent of M and В may as well be written as
B, = MTl + I Us>(M.M)^)dps.
Ju
In the general setting of Proposition A.5), we had to assume the finiteness of С
because Me, would have been meaningless otherwise. Here where we can define
MTi even for infinite T,, the reasoning of Proposition A.5) applies and shows that
W is a local martingale (see also Exercise A.26)).
Being the sum of two .3^"-local martingales, В is a local martingale and its
increasing process is equal to
\{.s>(M.M)^)ds +2 / \(S>(M
Jo
Because of the independence of Mj and ft, the last term vanishes and it is easily
deduced from the last proof that (MT,MT), — t A {M,M)X; it follows that
(В, В), — t which, by P. Levy's characterization theorem, completes the proof.
Remark. An interesting example is given in Lemma C.12) Chap. VI.
We may now complete Proposition A.26) in Chap. IV.

A.8) Proposition. For a continuous local martingale M, the sets {(A/, MH0<oo]
and {lim^oo M, exists) are almost-surely equal. Furthermore, lim/^ooA/, = +00
and lim/^ocA// = -oo a.s. on the set {(A/, A/>oc = ooj.
Proof. We can apply the preceding result to M — Mq. Since В is a BM, we have
lirrif-^oofif — +°° a-s- On the set (A/, M)^ = 00, we thus have lim/^ooA/7; = +00
a.s. But on this set, T, converges to infinity as t tends to infinity; as a result,
firn,^ooA/, is larger than lim,^^^^, hence is infinite. The same proof works for
the inferior limit.
Remarks. 1°) We write lim M, > lim MTi because the paths of MT could a priori
be only a portion of those of A/. We leave as an exercise to the reader the task of
showing that they are actually the same.
2°) Another event which is also a.s. equal to those in the statements is given
in Exercise A.27) of Chap. VI.
3°) This proposition shows that for a cont. loc. mart., the three following
properties are equivalent:
i) supf M, — 00 a.s., ii) inf, M, = —00 a.s., iii) (A/, M)x = 00 a.s.
We now turn to a multi-dimensional analogue of the Dambis, Dubins-Schwarz
theorem which says that if (M, N) = 0, then the BM's associated with A/ and N
are independent.
A.9) Theorem (Knight). Let M — (A/1 Md) be a continuous vector valued
local martingale such that A/o = 0, (Mk, Мк)ж = oo for every к and (Mk, A/') =
Ofor к ф I. If we set
Tk = inf{.y : (Mk,Mk)s > t]
and Bk = Mkt, the process В = (Bl Bd) is a d-dimensional BM.
As in the one-dimensional case, the assumption on (A/*, M*)oo may be re-
removed at the cost of enlarging the probability space. It is this general version that
we will prove below.
A.10) Theorem. If(Mk, Mk)x is finite for some k's, there is a BMd fi independent
of M on an enlargement of the probability space such that the process В defined
by
Rk = .,, for t < (Mk, Mk)
is a d-dimensional BM.
Proof. By the previous results, we know that each Bk separately is a linear BM, so
all we have to prove is that they are independent. To this end, we will prove that,
with the notation of Theorem C.6) of Chap. IV, for functions Д. with compact
support, E K^ — 1. Indeed, taking fk — X!f=i A.^" I jf/_,./,-], we will then obtain

that the random vectors (вк - В* V 1 < j < p, 1 < к < d, have the right
characteristic functions, hence the right laws.
In the course of this proof, we write Ak for (Mk, Mk). From the equality
which follows from Proposition A.5), we can derive, by the usual monotone class
argument, that
fk(s)dMkTi = / fk(Aku)dMk.
Jo
Consequently,
лоо лоо /»эо
/ fk(s)dBk = / fk(Aku)dMk + / fk(s + AkK)dpk.
Jo Jo Jo
Note that the stochastic integral on the right makes sense since A^ is independent
of pk. Passing to the quadratic variations, we get
/•ОС />OO />OO
/ fk2{s)ds= f^(Ak)dAku+ fk
Jo Jo Jo
/>OO
The process X = ?t /0 fk(Ak)dMk is a local martingale and, using the hypothesis
(Mk,M') =0 for*//, we get
By lto's formula, /, = exp{/X, + j(X, X),] is a local martingale. Since it is
bounded (by ехр{?^ НЛНг/^})* 't is 'n fact a martingale. Likewise
У, = exp
is a bounded martingale and E <C^ = E [/ооУос]-
Now, conditionally on M, У» has the law of exp {/Z + ^Var(Z)} where Z is
gaussian and centered. Consequently, E K^, = E [f^o] and, since / is a bounded
martingale, E '6„[ = E [/()] = I, which ends the proof.
Remarks. 1°) Let us point out that the DDS theorem is both simpler and somewhat
more precise than Knight's theorem. The several time-changes of the latter make
matters more involved; in particular, there is no counterpart of the filtration {.Щ,)
of the former theorem with respect to which the time-changed process is a BM.
2°) Another proof of Knight's theorem is given in Exercise C.18). It relies on
a representation of Brownian martingales which is given in Sect. 3.

An important consequence of the DDS and Knight theorems is that, to some
extent, a property of continuous local martingales which is invariant by time-
changes is little more than a property of Brownian motion and actually many
proofs of results on cont. loc. mart, may be obtained by using the associated BM.
For instance, the BDG inequalities of Sect. 4 Chap. IV can be proved in that way.
Indeed, if M, = B{MM)i,
and since (A/, M), is a stopping time for the filtration (¦>ет:) with respect to which
В is a Brownian motion, it is enough to prove that, if (.'?,) is a filtration, for a
(.^)-ВМ В and a (#,)-stopping time 7",
cp?[P/2]<?[(B;f]<C,,?[P/2].
The proof of this is outlined in Exercise A.23).
Finally, let us observe that, in Theorem A.6), the Brownian motion В is
measurable with respect to •>^CM, where, we recall from Sect. 2 Chap. Ill, (.J^x)
is the coarsest right-continuous and complete filtration with respect to which X is
adapted; the converse, namely that M is measurable with respect to .3^f is not
always true as will be seen in Exercise D.16). We now give an important case
where it is so (see also Exercise A.19)).
A.11) Proposition. If M is a cont. loc. mart, such that (M, М)ж = оо and
M, = x + I o(Ms)dP,
Jo
fora BM p and a nowhere vanishing function a, then M is measurable with respect
to ,Wj^ where В is the DDS Brownian motion of M.
Proof. Since a does not vanish, (Л/, М) is strictly increasing and
(M, M)Tl = / a2(Ms)ds = t.
Jo
Using Proposition A.4) with the time change T,, we get /0' o2(Bs)dTs — t, hence
T, = / a~2(Bs)ds.
Jo
It follows that T, is .TtB-measurable, (M, M), which is the inverse of G",) is
•J^f-measurable and M — B(M M) is consequently also .7^-measurable, which
completes the proof.
Remark. We stress that, in this proposition, we have not proved that M is (.3^
•^[m m) (-adapted (see Exercise D.16)).

186 Chapter V. Representation of Martingales
A.12) Exercise. Let С be a time change and D be a time change relative to
{¦7~t) — {¦'7c,). Prove that s —> Co, is a time change and ,7Ь, = .7c Di ¦
A.13) Exercise. 1°) Let Л be a right-continuous adapted increasing process. If X
and Y are two positive measurable processes such that
E \XT\(T<oc)\ — E [Yt\(t<oo)\
for every (..^-stopping time 7", in particular, if Y is the (.i^D-optional projection
of X (Theorem E.6) Chap. IV), then for every t < +oo,
[Hint: Use the same device as in Proposition A.4) for the time change associ-
associated with A.]
2°) If A is continuous, prove that the same conclusion is valid if the assump-
assumption holds only for predictable stopping times, hence, in particular, if Y is the
predictable projection of X.
[Hint: Use C,- instead of C, and prove that C, is a predictable stopping time.]
The result is actually true if A is merely predictable.
3°) If M is a bounded right-continuous positive martingale and Л is a right-
continuous increasing adapted process such that Ao = 0 and E[A,] < oo for every
t, then
Г [' 1
E[M,A,] = E \ MsdAs
Uo J
for every t > 0. The question 3°) is independent of 2°) .
# A.14) Exercise. (Gaussian martingales. Converse to Exercise A.35) Chap.
IV).
1°) If M is a cont. loc. mart, vanishing at zero and if (M. M) is deterministic,
then M is a Gaussian martingale and has independent increments.
[Hint: This can be proved either by applying Theorem A.6) or by rewriting in
that case the proof of P. Levy's characterization theorem.]
2°) If В is a standard BM1 and fi — H ¦ В where H is a (.>^B)-predictable
process such that \H\ — 1, prove that the two-dimensional process {B,f5) is
Gaussian iff H is deterministic.
A.15) Exercise. Let M be a continuous local martingale. Prove that on {(A/, M)x
= oo}, one has
Hm M,/ B(M, M), log,(M, M),)y2 = 1 a.s.
A.16) Exercise (Law of large numbers for local martingales). Let A e . -^+
be such that Ao > 0 a.s. and M be a continuous local martingale vanishing at 0.
We set

§ 1. Continuous Martingales as Time-changed Brownian Motions 187
г -1
Z, = / As dM
Jo
Iе) Prove that
f
i. — I (Z, — ZA dAs
Jo
M,
Jo
2°) It is assumed that lim,^^ Z, exists a.s. Prove that Iim/^OC(M,/A,) — 0
on the set {Ax = oc}.
3°) If/ is an increasing function from [0, oo[ into ]0, oof such that fx f(ty2dt
< oo, then
lim M,/f((M, M),) = 0 a.s. on {{M, M)x = oc}.
t—*cc
In particular lim,-^ M,/(M, M), = 0 a.s. on {{M, M)^ = oo}.
4°) Prove the result in 3°) directly from Theorem A.6).
# A.17) Exercise. Iе) If M is a continuous local martingale, we denote by /3(M)
the DDS BM of M. If С is a finite time-change such that Cx — oo and if M is
C-continuous, prove that fi(M) — fi(M).
2°) If h is > 0, prove that 0 (\M) = P(M)(h) where X{,h) = \Xhi,. Conclude
h h
* A.18) Exercise. Extend Theorems A.6) and A.9) to continuous local martingales
defined on a stochastic interval [0, T[ (see Exercises A.48) and C.28) in Chap.
IV).
* A.19) Exercise. In the situation of Theorem A.6), prove that .Vj^ is equal to the
completion of a ((Bs, Ts), s > 0). Loosely speaking, if you know В and T, you
can recover M.
* A.20) Exercise (Holder condition for semimartingales). If X is a cont. semi-
martingale and A the Lebesgue measure on M+ prove that
Л (It > 0 : hme-a\Xl+F - X,\ > ol j = 0 a.s.,
for every a < 1/2.
[Hint: Use the DDS and Lebesgue derivation theorems.]
A.21) Exercise. Let (X. Y) be a BM2 and H a locally bounded {.Ttx (-predictable
process such that J™ Hjds = oo a.s. Set M, = /0' HsdYs and call T, the inverse
of {M, M)t. Prove that the processes MT, and Xj, are independent.
% A.22) Exercise. In the notation of the DDS theorem, if there is a strictly positive
function / such that
(M, M), = / f{Ms)ds and {M. M)x = oo
then
f(Bu)xdu.
T,-f.
Jo

188 Chapter V. Representation of Martingales
A.23) Exercise. Let В be a (.y^")-Brownian motion and Г be a bounded (.^")-
stopping time.
1°) Using Exercise D.25) in Chap. IV and the fact that E [Bj] = E[T], prove
that, for p > 2, there is a universal constant Cp such that
By the same device as in Sect. 4 Chap. IV, extend the result to all /?'s.
2°) By the same argument, prove the reverse inequality.
3°) Write down a complete proof of the BDG inequalities via the DDS theorem.
# A.24) Exercise. This exercise aims at answering in the negative the following
question: if M is a cont. loc. mart, and И a predictable process, such that
fQ' Hjd{M, M)s < oc a.s. for every t, is it true then that
(*)
t
HsdMs
о
= E
whether these quantities are finite or not? The reader will observe that Fatou's
lemma entails that an inequality is always true.
1°) Let В be a standard BM and Я be a (.>?"B)-predictable process such that
/ Hfds < oo for every t < 1, but / Hjds =
Jo Jo
(the reader will provide simple examples of such Я'в). Prove that the loc. mart.
M, = /J HsdBs is such that
limM, = —oo, HmM, — oo a.s.
2°) For a e Ш, а ф 0, give an example of a cont. loc. mart. N vanishing at 0
and such that
i) for every t0 < 1, {N,. t < f0) is an L2-bounded martingale,
ii) N, = a for t > 1.
Prove furthermore that these conditions force
e\(N, N)\/2~\ - oo,
and conclude on the question raised at the beginning of the exercise. This provides
another example of a local martingale bounded in L2 which is nevertheless not a
martingale (see Exercise B.13)).
3°) This raises the question whether the fact that (*) is true for every bounded
H characterizes the L2-bounded martingales. Again the answer is in the negative.
For a > 0 define

§2. Conformal Martingales and Planar Brownian Motion 189
r > 1/2 :
ft ]
r > 1/2 : / A -syldB< =a\
Jl/2 i
By stopping /1/2A - s)~ldBs at time Sx for a suitable .J^-measurable r.v.
X, prove that there exists a cont. loc. mart. M for which (*) obtains for every
bounded H and E [M2] = E [(M, M)\] = oo. Other examples may be obtained
by considering filtrations (.5f) with non trivial initial a -field and local martingales,
such that (M, M), is .^-measurable for every t.
A.25) Exercise. By using the stopping times Ta+ of Proposition C.9) Chap. Ill,
prove that in Proposition A.5) the C-continuity cannot be omitted.
A.26) Exercise. A cont. loc. mart, with increasing process t л T where T is a
stopping time, is a BM stopped at T.
A.27) Exercise. A time-changed uniformly integrable martingale is a uniformly
integrable martingale, even if the time change takes on infinite values.
§2. Conformal Martingales and Planar Brownian Motion
This section is devoted to the study of a class of two-dimensional local martingales
which includes the planar BM. We will use the complex representation of 1R2; in
particular, the planar BM will be written В — B] +iB2 where (fi\ B2) is a pair of
independent linear BM's and we speak of the "complex Brownian motion". More
generally, we recall from Sect. 3 in Chap. IV that a complex local martingale is
a process Z = X + iY where X and Y are real local martingales.
B.1) Proposition. If Z is a continuous complex local martingale, there exists a
unique continuous complex process of finite variation vanishing at zero denoted
by (Z, Z) such that Z2 - (Z, Z) is a complex local martingale. Furthermore, the
following three properties are equivalent:
i) Z2 is a local martingale;
ii) (Z,Z)=0;
Ш) (X, X) = (Y, Y) and (X, Y) = 0.
Proof It is enough to define (Z, Z) by С х C-linearity, that is
(Z, Z) = (X + iY,X + iY) = (X, X) - (Y, Y) + 2i{X, Y).
Plainly, the process thus defined enjoys all the properties of the statement and
the uniqueness follows from the usual argument (Proposition A.12) in Chap. IV)
applied to the real and imaginary parts.
B.2) Definition. A local martingale satisfying the equivalent properties of the
above statement is called a conformal local martingale (abbreviated to conf. loc.
mart.).

190 Chapter V. Representation of Martingales
Obviously, the planar BM is a conf. loc. mart, and if Я is a complex-valued
locally bounded predictable process and Z a conf. loc. mart., then U, = /0' HsdZs
is a conf. loc. mart.
For a conf. loc. mart. Z, one sees that (Re Z, Re Z) = \{Z, Z); in particular,
{U, 0), = /J \Hs\2d(Z, Z)s. Moreover, Ito's formula takes on a simpler form.
Let us recall that ¦§-_ = \ (± - i-^) and J= = \ (± + /|Л and that a function
F : С -»¦ С which is differentiable as a function of both variables x and y, is
holomorphic if and only if Щ- = 0, in which case we set F' — |r •
B.3) Proposition. //"Z й a conf. loc. mart, and F a complex function on С which
is twice continuously differentiable (as a function of two real variables) then
—(zs)dzs+ / —(zorfz,
3z Jo dz
+ \ I AF(Z,)d(Z,Z),.
4 Л
In particular. ifF is harmonic, F(Z) is a local martingale and, ifF is holomorphic,
F(Z,) = F(Z0)+ [ F'(Z,)dZt.
Jo
Proof. Straightforward computations using Ito's formula. ?
Remark. If Z is conformal and F is holomorphic, F(Z) is conformal. We will
give shortly a more precise result in the case of BM2.
We now rewrite Theorem A.9) in the case of conf. loc. martingales.
B.4) Theorem. If Z is a conformal local martingale and Zo = 0, there exists
(possibly on an enlargement of the probability space) a complex Brownian motion
В such that
Z, — B{X.x)r
Proof Since (X. X) = (Y. Y) and (X. Y) = 0, Theorem A.9), applied to the 2-
dimensional local martingale (X, Y), implies the existence of a complex Brownian
motion В such that, for / < (X, X)^,
B, = XTl +iYTl
where T, — inf{« : (X, X)u > t}. The result follows as in the proof of Theorem
A.6). ?
The foregoing theorem has a very important corollary which is known as the
conformal invariance of complex Brownian motion.

§2. Conformal Martingales and Planar Brownian Motion 191
B.5) Theorem. If F is an entire and поп constant function, F{B,) is a time-
changed BM. More precisely, there exists on the probability space of В a complex
Brownian motion В such that
where (X. X), = /0' \F'{Bs)\2ds is strictly increasing and (X, X)x = oo.
Proof. If F is an entire function, F2 is also an entire function and by Proposition
B.3), F2(B) is a loc. mart. As a result, F{B) is a conformal local martingale to
which we may apply Theorem B.4). By the remarks before Proposition B.3) and
the Proposition itself, for X — Re F(B,), we have
(X,X),= f \F'(Bs)\2ds.
Jo
As F' is entire and non identically zero, the set Г of its zeros is countable;
therefore P (/0°° \r(Bs)ds = 0) = 1 and (X, X) is strictly increasing.
It remains to prove that (X, X)x = oo; the proof of this fact will require some
additional information of independent interest. What we have proved so far is that
F{B) has the same paths as a complex BM but possibly run at a different speed.
The significance of the fact that (X, X) is strictly increasing is that these paths are
run without gaps and this up to time {X, A')oo. When we prove that (X, A')oo — oo,
we will know that the paths of F(B) are exactly those of a BM. The end of the
proof of Theorem B.5) is postponed until we have proved the recurrence of planar
BM in Theorem B.8). П
We begin with a first result which is important in its own right. We recall
from Sect. 2 Chap. Ill that hitting times TA are stopping times so that the events
{TA < oo} are measurable.
B.6) Definition. For a Markov process with state space E, a Borel set A is said
to be polar if
P. [TA < oo] — 0 for every z 6 E.
B.7) Proposition. For the BM in Rd with d > 2, the one-point sets are polar sets.
Proof. Plainly, it is enough to prove the result for d — 2, and because of the
geometrical invariance properties of BM, it suffices to show that the planar BM
started at 0 does not hit the point set {(-1, 0)}. By what we already know, the
process Mt — exp(fi,) — 1 may be written BAi where В is a planar BM and
A, = f^expBXs)ds where X is the real component of B. The process A is
clearly strictly increasing. We also claim that A,*, = oo a.s. Otherwise, M would
converge in С as t tends to infinity; since |exp(B,)| = exp(X,) where X, as a
linear BM, is recurrent, this is impossible. As a result, the paths of M are exactly
the paths of a BM (run at a different speed) and, since exp(fi,) never vanishes,
the result is established.

192 Chapter V. Representation of Martingales
Remarks. 1°) For BM1, no non empty set is polar, whereas for BM2, d > 2, all
one-point sets, hence all countable sets are polar and, for instance, if we call Q2
the set of points in Ш2 with rational coordinates, the Brownian path {B,, t > 0}
is a.s. contained in K2\Q2. But, there are also uncountable polar sets even in the
case d — 2.
2°) Another more elementary proof of Proposition B.7) was given in Exercise
A.20) of Chap. I and yet another is given in Exercise B.14).
We may now state the recurrence property of the planar BM, another proof of
which is given in Exercise B.14).
B.8) Theorem. Almost surely, the set {t : В, е U) is unbounded for all open
subsets U o/R2.
Proof. By using a countable basis of open balls, it is enough to prove the result
whenever U is the ball B(z, r) where z = x + iy and r is > 0.
Since one-point sets are polar, we may consider the process M, = log \B, — z
which by the same reasoning as above is equal to pAi, where /3 is a linear BM
started at log|z| and A, = /0' \BU - z\~2du. But since supJ<r Ms is larger than
sups<, log |Xs —x\ which goes to infinity as t tends to +oo, it follows from Remark
3°) after Proposition A.8) that A, converges to infinity. As a result, inf, M, = +oo
a.s., hence M takes on values less than logr at arbitrary large times which ends
the proof.
Remarks. Г) One can actually prove that for any Borel set A with strictly positive
Lebesgue measure, the set {г : В, е A] is a.s. unbounded and in fact of infinite
Lebesgue measure (see Sect. 3 Chap. X).
2°) The above result shows that a.s., the Brownian path, which is of Lebesgue
measure zero, is dense in the plane.
We can now turn to the
B.9) End of the proof of Theorem B.5). If we had (X, X)oc < oc, then F(B,)
would have a limit as / tends to infinity. But since F is non constant, one can
find two disjoint open sets U\ and U2 such that F(U\) П F(t72) = 0 and as
{t : B, e U\} and {t : B, e U2] are unbounded, F(B,) cannot have a limit as t
tends to infinity. ?
We now state one more result about recurrence. We have just seen that the
BM in W1 is recurrent for d — 1 and d — 2. This is no longer true for d > 3, in
which case the BM is said to be transient. More precisely we have the
B.10) Theorem. If d > 3, lim,^^ \B,\ = +oo almost surely.
Proof. It is clearly enough to prove this when d = 3 and when В is started
at jc0 Ф 0. Since {0} is a polar set, by Ito's formula, \/\B,\ is a positive loc.
mart., hence a positive supermartingale which converges a.s. to a r.v. H. By
Fatou's lemma, EXo[H] < Hm?^0[l/|fi,|]. But, the scaling property shows that

§2. Conformal Martingales and Planar Brownian Motion 193
E.4U/\B,\] = Ofl/V?). As a result, H = 0 PXo-a.s. and the proof is now easily
completed. a
We close this section with a representation result for the complex BM 8 which
we study under the law Pa for а ф 0. Since 8, ф 0 for all t Pa-a.s., we may
choose a continuous determination в, (to) of the argument of B, (to) such that 9q(co)
is a constant and e'til> = a/|a|. We then have 5, — p, exp(/0,) and 0 is adapted
to the filtration of B,. The processes p, and в, may be analysed in the following
way.
B.11) Theorem. There is a planar BM (p\ y) such that
p, = \a\ exp(?c,)¦ в, = 6»o + Kc,,
w/геге C, = /0' p~2ds. Moreover, .7^ = .%?, hence у is independent of p.
Proof. Because В almost-surely never vanishes, we may define the conformal
local martingale H by
н,= f B;{dBs
Jo
and we have (Re H, Re H), — C,. Applying the integration by parts formula to
the product B, exp(—#,), it is easily seen, since (8, B) — 0, that 8, = aexp(#,).
By Theorem B.4), there is a planar BM which we denote by /5 + iy such that
H, = f$c, + iyc, which proves the first half of the statement.
The process /5 is the DDS Brownian motion of the local martingale Re H, =
log(p,/|a|). But with X = Re 8, Y = 1m B,
Jo
0 PJ Jo Pa
where Д is a real BM. We may rewrite this as
logp, =log|a|+ /
Jo
with a(x) = e~"; it follows from Proposition A.11) that .%? = -7X which is the
second half of the statement. ?
This result shows that, as one might expect, the smaller the modulus of Z, the
more rapidly the argument of Z varies. Moreover, as в, is a time-changed BM, it
is easy to see that
lim в, = —oo, lim 9, = +oo a.s.
Thus, the planar BM winds itself arbitrarily large numbers of times around 0, then
unwinds itself and does this infinitely often. Stronger asymptotic results on 9, or,
more generally, on the behavior of planar BM will be given in Chaps. VII and
XIII.

194 Chapter V. Representation of Martingales
At the cost of losing some information on the modulus, the preceding result
may be stated
B, = P,eiy"
where у is a linear BM independent of p and C, = /0' p~2ds. This is known as
the "skew-product" representation of two-dimensional Brownian motion.
It is interesting to stress the fact that we have worked under Pa with а ф 0. For
a — 0 and for t > 0, we may still, by the polarity of {0}, write a.s. unambiguously
B: = p, ¦ //,, with \Mt\ = 1. But we have no means of choosing a continuous
determination of the argument of '/6, adapted to (.%). This example hints at the
desirability of defining and studying semimartingales on open subsets of R+.
B.12) Exercise. Let A — {z : |Im z\ < n/2}; compute the law of BXT for the
complex Brownian motion Bl + iB2 started at 0.
[Hint: The exponential function maps A onto the half-plane H. Use the exit
distribution for H.]
B.13) Exercise (An important counterexample). 1°) Let В be the BM in M.d
with d > 3 started at x ф 0. Prove that \B,\2~d is a local martingale.
[Hint: \x\2-d is harmonic in Mrf\{0}.]
2°) Use Г) in the case d = 3 to give an example of a local martingale which
is bounded in L2 but is not a true martingale. This gives also an example of a uni-
uniformly integrable local martingale which is not a true martingale and an example of
a uniformly integrable -ipermartingale X such that the set {XT; T stopping time}
is not uniformly integrubt^. One can also find for every p > 1 a local martingale
bounded in Lp which is not a martingale, as is seen in Exercise A.16) of Chap.
XI.
3°) If В is the complex BM, let M = log|fi| and prove that for e > 0 and
a < 2
sup Ex [expa|W,|] < oo.
r<i<\
This provides an example of a local martingale with exponential moments which
is not a martingale. Furthermore, EX[{M, M)t] = oo, which bears out a remark
made • fter Corollary A.25) in Chap. IV.
B.14) Exercise. P) Retain the situation of 1°) in the preceding exercise and let
a and b be two numbers such that 0 < a < \x\ < b. Set Ra = Тщо.а) and
5/, — T/noj,)., and prove that
Px [Ra < Sb] = (\x\2-d - b2~d) I (a2-" - b2~«).
[Hint: Stop the local martingale \B,\2~d at Ra л Sh and proceed as in Chap.
П.]
Prove that Px[Ra < oo] = (a/\x\)d-2.
2°) Using the function log jx| instead of \x\2~d, treat the same questions for
d = 2. Base on this another proof that one-point sets are polar.

§2. Conformal Martingales and Planar Brownian Motion 195
3°) Deduce from 2°) that for d — 2, any open set U and any z, P:[Tu < oo] =
1.
4C) Let D\ and D2 be two disjoint disks and define inductively a double
sequence of stopping times by
7", = T[h, U\ = inf [t > TUB, e D2}.
Tn = inf{/ > ?/„_,, fi, efl,|, ?/„ = inf{f > Tn, В, е D2).
Prove that, for any z and every л,
Р:[Т„ < oo] = р.[?/„ < oo] = 1
and deduce therefrom another proof of Theorem B.8).
B.15) Exercise. In the situation of Theorem B.11) prove that the filtration gen-
generated by 0, is the same as that of B,.
[Hint: p is .^-adapted.]
B.16) Exercise. Let B, be the complex BM and suppose that fio = 1 a.s. For
r > 0, let
Tr = inf{f :\B,\=r}.
Prove that, if в, is the continuous determination of the argument of B, which
vanishes for t — 0, then 9ji is either for r < 1 or for r > 1, a process with
independent increments and that the law of От, is the Cauchy law with parameter
| log r |.
[Hint: Use the results in Proposition C.11) of Chap. III.]
B.17) Exercise. 1°) (Liouville's theorem). Deduce from the recurrence of BM2
and the martingale convergence theorem that bounded harmonic functions in the
whole plane are constant.
[Hint: See the reasoning in Proposition C.10) of Chap. X.]
2°) (D'Alembert's theorem). Let P be a non constant polynomial with com-
complex coefficients. Use the properties of BM2 to prove that, for any e > 0, the
compact set {z : |P(z)| S e] is non empty and conclude to the existence of a
solution to the equation P(z) = 0.
B.18) Exercise. 1°) Let Z — X + iY be the complex BM started at -1 and
T = inf {Г : Y, =0. X, > 0}.
Prove that the law of logXr has a density equal to Bn cosh(x/2))~'.
[Hint: See Exercise C.25) in Chap. HI.]
2°) As a result, the law of XT is that of C2 where С is a Cauchy r.v..
[Hint: See Sect. 6 Chap. 0.]

196 Chapter V. Representation of Martingales
*# B.19) Exercise. We retain the notation of Exercise B.18) in Chap. IV.
1°) Let Г be a holomorphic function in an open subset U of С and to the
differential form F(z)dz. If Z is a conf. loc. mart, such that P[3f > 0 : Z, ^
U] = 0, then
/ w= I F(Zs)dZs = I F(Zs)odZs a.s..
Jz(O.t) J0 J0
where о stands for the Stratonovich integral.
2C) In the situation of Theorem B.11), we have
9,-во= f
JZiO.t
with со — (x2 + y2) ' {xdy - ydx) = \m(dz/z)-
У) In the same situation, let 5, be the area swept by the segment [0, Zu],
0 < и < t (see Exercise B.18) in Chap. IV). Prove that there is a linear BM
S independent of p such that 5, = 8a, where A, = | /0 p2ds. Another proof is
given in Exercise C.10) Chap. IX which removes the condition а ф 0.
B.20) Exercise. Г) Let / be a meromorphic function in С and A be the set of
its poles. For zo ^ A, prove that /(8) is a time-changed BM2 started at /(zo)-
2°) Let z be a non-zero complex number. If T = mf[t : \B,| = 1} prove that
BT has the same law under P, and Р\ц.
B.21) Exercise (Exit time from a cone). Retain the situation and notation of
Theorem B.11) with a = 1 and set в0 — 0. For n, m > 0, define
T = inf [и : 9„ <? [—и, m]}, г = inf {v : yv ? [-и, m]}.
1°) For S > 0, prove that E [p2s] = E [expBS2r)]. Prove that consequently
E[TS] < oo implies 2S(n + m) < n.
[Hint: Use the result in Exercise C.10) 2°) of Chap. П.]
2°) From the equality T — ft expBpu)du, deduce that
E[TS] < IE [ts expB52r)].
Conclude that E[TS] < oo if and only if 2S(n +m) <n.
B.22) Exercise. Let С (со) - [B,(co).t > 0} where В is the BM2.
1°) Prove that C(co) is a.s. a Borel subset of Ш2. Call Л(ш) its Lebesgue
measure.
2°) Prove that
Л (со)
= / (s\ip\l:](B,))dx dy
Jr* V>o /
and conclude that A(co) = 0 a.s. Compare with the proof given in Exercise A.17)
Chap. I.

§2. Conformal Martingales and Planar Brownian Motion 197
B.23) Exercise. If Z is the complex BM, a complex C2-function / on R2 is
holomorphic if and only if
f(Z,) = f(Z0)+ [ HsdZs
Jo
for some predictable process H.
* B.24) Exercise. We say that Z e Y if for every e > 0, the process Z,+s, t > 0
is a (.>f+f)-conformal martingale (see Exercises A.42) and C.26) in Chap. IV).
1°) Prove that, if Z e Y, then a.s.,
ft) : lim IZ, (со) I < oo [ = \ со : lim Z, (со) exists in
uo I I /io
2°) Derive therefrom that for a.e. со, one of the following three events occurs
i) Hmf40 Z,(co) exists in C;
ii) lim,4o \Z,(co)\ — +00;
iii) for every S > 0, [Z,(co), 0 < / < 8} is dense in С
[Hint: For z e С and r > 0 define T — infj/ : \Z, — z\ < r) and look at
V, — (Zj - z) 1(г>о> which is an element of Y .]
Describe examples where each of the above possibilities does occur.
# B.25) Exercise. 1°) Let p, be the modulus of BM'', d > 2, say X, and v a
probability measure such that v({0}) = 0. Prove that, under Pv,
1 Г' , ,
B, = p, — Pa (d — \) 1 p. ds
2V Jo Hs
is a BM. In the language of Chap. IX, p, is a solution to the stochastic differential
equation
Pt = Po + B/ -\—(d — 1) I ps ds.
2 Jo
The processes p, are Bessel processes of dimension d and will be studied in Chap.
XL
2°) We now remove the condition on v. Prove that under Po the r.v. /0 p~[ds
is in L2 and extend Г) to v = ?0-
[Hint: p, - ps - \(d - 1) f' p;txdu = //(grad r(Xu),dXu) where r is the
distance to zero. Prove that the right-hand side is bounded in L2 as s tends to 0.]
* B.26) Exercise. Let / = (/', f2) be a continuous R2-valued deterministic func-
function of bounded variation such that /@) ф 0 and В = (X, Y) a planar BM@).
Set Z, = B, + f(t) and T = inf{/ : Z, = 0).
1°) Prove that /?, = /„' (Xj + ? (s))dXs + (Ys + f (s))dYs Js a ljnear BM
on [0, T[ (see Exercise C.28) Chap. IV).
2°) By first proving that E \f'l/n \Zs\-l\df(s)\~\ < 00, show that a.s.

198 Chapter V. Representation of Martingales
/'
Jo
Zs\ \df(s)\ < oo, for every t.
[Hint: Use the scaling property of В and the fact that, if G is a two-dimensional
Gaussian reduced r.v., then
sup?[|m + СГ1] < oo.]
3r) By considering log |Z|, prove that T is infinite a.s.. In other words, / is a
polar function in the sense of Exercise A.20) of Chap. I.
[Hint: Use Exercise A.18).]
If / is absolutely continuous, a much simpler proof will be given in Exercise
B.15) Chap. VIII.
B.27) Exercise. Let XisJ) be a complex Brownian sheet, namely Х(,,,} — W^s n +
iW*sl) where W[ and IV2 are two independent Brownian sheets. If y(s) —
(x(s), y(s)), s e [0, 1], is a continuous path in ]0, oo[x]0, oo[, write XY for
the process s —> X/ \.
^ (.VE).yU)j
1°) If x and v are both increasing (or decreasing), prove that the one-point
sets of С are polar for Xy.
2°) Treat the same question when x is increasing and у decreasing or the
reverse.
[Hint: Use Exercise A.13) in Chap. Ill and the above exercise.]
As a result, if у is a closed path which is piecewise of one of the four kinds
described above, the index of any a e С with respect to the path of XY is a.s.
defined.
§3. Brownian Martingales
In this section, we consider the filtration {-У[в) but we will write simply (.i^"). It
is also the filtration (->^?") of Sect. 2 Chap. Ill, where it was called the Brownian
filtration.
We call .7 the set of step functions with compact support in R+, that is, of
functions / which can be written
As in Sect. 3 of Chap. IV, we write <C* for the exponential of/0 f(s)dBs.
C.1) Lemma. The set lx?, f e j\ is total in L2(.Ko, P).

§3. Brownian Martingales 199
Proof. We show that if Y e L2(.WX, Р) and У is orthogonal to every ^, then the
measure У • P is the zero measure. To this end, it is enough to prove that it is the
zero measure on the <r-field а (ДГ| Bhi) for any finite sequence (t\, t,,).
The function <p(z\ z,,) — E exp EZ"=i ~j (вь ~ B>,i)) ' Y\ IS easi'y
seen to be analytic on C". Moreover, by the choice of У, for any A., € R, we
have
= 0.
V(k\ K,) = E expf ?a;(B,, -В,,.,)) У
Consequently, <р vanishes identically and in particular
E exp
У =0.
The image of У-P by the map a> -> (fir,(со) Bti(w) - Bti_,(w) ) is
the zero measure since its Fourier transform is zero. The measure vanishes on
a (Bh, ..., B, , - Bh,...)— a (Bu , Bh Bti!) which ends the proof.
C.2) Proposition. For any F e L2(.i^c, P), there exists a unique predictable
process H in L2(B) such that
F = E[F] +
f HsdBs.
Jo
Proof. We call -Ж the subspace of elements F in L2^^. P) which can be written
as stated. For F e Ж
[f>4
(*) E[Fl] = E[FY + E
Thus, if [F"} is a Cauchy sequence of elements of .Ж\ the corresponding sequence
{#"} is a Cauchy sequence in L2(B), hence converges to a predictable H e L~{B)\
it is clear that |F"} converges in L2(.K^, P) to
which proves that .WJ is closed.
On the other hand, Ж contains all the random variables Xj[ of Lemma C.1),
since, by Ito's formula, we have
/ X/
Jo
X/ = 1 + / X/f(s)dBs, for every / < oo.
Jo
This proves the existence of H. The uniqueness in L2(B) follows from the identity
(*). о

200 Chapter V. Representation of Martingales
Remark. If the condition H e L2(B) is removed, there are infinitely many pre-
predictable processes H satisfying the conditions of Proposition C.2); this is proved in
Exercise B.31) Chap. VI, but it may already be observed that by taking H = \[o,dT\
with dT — mf{u > T : Bu = 0}, one gets F = 0.
We now turn to the main result of this section, namely the extension of Propo-
Proposition C.2) to local martingales. The reader will observe in particular the fol-
following remarkable feature of the filtration {.^): there is no discontinuous (.%)-
martingale. Using Corollary E.7) of Chap. IV, this entails the
C.3) Corollary. For the Brownian filtration, every optional process is predictable.
The reader who is acquainted with the classification of stopping times will also
notice that all the stopping times of the Brownian filtration are predictable.
C.4) Theorem. Every {.lF~t)-local martingale M has a version which may be writ-
written
M, = C
LHs>
where С is a constant and H a predictable process which is locally in L2(B). In
particular, any (.^)-local martingale has a continuous version.
Proof. If M is an L2-bounded (.^)-martingale, by the preceding result, there is
a process H e L2(B) such that
M, = E [Мж | .Щ =
= E[MOO]+
HsdBs | .Ю
f HsdBs,
Jo
hence the result is true in that case.
Let now M be uniformly integrable. Since L2(.>^o) is dense in L'C^o) there
is a sequence of L2-bounded martingales M" such that lim,, E \\MX — M?-|] — 0.
By the maximal inequality, for every X > 0,
Thanks to the Borel-Cantelli lemma, one can extract a subsequence |M"'} con-
converging a.s. uniformly to M. As a result, M has a continuous version.
If now M is an (.i^~)-local martingale, it obviously has a continuous version
and thus admits a sequence of stopping times Tn such that MT" is bounded. By
the first part of the proof, the theorem is established. ?
It is easy to see that the above reasonings are still valid in a multidimensional
context and we have the

§3. Brownian Martingales 201
C.5) Theorem. Every (,F^B)-local martingale, say M, where В is the d-dimen-
sional BM (B' Bd) has a continuous version and there exist predictable pro-
processes H1, locally in L2(B), such that
Remarks. Iе) The processes H' are equal to the Radon-Nikodym derivatives
jt{M, B')t of (M, B') with respect to the Lebesgue measure. But, in most con-
concrete examples they can be computed explicitely. A fairly general result to this
effect will be given in Sect. 2 Chap. VIII. Exercises C.13), C.16) of this section
already give some particular cases. When / is harmonic, the representation of the
martingale f(B,) is given by Ito's formula.
2°) It is an interesting, and for a large part unsolved, problem to study the
filtration of the general local martingale obtained in the above results. The reader
will find some very partial answers in Exercise C.12).
The above results are, in particular, representation theorems for L2(.>^f). We
now turn to another representation of this space; for simplicity, we treat the one-
dimensional case. We set
Л„ = {(*, sn) eM"+:st > s2 > ...> s,,}
and denote by L2(A,,) the L2-space of Lebesgue measure on An. The subset En
of L2(An) of functions / which can be written
1
with fj € L2(R+) is total in L2(An). For / € ?„, we set
Mf) = f Msi)dBs, f ' f2(s2)dBS2.../""' fn{sn)dBSii.
Jo Jo Jo
This kind of iterated stochastic integrals has already been encountered in Propo-
Proposition C.8) in Chap. IV, and it is easily seen that
C.6) Definition. Forn > 1, the smallest closed linear subspace of L2(.K^) con-
containing Jn(En) is called the и-th Wiener chaos and is denoted by К„.
The map J,, is extended to L2(An) by linearity and passage to the limit. If /
is in the linear space generated by En it may have several representations as linear
combination of elements of Е„ but it is easy to see that Jn(f) is nonetheless defined
unambiguously. Moreover J,, is an isometry between L2(An) and Kn. Actually,
using Fubini's theorem for stochastic integrals (Exercise E.17) Chap. IV), the

202 Chapter V. Representation of Martingales
reader may see that J,,(f) could be defined by straightforward multiple stochastic
integration.
Obviously, there is a one-to-one correspondence between К„ and L2{An).
Moreover, the spaces Kn and Km are orthogonal if n ф m, the proof of which we
leave to the reader as an exercise. We may now state
C.7) Theorem. L2(.J^CB) = ф^° К„ where А" о is the space of constants. In other
words, for each Y e L2(.7^) there exists a sequence (/") where f" e L2(An)
for each n, such that
и = 1
in the L2-sense.
Proof. By Proposition C.8) in Chap. IV, the random variables %¦?, of Lemma C.1)
may be written Yl^ Jn{f") pointwise with f"(s\, ... ,sn) — f(s\)f(s2) ¦ ¦ . /(¦?„).
As / is bounded and has compact support it is easy to see that this convergence
holds in L2(.i^o). Thus, the statement is true for Kjc. It is also true for any linear
combination of variables K^. Since, by Lemma C.1), every r.v. Y e L2(.J/^) is
the limit of such combinations, the proof is easily completed.
Remark. The first chaos contains only Gaussian r.v.'s and is in fact the closed
Gaussian space generated by the r.v.'s B,, / > 0 (see Exercise C.11)).
We now come to another question. Theorem C.4) raises the following problem:
which martingales can be written as (H ¦ B), for a suitable Brownian motion B?
We give below a partial answer which will be used in Chap. IX.
C.8) Proposition. If M is a continuous local martingale such that the measure
d(M. M), is a.s. equivalent to the Lebesgue measure, there exists an {-^\M)-
predictable process f, which is strictly positive dt ® dP a.s. and an (-^M)-
Brownian motion В such that
d(M, M), = f,dt and M, = Mo + f f^l2dBs.
Proof By Lebesgue's derivation theorem, the process
/, = ]im n ((M, M), - (M. M),_i/,,)
n~>cc
satisfies the requirements in the statement. Moreover, (f,)~[/2 is clearly in L20C(M)
and the process
B, = f~u2dMs
Л
is a continuous local martingale with increasing process t, hence a BM and the
proof is easily completed. ?

§3. Brownian Martingales 203
If d{M. M), is merely absolutely continuous with respect to dt, the above
reasoning fails; moreover, the filtration (-У,м) is not necessarily rich enough to
admit an (:^M)-Brownian motion. However, if B' is a BM independent of .7^
and if we set
в, = f \a>0)f-l/2dMs+ 11,л=<»</я;.
Jo Jo
then, by Levy's characterization theorem, В is again a BM and M, — Mo +
/o' fs ^ dBs. In other words, the foregoing result is still true provided we enlarge
the probability space so as to avail ourselves of an independent BM. Using a little
Jinear algebra and an enlargement (Q,.%. P) of (J2,->^, P), this can be carried
over to the multi-dimensional case. We only sketch the proof, leaving to the reader
the task of keeping track of the predictability and integrability properties of the
processes involved when altered by algebraic transformations.
C.9) Theorem. Let M = (M1,..., Md) be a com. vect. loc. mart, such that
d{M'. M'), <g. dt for every i. Then there exist, possibly on an enlargement of the
probability space, a d-dimensional BM В and a d x d matrix-valued predictable
process a in 1^Ж(В) such that
M, — MQ+ I asdBs.
Jo
Proof. We may suppose that Mq — 0.
By the inequality of Proposition A.15) in Chap. IV, we have d{M', Af7), ^C
dt for every pair (г, j). The same argument as in the previous proof yields a
predictable process у of symmetric d x d matrices such that
'¦ J - [' U
' ~~ Jo Ys
and the matrix у is dP ®Л-а.е. semi-definite positive. As a result one can find a
predictable process /} with values in the set ofdxd orthogonal matrices such that
p = p'yfi is diagonal. Setting aJl = fi'J(pJJ)]/2 we get a predictable process such
that у = a'a = aa'. Of course some of the p^'s may vanish and the rank of a,
which is equal to the rank of y, may be less than d. We call ? the predictable
process which is equal to the rank of ys.
Define a matrix P( by setting p'^ — 1 if i; = j < ? and p'^ — 0 otherwise.
There exist a predictable process ф such that фх is a d x d orthogonal matrix
such that аф = афР,- and a matrix-valued process X such that Хаф — Р^. Set
N = X ¦ M; then N is a cont. vect. loc. mart, and (Л'', NJ), — 5' /Q' \[i<^\ds as
follows from the equalities
XyX' — Xotot'X' = Хафф'а'Х' = P(.
If we set X = (аф) ¦ N it is easily seen that {X - M, X - M) = 0, hence X = M.
If we now carry everything over to the enlargement (Q,.^\, P), we have at
our disposal a BMd W = (Wl, W2,..., W) independent of N and if we define

204 Chapter V. Representation of Martingales
w: =
Jo
then, by Levy's theorem, W is a BM''. As Д is an orthogonal matrix, В — E ¦ W
is again a BMrf (See Exercise C.22) Chap. IV) and M = (аф?г) • В. a
Remark. If the matrix j/ is «if ®Л-а.е. of full rank then, as in Proposition C.8),
there is no need to resort to an enlargement of the probability space and the
Brownian motion В may be constructed on the space initially given. Actually,
one can find a predictable process ty of invertible matrices such that d{M, M)s =
(\lrsij/l)ds and В = i/f • M.
C.10) Exercise. Prove that Proposition C.2) and Theorem C.4) are still true if В
is replaced by a continuous Gaussian martingale.
[Hint: See Exercise A.14).]
C.11) Exercise. 1°) Prove that the first Wiener chaos K\ is equal to the Gaussian
space generated by B, i.e. the smallest Gaussian space containing the variables
B,, t > 0.
2°) Prove that a .i^f-measurable r.v. is in K\ iff the system (Z, Bu t > 0) is
Gaussian. As a result there are plenty of .7%-measurable Gaussian r.v. which are
not in K\.
* C.12) Exercise. Let В be a BM'(O) and H an (.^"s)-progressive process such
that:
i) /0' Hjds < oo a.s. for every /;
ii) P [X [s : Hs — 0} = 0] — 1 where X is the Lebesgue measure on M.+. If sgn x =
1 for x > 0 and sgn x = — 1 for x < 0, prove that
-^(sgn H)B jrHB c T7-B
for every /. Observe that (sgn H) ¦ В is itself a Brownian motion.
[Hint: sgn H, — HS/\HS\; replace |tf| by a suitable -^HB-adapted process.]
# C.13) Exercise. 1°) Let / > 0 and let В be the standard linear BM; if / e
L2(R, g,(x)dx) prove that
P,sf)'(B,)dBs.
[Hint: Recall that || + ^^-f = 0 and look at Exercise A.11) in Chap. Ill
and Exercise A.20) in Chap. VIL]
2°) Let B' be an independent copy of B; for \p\ < 1 the process C, =
pB, + y/l — p2B't is a standard BM1. Prove that the process

§3. Brownian Martingales 205
has a measurable version Z and that if f f(x)gi(x)dx = 0
E[f{Bl)\Cl] = p / ZlP)dCs
Jo
where Zip) is the .^^-predictable projection of Z (see Exercise A.13)). Instead
of Z(/'\ one can also use a suitable projection in L2(ds dP).
3C) (Gebelein's Inequality) If (X, Y) is a centered two-dimensional Gaussian
r.v. such that E[X2] = E[Y2] = 1 and E[XY] = p, then for any / as in 2=) ,
E[(E[f(X)\Y]J]<p2E[f2(X)].
"The reader will look at Exercise C.19) for related results.
C.14) Exercise. Let В and B' be two independent standard BM1 and set for s,
/>0,
.Tsj=cj(Bii,u < s: B'v, v < t).
1°) Prove that, as / and g range through L2(R+), the r.v.'s
[' f(s)dBs) 6 (I g{s)dB'\
o /do WO /
are total in L2 (-i^coo)-
2°) Define a stochastic integral fQ /0 H(s, t)dBsdB't of suitably measurable
doubly indexed processes H, such that any r.v. X in L2 (.5^ oo) may be uniquely
written
/»OO /¦ OO /*OO /0
X = E[X]+ h(s)dBs+ ti(s)dB's+ / / H(s.t)dBsdBt.
Jo Jo Jo Jo
3°) Let Xt,, be a doubly-indexed process, adapted to .5^, and such that
i) sup,, E [X.y < +oo;
ii) E [XS'_,' | .Xs.r] = -X's.r a-s- whenever s < s' and ; < f'.
Prove that there is a r.v. X in L2 (.^.oc) such that X4if = ?" [X | .5^.,] a.s.
for every pair (s,i) and extend the representation Theorem C.4) to the present
situation.
C.15) Exercise. 1°) Prove that the family of random variables
/=i
where A., e R+ and the functions /, are bounded and continuous on R is total in
[Hint: The measures e, are the limit in the narrow topology of probability
measures whose densities with respect to the Lebesgue measure are linear combi-
combinations of exponentials.]
2°) Prove that Z has a representation as in Proposition C.2) and derive there-
therefrom another proof of this result.

206 Chapter V. Representation of Martingales
C.16) Exercise. Let / be fixed and ф be a bounded measurable function on Ш.
Find the explicit representation C.2) for the r.v. F = exp (/J (j>(B<)ds\.
[Hint: Consider the martingale E [F | .3^], и < I.]
C.17) Exercise. Let G be a centered Gaussian subspace of L2(E, К. P) and •'?
the sub-cr-algebra generated by G. Call Kn the closure in L2 of the vector space
generated by the set
[h,,(X):XeG,\\X\\2 = 1)
where hn is the n-th Hermite polynomial defined in Sect. 3 Chap. IV.
1°) The map X —> exp(X) is a continuous map from G into L2(E, '0 , P) and
its image is total in L2(E, l-0, P).
2°) If X and Y are in G and ||X||2 = ||K||2 = 1, prove that
E[hm(X)hn(Y)] = n!E[XY]" if m = n.
= 0 otherwise.
3°) Prove that
4°) If H is the Gaussian space of Brownian motion (see Exercise C.11)),
prove that the decomposition of 3°) is one and the same as the decomposition of
Theorem C.7).
C.18) Exercise (Another proof of Knight's theorem). Г) Let M be a contin-
continuous (.5^")-local martingale such that (M, M)x = oo and В its DDS Brownian
motion. Prove that, for any r.v. H e L2 (.^f), there is a (.^")-predictable process
К such that E [f™ K2d{M, M)s] < oo and H = E[H] + f™ KsdMs. As .Kjf
may be strictly larger than ,5^f, this does not entail that M has the PRP of the
following section.
[Hint: Apply C.2) to В and use time changes.]
2°) In the situation of Theorem A.9), prove that for H e L2(.s^) there exist
(.^-predictable processes K' such that
rK\dM\.
This can be proved as in 1 = ) by assuming Theorem C.4) or by induction from p).
3°) Derive Theorem A.9), i.e. the independence of the B''s from the above
representation property.
*# C.19) Exercise (Hypercontractivity). Let ц. be the reduced Gaussian measure of
density Bjr)~'/2exp(-jc2/2) on R. Let p be a real number such that \p\ < 1 and
set
Uf(y) = J f(py + л/1 - Р2х) ix(dx).

§3. Brownian Martingales 207
By making p = e~l/2, U is the operator given by the transition probability of the
Ornstein-Uhlenbeck process.
1°) Prove that for any p e [1. oo[ the positive operator U maps Lp(p.) into
itself and that its norm is equal to 1. In other words, U is a positive contraction
onL"(At).
This exercise aims at proving a stronger result, namely, ifl < p < q < qo =
(p — \)p~2 + 1, then U is a positive contraction from Lp(/jL) into Lq([i). We call
q the conjugate number of q i.e. \/q + \/q' = 1.
2°) Let (Y,, Y't) be a planar standard BM and set X, = pY, + sj\ - p2Y't. Prove
that X is a standard linear BM and that for / > 0,
Observe that (Xi, У1) is a pair of reduced Gaussian random variables with corre-
correlation coefficient p and that U could have been defined using only that.
3°) Let / and g be two bounded Borel functions such that f > s and g > s
for some e > 0. Set M = fp{Xx), N = g4'(Y\), a = \/p, b = \/q'. Using the
representation result C.2) on [0, 1] instead of [0, 00], prove that
E[M"Nh]<\\f\\p\\g\\q,
Since also E [M"Nb] = f gUf d/л derive therefrom the result stated in 1°).
4°) By considering the functions f(x) = ехр(гд:) where zeK, prove that for
q > qo, the operator U is unbounded from Lp{jx) into Lq(n).
5°) (Integrability of Wiener chaoses) Retaining the notation of exercise C.17)
prove that Uhn — p"hn and derive from the hypercontractivity property oft/ that
if 2 e Kn, then for every q > 2,
\\Z\\q < (q - \)"/2\\Z\\2.
Conclude that there is a constant a* > 0 such that
E [exp (aZ2'")] < 00 if a < a*,
?[exp(aZ2/")] = 00 if a > a*.
C.20) Exercise. Let (Q, ¦?,, P) be a filtered probability space, В a .?;-BM van-
vanishing at zero and (,7i) = (.7^B).
Г) If M is a (.'?,)-martingale bounded in L2, prove that X, = E [M, | .i^] is
a (.>^~)-martingale bounded in L2 which possesses a continuous version. It is the
version that we consider henceforth.
2°) Prove that (M, B), = /0' asds where a is ('^-adapted. Prove that the
process 1 -*¦ E [a, \ .Щ has a (..^-progressively measurable version H and that
Jo
X, = E[M0]+ / HsdBs.
Jo
[Hint: For the last result, compute E[M,Yt] in two different ways where Y
ranges through the square-integrable (.5^)-martingales.]

208 Chapter V. Representation of Martingales
3°) If U is (.'?,)-progressively measurable and bounded
e\ f UsdBs \.?\= f E[US
Uo J Jo
If B' is another (-?,)-BM independent of В then
E \ f UsdB's | .t\ = 0.
Uo J
C.21) Exercise. 1°) If В is a BM'(O) and we set .Vt — .^"B, prove that if -0 is
a non-trivial sub-ст-field of .Vx, there does not exist any (.^")-BM which is also
a (.i^V .'()-BM.
2°) In contrast, prove that there exists a non-trivial sub-u-field '¦$ of .7~Jf and
a (.>^")-martingale which is also a (.^"v '!?)-martingale.
[Hint: Let # = .Kif where fi is the DDS Brownian motion of 1<в<0) • В and
take M = l(B>0) • В.]
* C.22) Exercise (The Goswami-Rao Brownian filtration). Let В be the stan-
standard BM and define .3Kt as the ст-field generated by the variables <P(Bs,s < t)
where Ф is a measurable function on C([0, f], K) such that <P(B,.s < t) =
<P(—BS, s < t).
Г) Prove that the inclusions .>f|B| С -^ С J^"B are strict.
[Hint: For s < t, the r.v. sgn (BsBt) is .^-measurable but not .Ур -
measurable.]
2°) Let Y e L2(.7^D) with t < oo. Prove that in the notation of Theorem C.7),
ОС
E [Y | .Ж,] = E[Y] + J2 J2P(fp)(t)
/7=1
where the notation J,,(f")(t) indicates that the first integral in J,,(f") is taken
only up to time t.
Deduce therefrom that every (.^,)-martingale is a (.^"B)-martingale.
3°) Prove that consequently every (.Ж,)-martingale M may be written
M,=c+
/'
Jo
where f}t — /0' sgn (Bs)dB, is а (.Ж',)-ВМ and m is (.^)-predictable.
The filtration .Я^|В| is in fact generated by fi (Corollary B.2) Chap. VI); that
(¦y^t) is the natural filtration of a BM is the content of the following question.
4°) Let (f,,),,ez be an increasing sequence of positive reals, such that ln —> 0,
/i^ -OG
and tn —>• +oo. Prove that (.Ж,) is the natural filtration of the BM y, =
/J fisdBs, where ц, = sgn (Btii - ?,„_,), for s e]rw> ?„+,].

§4. Integral Representations 209
§4. Integral Representations
D.1) Definition. The cont. loc. mart. X has the predictable representation property
(abbr. PRP) if, for any (.^x)-local martingale M, there is an (.lFtx у predictable
process H such that
M, = Mo + / HsdXs.
Jo
In the last section, we proved that the Brownian motion has the PRP and, in
this section, we investigate the class of cont. loc. martingales which have the PRP.
We need the following lemma which is closely related to Exercise E.11) in
Chap. IV.
D.2) Lemma. If X is any cont. loc. mart., every {.^x)-continuous local martin-
martingale M vanishing at 0 may be uniquely written
M = H -X + L
where H is predictable and (X, L) — 0.
Proof. The uniqueness follows from the usual argument.
To prove the existence of the decomposition, let us observe that there is a
sequence of stopping times increasing to infinity and reducing both M and X.
Let T be one of them. In the Hilbert space Hq, the subspace G = {H ¦ XT;
H 6 5?j,(Xr)} is easily seen to be closed; thus, we can write uniquely
MT = H-XT +1
where L 6 GL. For any bounded stopping time 5, we have
E [XTSLS] = E [XTsE[lx | .Щ]] = E [XTSLX] = 0
since ХГл5 e G. It follows from Proposition C.5) Chap. II that X1L is a martin-
martingale, hence (XT, L) = (X, L)T = 0.
Because of the uniqueness, the processes H and L extend to processes H and
L which fulfill the requirements of the statement. ?
From here on, we will work with the canonical space W = C(R+, R). The
coordinate process will be designed by X and we put .7^° — a(Xs, s < t). Let
S% be the set of probability measures on W such that X is a local martingale
(evidently continuous). If P e .Ж, (.^p) is the smallest right-continuous filtration
complete for P and such that .3^"° с .Yt?. The PRP now appears as a property
of P: any (.i^"p)-local martingale M may be written M = H X where H is
GS^"/>)-predictable and the stochastic integration is taken with respect to P.
We will further designate by ,Ж the subset of -96 of those probability measures
for which X is a martingale. The sets .Ж and ^ are convex sets (see Exercise
A.37) in Chap. IV). We recall the

210 Chapter V. Representation of Martingales
D.3) Definition. A probability measure P of .76' (resp. .%) is called extremal if
whenever P = а Л + A -a)P2 with 0 < a < 1 and P\, P2 e Ж (resp. Ж) then
P = pl = p2.
We will now study the extremal probability measures in order to relate ex-
tremality to the PRP. We will need the following measure theoretical result.
D.4) Theorem (Douglas). Let (Q. .>*") be a measurable space, '/* a set of real-
valued .Js~ -measurable functions and *~/>* the vector-space generated by 1 and '/..
lf-Ж-/, is the set of probability measures /л on (J2, .W) such that % С L' (p.) and
J f d/i — 0 for every f e ^, then Ж% is convex and [i is an extremal point of
Ж/, if and only if '/** is dense in L'(//).
Proof, That .Ж/ is convex is clear.
Suppose that %* is dense in L'(/x) and that /x = av\ + A — a)v2 with
0 < a < 1. Since v, = h\\i for a bounded function ft,-, the space _S?* is dense in
L](vj) for / = 1,2. Since clearly v\ and v2 agree on %*, it follows that v\ = v2.
Conversely, suppose that /Z* is not dense in L'(^). Then, by the Hahn-Banach
theorem there is a non-zero bounded function h such that f hf d/i = 0 for every
/ 6 '/?* and we may assume that \\1г\\ж < 1/2. The measures v± — A ±Л)д are
obviously in .Ж</ and /x being equal to (v+ + v~)/2 is not extremal. a
If we now consider the space (W, .'K?) and choose as set '/, the set of r.v.'s
1A(X, - Xs) where 0 < s < t and A ranges through .i^"°, the set .Ж'/ of
Theorem D.4) coincides with the set Ж of probability measures for which X is
a martingale. We use Theorem D.4) to prove the
D.5) Proposition. If P is extremal in Ж, then any (.J^p)-local martingale has a
continuous version.
Proof. Plainly, it is enough to prove that for any r.v. Y e Ll(P), the cadlag
martingale EP[Y \ ¦7'tp\ has in fact a continuous version. Now, this is easily seen
to be true whenever Y is in 'A*. If Y e Ll(P), there is, thanks to the preceding
result, a sequence (Yn) in %* converging to Y in Ll(P). By Theorem A.7) in
Chap. 11, for every e > 0, every t and every и,
P Lp | EP [Yn | .Vsp] -EP[Y\ .^p}\ >e]<e-lE[\Yn-Y\].
By the same reasoning as in Proposition A.22) Chap. IV, the result follows. ?
D.6) Theorem. The probability measure P is extremal in Ж if and only if P has
the PRP and .7~/ is P-a.s. trivial.
Proof. If P is extremal, then .Wup is clearly trivial. Furthermore, if the PRP did
not hold, there would, by Lemma D.2) and the preceding result, exist a non
zero continuous local martingale L such that {X, L) = 0. By stopping, since
(X, LT) = (X, L)T, we may assume that L is bounded by a constant k; the

§4. Integral Representations 211
probability measures P, = A + (Lx/2k)) P and P2 = A - (Lx/2k)) P are both
in Ж as may be seen without difficulty, and P = (P\ + Pi)/2 which contradicts
the extremality of P.
Conversely, assume that P has the PRP, that .3^p is P-a.s. trivial, and that
p = aP\ + A - a)P2 with 0 < or < 1 and P, e Ж'. The P-martingale ffl\ ?-
has a continuous version L since P has the PRP and XL is also a continuous
P-martingale, hence (X, L) — 0. But since P has the PRP, L, = Lo +/J Я,ЛХ.„
hence (X, L), = /0' H,d{X,X)s; it follows that P-a.s. Hs = 0, d(X. XL-a.e.,
hence /„' HsdXs — 0 a.s. and L is constant and equal to Lo. Since -3^p is P-a.s.
trivial, L is equal to 1, hence P — P\ and P is extremal.
Remark. By the results in Sect. 3, the Wiener measure W is obviously extremal,
but this can also be proved directly. Indeed, let Q e .Ж be such that j2«ff.
By the definition of (X, X) and the fact that convergence in probability for W
implies convergence in probability for Q, (X, X), — t under Q. By P. Levy's
characterization theorem, X is a BM under Q, hence Q — W which proves that
W is extremal. Together with the results of this section, this argument gives another
proof of Theorem C.3).
We will now extend Theorem D.6) from Ж to .Ж. The proof is merely
technical and will only be outlined, the details being left to the reader as exercises
on measure theory.
D.7) Theorem. The probability measure P is extremal in Ж if and only if it has
the PRP and .Тйр is P-a.s. trivial.
Proof. The second part of the proof of Theorem D.6) works just as well for .Ж
as for .Ж; thus, we need only prove that if P is extremal in Ж, then .^p is a.s.
trivial, which is clear, and that P has the PRP.
Set Tn — inf {/ : |X,| > n}; the idea is to prove that if P is extremal in .Ж,
then the laws of XT" under P are extremal in Ж for each n and hence the PRP
holds up to time Т„, hence for every /.
Let n be fixed; the stopping time Т„ is a (.^"°)-stopping time and the о -algebra
•^n° is countably generated. As a result, there is a regular conditional distribution
Q(u>, ¦) of P with respect to .7~T®. One can choose a version of Q such that for
every to, the process X, — ХГлА, is a Q{co, -)-local martingale.
Suppose now that the law of XT" under P, say PT\ is not extremal. There
would exist two different probabilities n\ and л2 such that PT" — ая, +A -а)тт2
for some а е]0, 1[ and these measures may be viewed as probability measures
on J^-0. Because .Щ? v вт\.7?) = .^, it can be proved that there are two
probability measures P, on .5^° which are uniquely defined by setting, for two
r.v.'s H and К respectively .Щ® and 6^'('5^)-measurable,
I HK dPi = I TTi{dco)H{co) f Q(co,dco')K(a)').

212 Chapter V. Representation of Martingales
Then, under P,, the canonical process X is a continuous local martingale and
P = aP\ + A — a)/>2 which is a contradiction. This completes the proof.
The equivalent properties of Theorem D.7) are important in several contexts,
but, as a rule-in contrast with the discrete time case (see the Notes and Comments)
- it is not an easy task to decide whether a particular local martingale is extremal
or not and there is no known characterization - other than the one presented in
Theorem D.7) - of the set of extremal local martingales. The rest of this section
will be devoted to a few remarks designed to cope with this problem. As may be
surmised from Exercise C.18), the PRP for X is related to properties of the DDS
Brownian motion of X.
D.8) Definition. A cont. loc. martingale X adapted to a filtration (.^) is said to
have the (.^)-PRP if any (.^)-local martingale vanishing at zero is equal to H ¦ X
for a suitable (.^^-predictable process H.
Although .7^x С .7[, the (J^)-PRP does not entail the PRP as the reader will
easily realize (see Exercise D.22)). Moreover, if we define (.^")-extremality as
extremality in the set of probability measures Q such that (Xt) is an {.Vt, Q)-local
martingale, the reader will have no difficulty in extending Theorem D.7) to this
situation.
In what follows, X is a P-continuous local martingale; we suppose that
{X, X)co = oo and call В the DDS Brownian motion of X.
D.9) Theorem. The following two properties are equivalent
i) X has the PRP;
ii) В has the (.7~T*)-PRP.
Proof. This is left to the reader as an exercise on time-changes.
We turn to another useful way of relating the PRP of X to its DDS Brownian
motion.
D.10) Definition. A continuous local martingale X such that (X, X)^ = oo is
said to be pure if, calling В its DDS Brownian motion, we have
By Exercise A.19), X is pure if and only if one of the following equivalent
conditions is satisfied
i) the stopping time T, is .S^f-measurable for every t\
ii) (X, X), is J?f -measurable for every t.
One will actually find in Exercise D.16) still more precise conditions which
are equivalent to purity. Proposition A.11) gives a sufficient condition for X to be
pure and it was shown in Theorem B.11) that if p, is the modulus of BM2 started
at а Ф 0 then log pt is a pure local martingale.

§4. Integral Representations 213
Finally, the reader will show as an exercise, that the pure martingales are
those for which the map which sends paths of the martingale into paths of the
corresponding DDS Brownian motion is one-to-one.
The introduction of this notion is warranted by the following result, which can
also be derived from Exercise C.18). A local martingale is said to be extremal if
its law is extremal in Ж.
D.11) Proposition. A pure local martingale is extremal.
Proof. Let P be a probability measure under which the canonical process X is a
pure loc. mart. If Q e .36 and Q <? P, then (X, X), computed for P is a version
of {X, X), computed for Q and consequently the DDS Brownian motion of X for
P, say P, is a version of the DDS Brownian motion of X for Q. As a result, P and
Q agree on <т(Д,, s > 0) hence on the completion of this о -algebra with respect
to P. But, since P is pure, this contains <j(Xs, s > 0) and we get P — Q. a
The converse is not true (see Exercise D.16)) but, as purity is sometimes easier
to prove than extremality, this result leads to examples of extremal martingales
(see Exercise C.11) in Chap. IX).
D.12) Exercise. With the notation of this section a probability measure P e .7%
is said to be standard if there is no other probability measure Q in .Ж' equivalent
to P. Prove that P is standard if and only if P has the PRP, and ,7^p is f-a.s.
trivial.
D.13) Exercise. Let (fi1, B2) be a two-dimensional BM. Prove that X, =
/„' BlsdB2 does not have the PRP.
[Hint: (BfJ — t is a (.^"x)-martingale which cannot be written as a stochastic
integral with respect to X.]
D.14) Exercise. Let X be a cont. loc. mart, with respect to a filtration {.^).
1°) Let Л be the space of real-valued, bounded functions with compact support
in K+. Prove that if, for every /, the set of r.v.'s
«-*(?
f(s)dXi\ ,
is total in L2(.7i. P), then .V/ is P-a.s. trivial and X has the (.i^")-
2°) Prove the same result with the set of r.v.'s
Hn (f f(s)dXs, f f2(s)d{X,
/ 6 A, n e N.
where Hn is the Hermite polynomial defined in Sect. 3 Chap. IV.
D.15) Exercise. 1°) If В is a standard linear BM, (,Ц) is the Brownian filtra-
filtration and Я is a (,j^)-predictable process a.s. strictly positive with the possible
exception of a set of Lebesgue zero measure (depending on со), then the martingale

214 Chapter V. Representation of Martingales
M, = I HsdBs
has the PRP.
[Hint: Use ideas of Proposition C.8) to show that .7^ — -VJ?.]
2°) In particular, the martingales M" = /0' B"dBs, n e N, are extremal. For
n odd, these martingales are actually pure as will be proved in Exercise C.11)
Chap. IX.
3°) Let Г be a non constant square-integrable (.3^")-stopping time; prove that
Y, = B, — Bj is not extremal.
[Hint: Г is a .^'''-stopping time which cannot be expressed as a constant plus
a stochastic integral with respect to Y.]
*# D.16) Exercise. (An example of an extremal local martingale which is not
pure). 1°) Let (.5f) and ('.?,) be two filiations such that .5^" с -Pt for every t.
Prove that the following two conditions are equivalent:
i) every (,3^)-martingale is a (&,)-martingale;
ii) for every t, the ст-algebras -7^ and '5, are conditionally independent with
respect to .5^.
If these conditions are in force, prove that -3^" = .'?, n.J^.
2°) Let M be a continuous (Jf Hoc. mart, having the (.>^")-PRP. If (.'^) is a
filtration such that J^"c c^ С -З^о for every t and M is a (.'?,)-local martingale,
then (.3T) = (¦%).
3°) Let M bea cont. loc. mart, with (M, M)x = oo and В is its DDS
Brownian motion. Prove that M is pure if and only if (M, M), is, for each t, an
(.5^"B)-stopping time.
[Hint: Prove first that, if M is pure, then ,TtB = .Утм.]
Prove further that if M is pure, (M, M)T is a (.5^B)-stopping time for every
(.i?"Af)-stopping time T (this result has no bearing on the sequel).
4°) Prove that M is pure if and only if .VtM = .J^ M for every t.
5°) From now on, fi is a standard BM and we set /3, — fQ sgn(ps)dps. In
Sect. 2 of Chap. VI it is proved that ft is a BM and that ,V^ — .5?"l/f| for every t.
Prove that fi has the (.5^)-PRP.
6°) Set T, = /J B + (Д5/ A + \ps\)))ds and observe that .VJ = .3^ for
every t. Let A be the inverse of T and define the (.VA^)-\oc. mart. A/, = J5A,;
prove that M has the PRP, but that M is not pure.
[Hint: Prove that .TTM = .5^ and use Theorem D.9).]
D.17) Exercise. Prove that the Gaussian martingales, in particular а В for any
real a, are pure martingales.
D.18) Exercise. If M is a pure cont. loc. mart, and С an (.3f M)-time-change such
that M is C-continuous, then M = Mc is pure.

§4. Integral Representations 215
D.19) Exercise. Let M be a cont. loc. mart, with the (.ЯП-PRP.
1°) Prove that S = inf {r : Mt+U = M, for every и > 0} is a (.^-stopping
time.
2°) If #(&>) is the largest open subset of K+ such that M,(w) is constant on
each of its connected components, prove that there are two sequences (Sn), (Т„)
of (.^-stopping times such that H = UJ^w. Т„[ and the sets ]Sn, Т„[ are exactly
the connected components of H.
[Hint: For e > 0, let ]S,^, 7Д be the и-th interval of H with length > e.
Observe that S? + e is a stopping time and prove that Sen is a (.3^")-stopping time.]
.{4.20) Exercise. 1°) If M is an extremal cont. loc. mart, and if (¦?*,M) is the
filtration of a Brownian motion, then d(M, M}s is a.s. equivalent to the Lebesgue
measure.
** 2°) Let F be the Cantor middle-fourths set. Set
M, = f lF,(Bs)dB<
Jo
where В is a BM'(O). Prove that (-WtM) = (.3^B) and derive that M has not the
PRP.
D.21) Exercise. Let /3 be the standard BM1 and set M, = /„' Up,>o)dBs. Let
f = l(^i <<» + °°l(ft>0) and
= a +inf|e : / Up,>o)ds > 0
I J\ \
By considering the r.v. exp(—т), prove that M does not have the PRP.
* D.22) Exercise. 1°) Let {.Yt) and (¦'?,) be two filiations such that ¦'?, с .5^for
every t. If M is a continuous (J^)-Iocal martingale adapted to (•'?•,) and has the
(.^)-PRP prove that the following three conditions are equivalent
i) M has the (-'?,)-PRP;
ii) every ('^-martingale is a (.j^")-martingale;
iii) every (.'?,)-martingale is a continuous (.5T)-semimartingale.
2°) Let В be the standard linear BM and (.?T) be the Brownian filtration. Let
to be a strictly positive real and set
= f
Jo
B + sgn (B,o)) l(,0<s)] sgn(Bs)dBs.
Prove that N has the (.5f)-PRP and has not the PRP.
[Hint: The process H, = (sgn(fi,0))l(,0<,) is a discontinuous (.^
gale.]
D.23) Exercise. Let в be a (.Wt )-Brownian motion and suppose that there exists
a continuous strictly increasing process A,, with inverse т,, such that A, > t and
. Set X, =/V

216 Chapter V. Representation of Martingales
Iе) Prove that X is pure if and only if J^f = -5^c-
2°) Prove that X is extremal if and only if /5 has the (.i^")-
** D.24) Exercise. Iе) Retain the notation of Exercise C.29) Chap. IV and prove
that the following two conditions are equivalent
i) X has the (.^
ii) for any Q such that A" is a (Q, />)-local martingale there is a constant с such
that Q = cP.
2°) Use Exercise C.29) Chap. IV to give another proof of the extremality of
the Wiener measure, hence also of Theorem C.3).
# D.25) Exercise (PRP and independence). 1°) Prove that if two (.^-continuous
loc. mart. M and N are independent, then they are orthogonal, that is, the product
MN is a (.>f )-loc. mart, (see Exercise B.22) in Chap. IV). Give examples in
adequate filtrations of pairs of orthogonal loc. martingales which, nonetheless, are
not independent.
2°) Prove that if M and N both have the PRP, then they are orthogonal iff
they are independent.
3°) Let В be the standard BM and set
Mi = f \{B<>0)dBs. N,= f \{Bi<0)dBs.
Jo Jo
Prove that these martingales are not independent and conclude that neither has the
PRP. Actually, there exist discontinuous (.^"M)-martingales.
Notes and Comments
Sect. 1. The technique of time-changes is due to Lebesgue and its application
in a stochastic context has a long history which goes back at least to Hunt [1],
Volkonski [1] and lto-McKean [1]. Proposition A.5) was proved by Kazamaki [1]
where the notion of C-continuity is introduced (with a terminology which differs
from ours.)
Theorem A.6) appears in Dubins-Schwarz [1] for martingales with no inter-
intervals of constancy and Dambis [1]. The formulation and proof given here borrow
from Neveu [2]. Although a nice and powerful result, it says nothing about the
distribution of a given continuous martingale M: this hinges on the stochastic
dependence between the DDS Brownian motion associated with M and the in-
increasing process of M. Let us mention further that Monroe [1] proves that every
semimartingale can be embedded by time change in a Brownian motion, allowing
possibly for some extra randomisation. Proposition A.8) is from Lenglart [1] (see
also Doss-Lenglart [1]).
The proof of Knight's theorem (Knight [3]) given in the text is from Cocozza
and Yor [1] and the proof in Exercise C.18) is from Meyer [3]. Knight's theorem

Notes and Comments 217
has many applications as for instance in Sect. 2 and in Chap. VI where it is used
to give a proof of the Arcsine law. Perhaps even more important is its asymptotic
version which is discussed and used in Chap. XIII Sect. 2. We refer the reader to
Kurtz [I] for an interesting partial converse.
Exercise A.16) is taken from Lepingle [I], Exercise A.21) from Barlow [1]
and Exercise A.12) from Bismut [2].
Sect. 2. Conformal martingales were masterfully introduced by Getoor and Sharpe
[1] in order to prove that the dual of #' is BMO in the martingale setting. The
proof which they obtained for continuous martingales uses in particular the fact
that if Z is conformal, and a > 0, then \Z\a is a local submartingale. The extension
to non-continuous martingales of the duality result was given shortly afterwards
by P. A. Meyer. The first results of this section are taken from the paper of Getoor
and Sharpe.
The conformal invariance of Brownian motion is a fundamental result of P.
Levy which has many applications to the study of the 2-dimensional Brownian
path. The applications we give here are taken from B. Davis [1], McKean [2] and
Lyons-McKean [1]. For the interplay between planar BM and complex function
theory we refer to the papers of B. Davis ([1] and [3]) with their remarkable
proof of Picard's theorems and to the papers by Carne [2] and Atsuji ([1], [2]) on
Nevanlinna theory.
For the origin of the skew-product representation we refer to Galmarino [1]
and McKean [2]. Extensions may be found in Graversen [2]. For some examples
involving Bessel processes, see Warren-Yor [1].
The example of Exercise B.13) was first exhibited by Johnson and Helms [1]
and the proof of D'Alembert's theorem in Exercise B.17) was given by Kono [1].
Exercise B.19) is from Yor [2]; more general results are found in Ikeda-Manabe
[I]. The results and methods of Exercise B.14) are found in Ito-McKean [1].
Exercise B.16) is from Williams [4]; they lead to his "pinching" method (see
Messulam-Yor [1]).
Exercise B.24) is from Calais-Genin [1] following previous work by Walsh
[1]. Exercise B.25) originates in McKean [1] and Exercise B.21) is attributed
to H. Sato in Ito-McKean [1]. Exercise B.21) is from Burkholder [3], but the
necessary and sufficient condition of 2°) is already in Spitzer [1].
The subject of polar functions for the planar BM, partially dealt with in Exer-
Exercise B.26) was initiated in Graversen [1] from which Exercise A.20) Chap. I was
taken. Graversen has some partial results which have been improved in Le Gall
[6]. Despite these results, the following questions remain open
Question 1. What are the polar functions of BM2?
The result of Exercise B.26) may be seen as a partial answer to the following
Question 2. Which are the two-dimensional continuous semimartingales for which
the one-point sets are polar?
Some partial answers may be found in Bismut [2] and Idrissi-Khamlichi [1].
The answer is not known even for the semimartingales the martingale part of

218 Chapter V. Representation of Martingales
which is a BM2. The result of Exercise B.26) is a special case and another is
treated in Sznitman-Varadhan [1].
Exercise B.27) is taken from Yor [3] and ldrissi-Khamlichi [1]. The paper of
Yor has several open questions. Here is one of them. With the notation of Exercise
B.27), if a is polar for Xy the index of Xy with respect to a is well-defined and
it is proved in Yor [3] that its law is supported by the whole set of integers.
Question 3. What is the law of the index of Xy with respect to a?
This is to be compared to Exercise B.15) Chap. Vlll.
Sect. 3. The first results of this section appeared in Doob [1]. They were one
of the first great successes of stochastic integration. They may also be viewed
as a consequence of decompositions in chaoses discovered by Wiener [2] in the
case of Brownian motion and generalized by lto to processes with independent
increments. The reader may find a more general and abstract version in Neveu [1]
(see also Exercise C.17)).
Theorem C.5) and the decomposition in chaoses play an important role in
Malliavin Calculus as well as in Filtering theory. Those are two major omissions of
this book. For the first we refer to lkeda-Watanabe [2], Nualart [2], and Stroock [3],
for the second one to Kallianpur [1]; there is also a short and excellent discussion
in Rogers-Williams [1]. A few exercises on Filtering theory are scattered in our
book such as Exercise E.15) Chap. IV and Exercise C.20) in this section which
is taken from Lipster-Shiryaev [1].
Our exposition of Theorem C.9) follows Jacod [2] and Exercise C.12) is taken
from Lane [1]. Exercise C.13) is inspired from Chen [1]. Exercise C.14) is due to
Rosen and Yor [1] and Exercise C.19) to Neveu [3], the last question being taken
from Ledoux and Talagrand [1]. The source of Exercise C.22) is to be found in
Goswami and Rao [1]; question 4°) is taken from Attal et al. [1].
Sect. 4. The ideas developed in this section first appeared in Dellacherie [1] in
the case of BM and Poisson Process and were expanded in many articles such as
Jacod [1], Jacod-Yor [1] and Yor [6] to mention but a few. The method used here
to prove Theorem D.6) is that of Stroock-Yor [1]. Ruiz de Chavez [1] introduces
signed measures in order to give another proof (see Exercise D.24)). The notion
of pure martingales was introduced by Dubins and Schwarz [2].
Most of the exercises of this section come from Stroock-Yor ([1] and [2]) and
Yor [9] with the exception of Exercise D.20) taken from Knight [7] and Exercise
D.19) which comes from Strieker [2]. Exercise D.12) is from Yan and Yoeurp [1].
The results in Exercise D.25) are completed by Azema-Rainer [1] who describe
all (.*^M) martingales.
The unsatisfactory aspect of the results of this section is that they are only
of "theoretical" interest as there is no explicit description of extremal martingales
(for what can be said in the discrete time case, however, see Dubins-Schwarz [2]).
It is even usually difficult to decide whether a particular martingale is extremal or
pure or neither. The exercises contain some examples and others may be found in
Exercise D.19) Chap. VI and the exercises of Chap. IX as well as in the papers
already quoted. For instance, Knight [7] characterizes the harmonic functions /

Notes and Comments 219
in Rd, d > 1, such that f(Bt) is pure. However, the subject still offers plenty of
open questions, some of which are already found in the two previous editions of
this book.
Here, we discuss the state of the matter as it is understood presently (i.e.:
in 1998) thanks mainly to the progress initiated by Tsirel'son ([1], [2]), and co-
workers.
First, we introduce the following important
Definition. A filtration (.5^) on the probability space (Q, .P~, P) such that -^ is
P-a.s. trivial is said to be weakly, resp. strongly, Brownian if there exists a {.if, )-
BM1 P such that P has the (.>^)-PRP, resp. .71 = .7/.
We will abbreviate these definitions to W.B. and S.B.
Many weakly Brownian filtrations may be obtained through "mild" perturba-
perturbations of the Brownian filtration. Here are two such examples:
(PI) (Local absolute continuity). If {.Vt) is W.B., and, in the notation of Chapter
VIII, Q <\ P, then СЯ") is also W.B. under Q.
(P2) (Time change). If {,7,) is W.B., and if (a,) is the time-change associated
with A, = /J H,ds, where Ял is > 0 dP ds-a.s. and Ax — oo a.s., then (-K,) is
also W.B. under P.
A usually difficult, albeit important, question is:
(Q) Given a W.B. filtration (.7,), is it S.B. and, if so, can one describe explicitly
at least one generating BM1?
Tsirel'son [1] gives a beautiful, and explicit, example of a W.B. filtration for
which a certain Brownian motion is not generating. See Prop. C.6), Chap. IX, for
a proof. Feldman and Smorodinsky [1] give easier examples of the same situation.
Emery and Schachermayer [2] prove that the filtration in Tsirel'son's example is
S.B. Likewise, Attal et al. [1] prove that the Goswami-Rao filtration is S.B. (see
Exercise C.22)).
A deep and difficult study is made by Dubins et al. [1] to show that the
filtration on the canonical space C(R+, R) is not S.B. under (many) probabilities
Q, equivalent to the Wiener measure, although under such probabilities, {-7t) is
well-known to be W.B. (see (PI) above). The arguments in Dubins et al. [1] have
been greatly simplified by Schachermayer [1] and Emery [4].
Tsirel'son [2] has shown that the filtration of Walsh's Brownian motion with
at least three rays, which is well-known to be W.B., is not S.B. The arguments
in Tsirel'son [2] have been simplified by Barlow, Emery et al. [1]. Based on
Tsirel'son's technique, it is shown in this paper that if .7t = >fs, for a BMrfS
(d > 1), and if L is the end of a (.>f) predictable set Г, then .7~L+ differs from .7[~
by the adjunction of one set A at most, i.e.: ,3^+ = a {¦?[', A}, a property of the
Brownian filtration which had been conjectured by M. Barlow. Watanabe [6] shows
the existence of a 2-dimensional diffusion for which this property does not hold.
Warren [2] shows that the filtration generated by sticky Brownian motion and its

220 Chapter V. Representation of Martingales
driving Brownian motion (for the definition of this pair of processes, see Warren
[1]) is not S.B. Another simplification of Tsirel'son's work is presented by de
Meyer [1]. Emery and Schachermayer [1] show the existence of a pure martingale
(M,) with bracket (M, M), such that the measure d(M, M), is equivalent to dt,
and nonetheless, the filtration of (Л/,) is not S.B., although it is W.B. (see (P2)
above).
Let us also recall the question studied by Lane [1].
Question 4. If В is a BM1 and H a (.^"B)-predictable process, under which
condition on H is the filtration of M, — /J HsdBs that of a Brownian motion?
Under which conditions are all the (.><fM)-martingales continuous?
In the case of Hs = f(Bs), Lane [1] has partial and hard to prove results.
There are also partial answers in Knight [7] when / is the indicator function of a
set (see Exercise D.20)).
We also list the
Question 5. Which of the martingales of the previous question are extremal or
pure?
Even for Hs — B", the answer to the question of purity is not known when n
is even (see Exercise C.11) Chap. IX); Stroock and Yor [2] give a positive answer
for n odd. When H is > 0-a.s., then /O' HsdBs is extremal (see Exercise D.15) but
we have the following question, which is a particular case of the previous one:
Question 6. Does there exist a strictly positive predictable process H such that
the above stochastic integral is not pure?

Chapter VI. Local Times
§1. Definition and First Properties
With Ito's formula, we saw how C2-functions operate on continuous semimartin-
gales. We now extend this to convex functions, thus introducing the important
notion of local time.
In what follows, / is a convex function. We use the notation and results of
Sect. 3 in the Appendix. The following result will lead to a generalization of Ito's
formula.
A.1) Theorem. If X is a continuous semimartingale, there exists a continuous
increasing process A? such that
fix,) = f(x0) + f fL(xs)dxs + ]-a{
Jo z
where f'_ is the left-hand derivative of f.
Proof. If/ is C2, then this is Ito's formula and a{ = /„' f"(Xs)d(X, X)s.
Let now j be a positive C°°-function with compact support in ] — oo, 0] such
that f°ooj(y)dy = 1 and set fn(x) = n /Отс f(x + y)j(ny)dy. The function /
being convex, hence locally bounded, /„ is well defined for every n and, as n
tends to infinity, /„ converges to / pointwise and f'n increases to f'_. For each n
MX,) = fn(Xo) + [ f;,{Xs)dXs + X-Af,\
Jo 2
and fn(Xi) (resp. /„(Xo)) converges to f(X,) (resp. f(X0)). Moreover, by stop-
stopping, we can suppose that X is bounded and then f'_(Xs) also is bounded. By the
dominated convergence theorem (Theorem B.12) Chap. IV) for stochastic inte-
integrals /0' f'n(Xs)dXs converges to /0' f_(Xs)dXs in probability uniformly on every
bounded interval. As a result, Af" converges also to a process A1 which, as a
limit of increasing processes, is itself an increasing process and
/(*,) = f(X0) + f fL(X,)dXs + V.
Jo 2
The process a{ can now obviously be chosen to be a.s. continuous, which ends
the proof. ?

222 Chapter VI. Local Times


The problem is now to compute AI in an explicit and useful way making
clear how it depends on f. We begin with the special cases of Ixl, x+ = x V 0
and x- = -(x 1\ 0). We define the function sgn by sgn(x) = 1 if x > 0 and
sgn(x) = -1 if x :s o. If f(x) = Ixl, then f
(x) = sgn(x).


(1.2) Theorem (Tanaka formula). For any real number a, there exists an increas-
ing continuous process U called the local time of X in a such that,
IXo-al+ l' sgn(X,-a)dX,+L
,
+ [' 1 a
(X o - a) + Jo l(X,>a)dX, + '2Lt,
[' 1
(X o - a)- - Jo I(X,::;a)dX s + '2L
.


IX , -al


(X t - a)+


(X t - a)-


ln particular, IX - al, (X - a)+ and (X - a)- are semimartingales.
Proof The left derivative of f(x) = (x - a)+ is equal to l]a.oo[; by Theorem
(1.1), there is a process A + such that


+ + [' 1 +
(X, - a) = (X o - a) + Jo l(X,>a)dX s + '2At .
ln the same way


(X t - a)- = (X o - a)- -11 1 (X,::;a)dX, +
A;.
By subtracting the last identity from the previous one, we get


l t 1
Xt = Xo + dX s + - (Ai - A;).
o 2
It follows that Ai = A; a.s. and we set L
 = Ai.
By adding the same two identities we then get the first fonnula in the statement.
o


We will also write L
 (X) for the local time in a of the semimartingale X when
there is a risk of ambiguity.
Remark. The lack of symmetry in the last two identities in the statement is due to
the fact that we have chosen to work with lefl derivatives. This is also the reason
for the choice of the function sgn. See however Exercise (1.25).


With the increasing process L
, we can as usual associate a random measure
dL
 on JR+. To some extent, it measures the "time" spent at a by the semimartin-
gale X, as is shown in the following result.
(1.3) Proposition. The measure dL
 is a.s. carried by the set {t : X, = a}.




91. Definition and First Properties 223


Proof By applying Itô's fonnula to the semimartingale IX - al, we get


(XI - a)2 = (X o - a)2 + 21 1 IX, - ald(IX - al), + (IX - al, IX - al)t


and using the first fonnula in Theorem (1.2) this is equal to
(X o - a)2 + 21 ' IX" - al sgn(X" - a)dX" + 21 t IX" - aldL
 + (X, X)t.
If we compare this with the equality, also given by Itô's fonnula,
(X t - a)2 = (X o - a)2 + 21 ' (x" - a)dX,. + (X, X)t


we see that J
 IX" - aldL
 = 0 a.s. which is the result we had to prove. 0
Remarks. 1°) This proposition raises the natural question whether {t : XI = a}
is exactly the support of d L
. This is true in the case of BM as will be seen in
the following section. The general situation is more involved and is described in
Exercise (1.26) in the case of martingales.
2°) If Œ a = sup{t : X, = a}, then L':x, = L
a' a fact which comes in handy in
some proofs.


We are now going to study the regularity in the space variable a of the process
L
. We need the following
(1.4) Lemma. There exists a ,A>(JR) @ ,
-measurable process [ such that, for
each a, [(a, ., .) is indistinguishablefrom U.
Proof Apply Fubini's theorem for stochastic integrals (Exercise (5.17) of Chap.
IV) to the process H(a, s, .) = l(X,>a).
Consequently, we henceforth suppose that Lais ,),»(JR) @ ,
-measurable and
we will use the measurability in a to prove the existence of yet another better
version. We first prove two important results. We recall that if f is convex, its
second derivative fil in the sense of distributions is a positive measure.


(1.5) Theorem (Itô- Tanaka formula). If f is the difJerence of two convex func-
tions and if X is a continuous semimartingale


f(X t ) = f(Xo) + l' f
(X")dX,, +
 l L
 f"(da).


ln particular, f(X) is a semimartingale.




224 Chapter VI. Local Times
Proof. It is enough to prove the formula for a convex /. On every compact subset
of R, / is equal to a convex function g such that g" has compact support. Thus by
stopping X when it first leaves a compact set, it suffices to prove the result when
/" has compact support in which case there are two constants a, fi such that
f{x) =ax + p + ~ [ \x- a\f"(da).
Thanks to the previous results we may write
f(X,) = aX,+p + ^j\Xt-a\f"(da)
= a(X, - Xo) + f(X0) + f \(f sgn(*< - a)dX> + L1) /"Wfl).
From Sect. 3 in the Appendix and Lemma A.4), we see that
\{ I Щп{Х5 - a)dXJ"(da) = [ fL(Xs)dXs - a(X, - Xo)
1 J& Jo Jo
w
hich completes the proof.
A.6) Corollary (Occupation times formula). There is a P'-negligible set outside
of which
[ 0(Xs)d(X,X).,= [ <P(a)L<'da
Jo J-x
for every t and every positive Borel function Ф.
Proof. If Ф = f" with / in C2, the formula holds for every t, as follows from
comparing Ito-Tanaka's and Ito's formulas, outside a P-negligible set Гф. By
considering a countable set (Ф„) of such functions dense in C0(IR) for the topology
of uniform convergence, it is easily seen that outside the F-negligible set Г =
Un ^~*»' me formula holds simultaneously for every / and every Ф in Co(R). An
application of the monotone class theorem ends the proof.
Remarks. 1°) The time t may be replaced by any random time S.
2°) These "occupation times" are defined with respect to d{X, X)s which may
be seen as the "natural" time-scale for X. However the name for this formula is
particularly apt in the case of Brownian motion where, if Ф = 1 д for a Borel set
Л, the left-hand side is exactly the amount of time spent in A by the BM.
3°) A consequence of these formulas is that for a function which is twice
differentiable but not necessarily C2, the И6 formula is still valid in exactly the
same form provided /" is locally integrable; this could also have been proved
directly by a monotone class argument.
We now turn to the construction of a regular version of local times with which
we will work in the sequel.

§ 1. Definition and First Properties 225
A.7) Theorem. For any continuous semimaningale X, there exists a modification
of the process {L"\ a eR,l e R+] such that the map (a, t) -*¦ Lat is a.s. continuous
in t and cadlag in a. Moreover, if X = M + V, then
Li - La~ = 2 f \iX,=a)dVs = 2 f
Jo Jo
Thus, in particular, ifX is a local martingale, there is a bicontinuous modification
of the family L" of local times.
Proof. By Tanaka's formula
L?=2F(X, -a)+-(Xo-a)+- / Ux,>a)dM,-J \lXl>a)dVA.
Using Kolmogorov's criterion (Theorem B.1) Chap. 1) with the Banach space
C([0, /], K), we first prove that the stochastic integral
к = f
Jo
possesses a bicontinuous modification. Thanks to the BDG-inequalities of Chap.
IV and Corollary A.6), we have, for any к > 1,
E sup M,a - M? <
L ' J
CkE
= CkE
= Ck(b-a)kE
V
1 f" \"
—ah L-dX)
By Fubini's theorem, this is less than
C*(ft-a)*sup?;[(l4)*].
Now, L) = 2 j(X, - x)+ - (Xo - x)+ - /0' l,^>A)dX, J and since \(X, - x)+ -
(Xo — x)+\ < \X, - Xo\, there is a universal constant dk such that
E [(Z4)*] < dkE sup \X, - X0\k + (f°° \dV\s\ + (M, M)
k/2
oc
Remark that the right-hand side no longer depends on x. If it is finite for some
к > 1, we have proved our claim. If not, we may stop X at the times

226 Chapter VI. Local Times
Т„ = inf \t : sup \XS - X0\k + ( [' \dV\s\ + (M, M)k/2 > n
The martingales (Ma)Tl< have bicontinuous versions, hence also Ma.
To complete the proof, we must prove that
К = f llXi>B)dVs
Jo
is jointly cadlag in a and continuous in t. But, by Lebesgue's theorem
V?~ = lim f \{x^b)dVs = [ \lXt>a)dVs.
*ta Jo JO
It follows that Lat - Lat- = 2 (v?~ - vA = 2/0' \(x,=a)dV<. In the same way
V,a+ - lim f \lX,>b
hi" JO
v,a
hi" JO JO
so that Ц = L't'+.
Finally, the occupation times formula implies that
f \(Xk=a)d{M,M)s= I \{x,=a)d{X.X)s=Q
Jo Jo
so that /Q' \(x,=U)dMs • л which ends the proof. П
Remark. If X is of finite variation, then L"{X) = 0. Indeed by the occupation
times formula, L" — 0 for Lebesgue almost every a and, by the right-continuity in
a, L"{X) — 0 for every a. However, a semimartingale may have a discontinuous
family of local times, or in other words, the above theorem cannot be improved
to get continuity in both variables, as is shown in Exercise A.34).
As a by-product of the use, in the preceding proof, of Kolmogorov's criterion
we see that for local martingales we may get Holder properties in a for the family
L". TLis is in particular the case for Brownian motion where
< Ск\а-Ь\Чк'2
for any fixed time t and for every к. Thus we may now state the
A.8) Corollary (Continuity of martingale local times). The family L" may be
chosen such that almost surely the map a -*¦ Lat is Holder continuous of order a
for every a < 1 /2 and uniformly in t on every compact interval.
Proof In the case of BM, only the uniformity in t has to be proved and this
follows from Exercise B.10) Chap. I. The result for local martingales is then a
consequence of the DDS Theorem (see Exercise A.27)). The details are left to the
reader. ?

§ 1. Definition and First Properties 227
From now on, we will of course consider only the version of the local time
L"{X) which was exhibited in Theorem A.7). For this version, we have the fol-
following corollary which gives another reason for the name "local time".
A.9) Corollary. lfX is a continuous semimartingale, then, almost-surely,
Lat(X) = \\mX- f \[u.a+F[(Xs)d(X, X)s
for every a and t, and if M is a continuous local martingale
= lim-!- / \]a.eM+F[(Ms)d(M.M)s.
«iO IS Jo
The same result holds with any random time S in place oft.
Proof. This is a straightforward consequence of the occupation times formula and
the right-continuity in a of L"(X). ?
For BM we have in particular
which proves that Ц is adapted to the completion of cr(\Bs\, s < t). This will be
taken up in the following section.
The above corollary is an "approximation" result for the local time. In the
case of BM, there are many results of this kind which will be stated in Chap.
XII. We begin here with a result which is valid for all the semimartingales of Ур
(Exercise D.13) Chap. IV). Let X be a continuous semimartingale; for e > 0,
define a double sequence of stopping times by
aFQ = 0, rof = inf{/ : X, = e],
< = inf{r > <_, : X, = 0}. < - inf{/ > < : X, = s].
We set dF(t) — max{« : a* < t}\ this is the number of "downcrossings" of X
(see Sect. 2 Chap. 11) from level s to level 0 before time t. On Figure 4, we have
4@ - 2.
For simplicity, we will write only a,, and т„ instead of of and xf and L, for
Ц-
A.10) Theorem. lfX = M + V is in .%, p > 1, i.e.
E Г(М, M)^1 + QH IrfVI,)' 1 < oo.
then
lim E sup \sde(t) - -L,\p = 0.

228 Chapter VI. Local Times
Fig. 4.
Proof. By Tanaka's formula
+ + - Г '
Xz,,AI - Xa,,Ai — I Uxt>0)dXs + - (LZuA, - LOllA,) .
J ]o,i At Л,, At] ^
Because X does not vanish on [т„, от,-и[, we have LTiiAt — LanA, = Lnn^A, — i
As a result
where <9f is the predictable process ?,, l]a,,.i,,](*)l]0.6-](^)- But Х+л, - Х+л, = е
on {r,, < t}; if/j(f) = inf{« : г„ > f}, the left-hand side of the above equality is
equal to edF(t) + u(s) where 0 < u{e) = X+ - Xff,,,,|A, < e. Thus the proof will
be complete if
lim?jsup| f e^dXAp\ =0.
f^° L ' Jo ' J
But, by the BDG inequalities, this expectation is less than
CPE
, X)s
%\dV\,
and since 0e converges boundedly to zero, it remains to apply Lebesgue's domi-
dominated convergence theorem. ?
With the same hypothesis and notation, we also have the following proposition
which is the key to Exercise B.13) Chap. XIII.
A.11) Proposition.
limE sup ?-' / e*d(X, X), - -L, = 0.
fi° L>o Jo ' 2 J

§ 1. Definition and First Properties 229
Proof. Using Ito's and Tanaka's formulas and taking into account the fact that
dhs does not charge the set |л- : Xs ф 0}, one can show that
f
J]v
liXt>0)d(X,X),
which entails that
s2dl
Therefore
M2(e)=2 / 0csX,dXs+ / 0/
Jo Jo
d(X,X)s.
9?sd(X,X)s -ed?{t)-2e
-l
f 6esXsdXs
Jo
e.
and it is enough to prove that e ' /J 0FsXsdXs converges to zero uniformly in Lp.
But using the BDG inequalities again, this amounts to showing that
e~pE
P/2
Of\Xs\\dVU
converges to zero as e converges to zero, and since \XS\ < s on [вI > 0), this
follows again from Lebesgue's theorem. П
Remark. By stopping, any semimartingale can be turned into an element of .S^.
Therefore, the last two results have obvious corollaries for general semimartin-
gales provided one uses convergence in probability on compact sets instead of
convergence in Lp.
We close this section with a more thorough study of the dependence of L" in
the space variable a and prove that La, as a function of я, has a finite quadratic
variation.
For each t, the random function a ->• Lat is a cadlag function hence admits
only countably many discontinuities. We denote by AL" the process L" - Lat~. If
X = M + V,
Consequently, for a < b,
J2 \AL*\<2['\dV\s
< oo
and, a fortiori,
E
a<x<h
< oo.
For our later needs, we compute this last sum. We have

230 Chapter VI. Local Times
and since, by the continuity of V, the measure dV ® dV does not charge the
diagonal in [0, t]2,
(АЦJ = 8 / \tXt=x)dVs [ \a.=x,)dVu
Jo Jo
= 4 / AL^Ux^dV,.
Jo
Since there are at most countably many x ? ]a, b[ such that ALls > 0 for some
s e [0, t], we finally get
V (AL
a<x<b
*J=4 f AL^\ia<XtSb)dVs.
We may now state
A.12) Theorem. Let (An) be a sequence of subdivisions of [a, b] such that \A,,\
0. For any поп negative and finite random variable S
las+l-Las) =4 / L\dx+ ?И1$.)
lim
Л, •'я «<x<fo
ш probability.
Proof. The case of a general S will be covered if we prove that X!^ (^/*' ~~ L"')
converges in probability, uniformly for / in a compact subinterval of M.+, to
4/j" L*dx + T.a<x<b(AL"J- To this end' we develop - (L^1 - La;J with the
help of Tanaka's formula, namely,
X-La,= (X, - a)+ - (X0 - я)+ - Щ - V,"
where we define Z" = /0' l(x,>a)^Zv for Z = M. V, X.
The function ф,(я) = (X, — я)+ — (Xo — я)+ is Lipschitz in я with constant
1 which implies that
«->ОС L—l
Likewise, using the continuity of M" as was done in Sect. 1, Chap. IV,
/7—*OG
lim V @,(a,-+i) - 0,(a,)) (m,"'+1 - мМ = О.
Finally

(jl. Definition and First Properties 231
(V,"- - V,"')
< \An\ / \dV\s
J
which goes to zero as n tends to infinity.
As a result, the limit we are looking for is equal (if it exists) to the limit in
probability of
2
Ito's formula yields
A,,
J / , X)s.
The occupation times formula shows that the second term in the right-hand side
is equal to fa Lxtdx. We shall prove that the first term converges to the desired
quantity.
By localization, we may assume that X is bounded; we can then use the
dominated convergence theorem for stochastic integrals which proves that this
term converges in probability uniformly on every compact interval to
( ) [
Jo v ' Jo
By the computation preceding the statement, it remains to prove that
f
Jo
But as already observed, there are only countably many x e ]a.b[ such that
AL"S > 0 for some s e [0, t] and moreover f-a.s., /0' l\xK=.x)d(M, M)s = 0 for
every x. Thus the result follows. о
The following corollary will be used in Chap. XI.
A.13) Corollary. If the process x -*• L\, x > 0, is a continuous semimartingale
(in some appropriate filtration), its bracket is equal to 4/0* Lxsdy.
A.14) Exercise. Let M be a continuous local martingale vanishing at 0 and L its
local time at 0.
1°) Prove that infj/ : L, > 0} - infj? : (M, M), > 0} a.s.. In particular M = 0
if and only if L = 0.
2") Prove that for 0 < a < 1, and M ф 0, \M\a is not a semimartingale.

232 Chapter VI. Local Times
A.15) Exercise (Extension of the occupation times formula). If X is a contin-
continuous semimart., then almost-surely, for every positive Borel function h on R+ x K,
f h(s,Xs)d{X1X)s = f da f h(sia)dLas{X).
Jo J-oc Jo
Extend this formula to measurable functions h on R+ xfixl,
# A.16) Exercise. 1D) Let X and Y be two continuous semimartingales. Prove that
/ \(X^Yt)d(X,Y)s= f l(Xt=Ys)d(X,X)s= f llX,=Y,)d(Y.Y)s.
Jo Jo Jo
[Hint: Write (X, Y) = (X, X) + {X,Y - X).]
2°) If X — M + V and A is a continuous process of finite variation
/ l(X,=A,)dXs= f l(X^)dVs.
Jo Jo
3°) If X — M + V is > 0 and Mo = 0, its local time at 0 is equal to
2/0 l(xt=0)dVs = 2/0 l(x,=0)<^.5- As a result, if dVs is carried by the set [s :
Xs — 0}, then V is increasing; moreover if M = -N and V, = sups<; Ns, the
local time of X at 0 is equal to 2V.
A.17) Exercise. 1°) If X is a continuous semimartingale, prove that
Lat{\X\) = L^(X) + L(~a)~(X) ifa>0, L°(|X|) = 0 ifo<0,
and that L^(X+) = L°(X).
2°) If X is a continuous semimartingale, prove that
La,{-X) = Ц-а)-(Х).
3°) Prove a result similar to Theorem A.10) but with upcrossings instead of
downcrossings.
A.18) Exercise. 1°) If X and Y are two cont. semimarts., prove that X v Y and
X л Y are cont. semimarts. and that
L\X v Y) + L°(A: л Y) = /."(Я) + ?°(У).
[Hint: By the preceding exercise, it is enough to prove the result for positive
X and Y. Use Exercise A.16), 3°) .]
2°) Prove further that
L°(XY) = X+ ¦ L°(Y) + Y+ ¦ L°(X) + X~ ¦ L°" (Y) + Y'-L°~(X).

§1. Definition and First Properties 233
** A.19) Exercise. 1°) If X is the BM, prove that, with the notation of Theorem
A.10),
hm Ede(-) — -L a.s.
f|0 2
[Hint: Prove the almost-sure convergence for the sequence ?„ = n~2, then use
the fact that de(t) increases when e decreases. Another proof is given in Sect. 1
Chap. XII.]
2°) More generally, if for a < 0 < b we denote by da,b the number of
downcrossings from b to a, then
if ?(?« -««) < oo.
1
lim (Ь„ - a,,)da ь„(.() — т^-'- a-s-
п—*оо ' 2
A-20) Exercise. (A generalization of P. Levy's characterization theorem). Let
f(x, t) be a fixed solution to the heat equation i.e. be such that // + jf? = 0.
We recall that such a function is analytic in x for / > 0.
1°) If ? is a (.'?,)-BM and A a ('?,)-adapted continuous increasing process,
prove that a.s.
up to m -negligible sets where m is the Lebesgue measure.
[Hint: Apply to the semimartingale Y, = f't(B,,At) the fact that d(Y, X),
does not charge the sets {t : Y, — a}.] Arguing inductively prove that if // is not
identically zero then m[t : /r'(B,, A,) = 0} = 0.
[Hint: If not, there would be a point x where all the spatial derivatives of //
vanish.]
2°) Let X be a continuous local martingale such that f(Xt, t) is a local mar-
martingale and m{t : ft'(X,, t) = 0} = 0; prove that X is a BM.
[Hint: Observe that /J f's(Xs, s)d{X, X)s = /0' /S'(XS, s)ds.]
A.21) Exercise. 1°) Let X1 and X2 be two continuous semimartingales vanishing
at 0. Prove that
L°(X] v X2) = /' \{X2<0)dL°s{X]) + L°(X2+ - Xl + ).
Jo
[Hint: Use Exercise A.16) 3°) and the equality (XlvX2)+ = (X2+-Xl + )+ +
2°) Suppose henceforth that L°(X2-X') = 0. Pick a real number a in ]0. l/2[
and set Z' = X1 - 2aXi+. After observing that, for X2 > X1,
A -2a)(X2 - X1) < Z2 -Z1 < X2 - X\
prove that L°(Z2 -Z1) = 0.
[Hint: Use Theorem A.10).]

234 Chapter VI. Local Times
3°) From the equality 2a(X2+ - X1 + )+ = (X2 - X])+ - (Z2 - Z')+ derive
that
L?(X'vX2)= f \(X:<0}dL°(Xl)+ f \{Xl<O)dL'l(X2).
Jo Jo
A.22) Exercise. Prove that for the standard BM
sup Lf- = supL^.
\<t a
[Hint: One may use the Laplace method.] The process LB is further studied
in Exercise B.12) Chap. XI.
A.23) Exercise. Let / be a strictly increasing function on R, which moreover is
the difference of two convex functions. Let X be a cont. semimart.; prove that for
every /, the equality
Lftta){f(X)) = f'+(a)L«(X)
holds a.s. for every a.
** A.24) Exercise. Let X — M + V be a continuous semimartingale and A a con-
continuous process of bounded variation such that for every ?,
i) I l(xt=At)dXs = 0. ii) / 1(дг,=л,,</Л, = 0,
./о ./о
(see Exercise A.16) above). If (Lxt) is the family of local times of X and A, is
the local time of X — A at zero, prove that for any sequence (An) of subdivisions
of [0, t] such that \An\ ->• 0, and for any continuous adapted process Я,
/ H,dAs = lim V H,, (L^ - L?'1
Jo "^xx V
in probability. This applies in particular when A — V, in which case we see that
the local time of the martingale part of X is the "integral" of the local times of X
along its bounded variation part.
A.25) Exercise (Symmetric local times). Let X be a continuous semimartingale.
Iе) If we take sgh(.v) to be 1 for x > 0, -1 for x < 0 and 0 for x = 0, prove
that there is a unique increasing process L" such that
\X, - a\ = \X0 - a\ + / sgn(Xv - a)dXs + L</.
Jo
Give an example in which L" differs from La. Prove that L" = (La +L"~)/2 and
Ц = lim i- / \]a-e,a+s[(Xs)d(X, X)s.
?j.O IE Jo
2°) Prove the measurability of L" with respect to a and show that if / is
convex

§1. Definition and First Properties 235
f(X,) = /(X()) + [ \(f'+ + fL)(X,)dXs + \ I L<;f"(da).
Jo z ¦? J
Write down the occupation times formula with L" instead of L".
¦* A.26) Exercise (On the support of local times). Let M be a continuous (.7i)-
martingale such that Mo — 0 and set Z(w) = [t : M,(w) — 0). The set Z(w) is
closed and Z(oj>)c is the union of countably many open intervals. We call R(w)
(resp. G(co)) the set of right (resp. left) ends of these open intervals.
1°) If T is a (.^-stopping time and
DT(w) = inf{/ > T{со) : М,(ш) = 0},
prove that if M is uniformly integrable, the process
Yt — \Mt+i\\[o<i<dt-t]
is a (^^il
2C) Prove that there exists a sequence of stopping times Tk such that R(oj) С
LJt|7"/c(w)}, that for any stopping time T, P[co : T(w) e G(w)] = 0, and finally
that Z(w) is a.s. a perfect set (i.e. closed and without isolated points).
We now recall that any closed subset Z of IK is the union of a perfect set and
a countable set. This perfect set may be characterized as the largest perfect set
contained in Z and is called the perfect core of Z.
о
3°) Prove that the support of the measure dL, is the perfect core of Z\ Z
о
where Z is the interior of Z. Thus in the case of BM, the support is exactly Z, as
is proved in the following section.
[Hint: Use Exercise A.16).]
4°) Prove the result in 3°) directly from the special case of BM and the DDS
theorem of Sect. 1 Chap. V.
# A-27) Exercise. Let г be a process of time changes and X a т-continuous semi-
martingale; prove that
L°(XT) = L"Ti{X).
Consequently, if X is a continuous local martingale, and В its DDS BM, prove
that LI{X) = La{XX)i(B).
Prove further that if X is a local martingale then for any real a, the sets
{lim^c;, X, exists} and {L^ < oo} are a.s. equal.
[This Exercise is partially solved in Sect. 2 Chap. XI.]
* A.28) Exercise. Let X be a continuous semimartingale and L" the family of its
local times.
1°) Let S be a positive r.v. Prove that there exists a unique vector measure on
the Borel sets of Ж with values in the space of random variables endowed with
the topology of convergence in probability, which is equal to

236 Chapter VI. Local Times
on the step function J] / l]a,.a,+]]. We write f^x f(a)daLas for the integral with
respect to this vector measure.
2°) If / is a bounded Borel function and F{x) = /Ql f(u)du, prove that
/ f{Xu)dXu - - / f(a)daLas.
Jo l J-x
Extend this formula to locally bounded Borel functions /.
3°) Prove that if a -> L"s is a semimartingale (see Sect. 2 Chap. XI) then
/!?o f(a)daLas is equal to the stochastic integral of / with respect to this semi-
semimartingale.
* A.29) Exercise (Principal values of Brownian local times). 1°) For the linear
BM, prove that for a > — 1 the process
(Г= f'\B5
Jo
A'ds
Jo
is finite valued, hence
X(ta) = / |fi,|"(sgn Bs)ds
Jo
is also well-defined. Prove that these processes have continuous versions.
2°) Prove that
H, = lim / B~l\uB,\>e)ds
?" J, 0 /n
or more generally that for a > —3/2
H,{a) =lim / |ft|a(sgn &)l(|e,,>e)rfj
exists a.s. Prove that these processes have continuous versions.
[Hint: Use the occupation times formula and Corollary A.8).]
3°) Setting B, = B,-H,, prove that B,Bt is a local martingale. More generally,
if h : E x E+ -> E is a solution of the heat equation i ^-| + ^7 = 0, then
z dx 0t
h(B,, t)Bt is a local martingale.
4°) if / = Yl=\^khk.tM\-
the process B,%/ is a local martingale. Derive therefrom that
This discussion is closely related to that of Exercise C.29), Chap. IV.

§1. Definition and First Properties 237
A.30) Exercise. 1°) Using the same notation as in the previous exercise prove
that for any С' -function ф we have
В,ф(Н,)= I ф(Нг)ёВ,+ I <p'{Hs)ds.
Jo Jo
2°) For every a e R, define a process la by the equality
Bt\{H,>a) — I \{Ht>u)dBs + A".
Jo
Show that there is a jointly measurable version which is again denoted by Л. Then
prove that for any positive Borel function /,
f
Jo
f{a)\a,da.
3°) Show that there is a version of A" which is increasing in t and such that
A" - B,\{H,>a) is jointly continuous.
A.31) Exercise. If X is a continuous semimartingale, the martingale part of which
is a BM, prove that for any locally integrable function / on R+,
/ ds f( j l(Xu<x,)du) = j f(v)dv a.s..
./o \./o / ./o
More generally, prove that, for every locally integrable function h on R+,
f ds f( f du 1(х„<хл) A(*s) = I dzh (a,(z)) /(г),
where a,(z) = inf j}> : /0' du l(x,,<y) > Z J.
*# A.32) Exercise. (Holder properties for local times of semimartingales). Let
X = M + V be a continuous semimart. and assume that there is a predictable
process v such that
i) Vt=fivsd(M,M)s;
ii) there exists a p > 1 for which
/\vs\pd(M, M)s < oo a.s., for every I> 0.
Call Lat, a e Ш, I > 0, the family of the local times of X.
1°) Prove that after a suitable localization of X, there exists, for every N > 0,
к 6 N and 7" > 0, a constant С = C(N, к, Т) such that
E Fsup \L* - L,v|*l < С (\x - y\
\k'2 + \x-

238 Chapter VI. Local Times
for every pair (x,y) for which \x\ < N and \y\ < N. As a result, there is a
bicontinuous modification of the family L", a e R, / > 0.
2°) Prove that, consequently, there exists a r.v. Dj such that
sup IL* - Ц| < DT\x- y\a
t<T
for every a < 1 /2 if p > 2 and every a < (p - 1 )/p if /? < 2.
Apply these results to X, = B, + c/m where В is a BM@), ceR and m < \.
The following exercise shows that the conditions stated above are sufficient
but not necessary to obtain Holder properties for local times of semimarts.
# A.33) Exercise. Let В be a BM@) and / its local time at 0. Let further X = B+cl
where с е К and call La the family of local times of X. Prove that for every T > 0
and к > 0, there is a constant Ст.к such that
Tsup|Z.--Z.?|*l<Cr.t|a-ft|*/2.
Ust J
and that consequently the Holder property of the preceding exercise holds for
every a < 1/2. Prove the same result for X = \B\+ cl.
A.34) Exercise. Let В the standard BM1 and a and b two different and strictly
positive numbers. Prove that the local time for the semimartingale X = aB+—ЬВ~
is discontinuous at 0 and compute its jump.
* A.35) Exercise. For the standard BM В prove the following extensions of The-
Theorem A.12):
HudL"u\ =
whenever either one of the following conditions applies:
i) Я is a (.J^~B)-cont. semimart.;
ii) H = f(B) where / is of bounded variation on every compact interval and S
is such that x —> Lxs is a semimartingale (see Sect. 2, Chap. XI).
Examples of processes H for which the above result fails to be true are found
in Exercise B.13) Chapter XI.
§2. The Local Time of Brownian Motion
In this section, we try to go deeper into the properties of local times for standard
linear BM. By the preceding section, it has a bicontinuous family L" of local
times. We write L, instead of Lj1 as we focus on the local time at zero.
We first introduce some notation, which is valid for a general semimartingale
X and will be used also in Sect. 4 and in Chap. XII. We denote by Z(co) the

§2. The Local Time of Brownian Motion 239
random set {s : X5(co) = 0}. The set Zc is open and therefore is a countable union
of open intervals.
For / > 0, we define
g, = supj.T </; X, =0}
with the convention that sup@) = 0, hence in particular g0 = 0. The r.v.'s g, are
clearly not stopping times since they depend on the future. We also define
d, = inf{5 > / : X, = 0}
with the convention inf@) = +oo. Those are stopping times which are easily seen
to be predictable and for / > 0, we have
(gt < «) = (X, ф 0, и < s < /) = (du > /).
Finally, we observe that Lgi — L, = L^ because as was proved in the last section,
dL, is carried by Z. In the case of BM we will prove shortly a more precise result
(see also Exercise A.26)).
Our first goal is to find the law of the process L. We shall use Ito-Tanaka's
formula
|B,|= / sgn(Bs)dBs + L,.
Jo
We saw in Exercise A.14) of Chap. Ill that |B,| is a Markov process (see also
Exercise B.18) in this section). It is also clearly a semimartingale and its local
time at 0 is equal to 2L (see Exercises A.17) and B.14)). A process having the
law of \B\ is called a reflecting Brownian motion (we shall add "at zero" if there
is a risk of ambiguity).
To analyse L, we need the following lemma which is useful in other contexts
as well (see Exercise B.14) Chap. IX).
B.1) Lemma (Skorokhod). Let у be a real-valued continuous function on [0, oo[
such that y@) > 0. There exists a unique pair (г, a) of functions on [0, oo[ such
that
i) z = y+a,
H) z is positive,
Hi) a is increasing, continuous, vanishing at zero and the corresponding measure
das is carried by {s : z(s) = 0).
The function a is moreover given by
a(t) = sup (-y(s)vO).

240 Chapter VI. Local Times


Proof We first remark that the pair (a, z) defined by
a(t) = sup ( - y(s) V 0), Z = y + a
s
t


satisfies properties i) through iii).
To prove the uniqueness of the pair (a, z), we remark that if (a, z) is another
pair which satisfies i) through iii), then z - z = a - a is a process of bounded
variation, and we can use the integration by parts fonnula to obtain


O:s (z - Z)2(t) = 21 t (z(s) - z(s»)d(a(s) - a(s»).
Thanks to iii) this is further equal to
-21 t z(s)da(s) - 21 ' z(s)da(s)
which by ii) and iii) is :s o. 0
(2.2) CoroHary. The process {3t = J
 sgn(B,,)d B" is a standard BM and ,
fJ =
.:3i1fIBI. Moreover, Lt = sup( -{3,,).
J
t


Proof That {3 is a BM is a straightforward consequence of P. Lévy's character-
ization theorem. The second sentence follows at once from the previous lemma
and Tanaka's fonnula
IBI, = {3t + Lt.
It is now obvious that ,:3i1fIBI C ,?;f! and, since it follows from Corollary (1.9) that
.?;L C .
IBI, we have .?;f! C .
IBI; the proof is complete.
Remark. Another proof of the equality ,
IBI = .
fJ will be given in Exercise
(3.16) Chap. IX.


This corollary entails that the processes Lt and SI have the same law; in
particular, by Proposition (3.8) Chap. III, Lt is the inverse of a stable subordinator
of index 1/2. Another proof of this is given in Exercise (1.11) Chap. X. The
equality of the laws of Lt and S, can be still further improved.
(2.3) Theorem (Lévy). The two-dimensional pro cesses (St - Bt, St) and (1 Bt 1, L,)
have the same law.


Proof On one hand, we have by Tanaka's fonnula, IB,I = {3t + LI; on the other
hand, we may trivially write St - B, = - Bt + St. Thus, Lemma (2.1) shows
that one gets Sand S - B (resp. Land 1 B 1) from - B (resp. {3) by the same
detenninistic procedure. Since - Band {3 have the same law, the proof is fini shed.


Remark. The filtration of S - Bis actually that of B (see Exercise (2.12» hence,
the filtration of (S - B, S) is also (.?;B) whereas the filtration of (IBI, L), which
is (.
IBI), is strictly coarser.




92. The Local Time of Brownian Motion 241


The following corollary will be important in later sections.
(2.4) Corollary. P [L':x, = 00 for every a] = 1.
Proof By the recurrence properties of BM, we have obviously P [Soo = 00] = 1;
thus P [L
 = 00] = 1 follows from Theorem (2.3). For every a i= 0, we then
get P [L':x, = 00] = 1 from the fact that BTa+t - a is a standard BM.
Therefore pl L7x, = 00 for every q E Q] = 1 and the result follows from the
almost-sure lower semi-continuity in a of L':x,. 0
We now tum to the result on the support of the measure dL" which was
announced earlier. We cali (Tt) the time-change associated with L" i.e.
T, = inf {s > 0 : L" > t} .
By the above corollary, the stopping times Tt are a.s. finite. We set
6(w) = U ]Ts- (w), T,,(W)[.
,,:,:0


The sets ]Ts-, T" [ are empty unless the local time L has a constant stretch at level
sand this stretch is then precisely equal to [Ts-, T,,]. The sets ]1's- , T" [ are therefore
pairwise disjoint and 6(w) is in fact a countable union. We will prove that this set
is the complement of Z(w). We recall from Sect. 3 in Chap. III that Z has almost
surely an empty interior and no isolated points; the sets Jr..-, T" [ are precisely the
excursion intervals defined in Chap. III.
(2.5) Proposition. The following three sets
i) Z(w), ii)
(w)C, iii) the support .E(w) of the measure dLt(w),
are equal for almost every w.
Proof An open set has zero dL,-measure if and only if L is constant on each
of its connected components. Thus, the set (J(w) is the largest open set of zero
measure and .E(w) = f'i(wY.
We already know from Proposition (1.3) that .E(w) C Z(w) a.s. To prove the
reverse inclusion, we first observe that Lt > 0 a.s. for any t > 0 or in other words
that TO = 0 a.s. Furthennore, since d, is a stopping time and B d , = 0, Tanaka's
fonnula, for instance, implies that Ld t +" - L d " s :::: 0, is the local time at zero of the
BM Bd,+,n S :::: 0 and therefore LdtH - L d , > 0 a.s. for every s > O. We conclude
that for any fixed t the point dt(w) is in .E(w) for a.e. w and, consequently, for
a.e. w the point dr(w) is in .E(w) for every r E Q+.
Pick now s in Z(w) and an interval 1
 s. Since Z(w) is a.s. closed and has
empty interior, one may find r such that r < s, r E Q+ n 1 and r fj. Z(w); plainly
d r :S s, thus sis the limit of points of the closed set .E(w), hence belongs to .E(w)
and we are done.


Remarks. 1°) The fact that TO = 0 a.s. is worth recording and will be generalized
in Chap. X.
2°) The equality between Z(w) and .E(w) is also a consequence of Exercise
(1.26) and Proposition (3.12) in Chap. III.




242 Chapter VI. Local Times
B.6) Corollary. P(S/.s > 0, BTi = Bz_ = 0) = 1. Conversely, for any и е Z,
either и = rv or и — rv for some s.
Proof. The first statement is obvious. To prove the second let и > 0 be a point in
Z = Z; then, either Lu+E — Lu > 0 for every s > 0, hence и = inffr : L, > Lu)
and и — rs for s — Lu, or L is constant on some interval [и, и + s], hence
Lu — Lu-n > 0 for every ц > 0 and м is equal to rs- for .v — Lu.
Remark. We have just proved that the points of Z which are not left-ends of
intervals of f- are points of right-increase of L.
We close this section with P. Levy's Arcsine law which we prove by using the
above ideas in a slightly more intricate context. The following set-up will be used
again in an essential way in Sect. 3 Chap. ХШ.
We set
*,+ = / \(B,>0)ds, At = /
Jo Jo
and call ar+ and at the associated time-changes. Our aim is to find the law of the
r.v. A~l and since [A~l > t\ = {ar+ < l}, this amounts to finding the law of ar+.
But since и — A+ + A~ entails ar+ = / + A~(ar+), we will look for the law of
A~(af). The following considerations serve this particular goal.
The processes A± are the increasing processes associated with the martingales
,+ = I 1,b,>o,^S5. M,- = I 1(B,<O
Jo Jo
Obviously (A/+, M~) — 0, and so, by Knight's theorem (Theorem A.9) Chap.
V), since A^ — Ax = со, there exist two independent Brownian motions, say S+
and S~, such that M±{a±) — <5± and by the definition of the local time L of B,
B + (a+) =8+ + -L(a+), В "(or;) = -S~ + -Ца~),
where B+ and В are simply the positive and negative parts of В (and not other
processes following the ± notational pattern).
Using Lemma B.1) once more, we see that (B±(a±), ^L(a±)) has the same
law as (| В|, L) and moreover that
-L{a?) = sup(-<$+), -L{a;) = sup(<5;).
? .s<r I %<t
We now prove the
B.7) Theorem. The law of A\ is the Arcsine law on [0, 1].

§2. The Local Time of Brownian Motion 243
Proof. For a > 0 let us put
Ts'(a) = T^=M[t:S; > a}.
We claim that for every /, we have A~ = Ts~ (t/2). Indeed, by definition of a,",
в- = -s~(a;) + l,/2
whence B~ (r,) = 0 = —8~(A~) + t/2. Moreover, r is a.s. a point of right increase
of L (see Corollary B.6)), hence there is a sequence (sn) decreasing to r, such
that B~ = 0 and LSn > LTi = t and consequently 8~(A~) > t/2. It follows that
A~ > Ts (t/2). Now, if и < A~, then и — A~ for some v < r,. If v < r,- then
If r,- < v < т„ then A~ increases between r,- and r, which implies that В is
negative on this interval, hence В is > 0 and again &~(A~) is less than i/2,
which proves the reverse inequality A~ < T5 (t/2).
Moreover A~(a+) = A~ (хца*)У, indeed tl,. = v if v is a point of increase
of L and if v belongs to a level stretch of L and v — a,+ for some /, then В is
positive on this stretch and A~(tiv) = A~.
Combining these two remarks we may write that for each fixed /,
Now, jL(af) is independent of S since it is equal to supJ<((— <5S+). Thus we
have proved that A~(af) = Ts (S,) where 5, is the supremum process of a BM
/6 independent of S ~. By the scaling properties of the family T^ and the reflection
principle seen in Sect. 3 of Chap. Ill we get
ЙГГ2
where С is a Cauchy variable with parameter 1.
By the discussion before the statement, a^ — t(] + C2) and
Hence, the r.v. Aj1" follows the law of A + C2)~' which is the Arcsine law as can
be checked by elementary computations (see Sect. 6 Chap. 0).
The reader ought to ponder what the Arcsine law intuitively means. Although
the BM is recurrent and comes back infinitely often to zero, the chances are that
at a given time, it will have spent much more time on one side of zero than on
the other.

244 Chapter VI. Local Times
# B.8) Exercise. 1°) For the linear BM and a < x < у < b prove that
E* [^„at-J =2u(x, v)
where u(x, y) = (x — a)(b — y)/(b — a).
2°) For a < у < x < b set u(x, y) = u(y. x) and prove that for any positive
Borel function /
-Т„лТ,, Л pb
Г гТ„лТ,, "I pb
\j f(Bs)ds\=2j u(x,y)j\y)dy.
This gives the potential of BM killed when it first hits either a or b.
B.9) Exercise. Let (P,) be the semi-group of BM and put f(x) — \x\. Prove that
1 V
L, = lim - / (PhfiBs) - f(Bs))ds a.s.
[Hint: Write the occupation times formula for the function P,,f - /.]
# B.10) Exercise. 1°) Prove that for a > 0, the processes
E, - B, +a"')exp(-o'5,) and (\B,\+a~[)sxp(-aLt)
are local martingales.
2°) Let Ux = inf{t : S, - B, > x] and fx = inf{/ :\B,\>x]. Prove that both
Su, and Lf follow the exponential law of parameter x~x. This can also be proved
by the methods of Sect. 4.
# B.11) Exercise (Invariance under scaling). Let 0 < с < oo.
P) Prove that the doubly-indexed processes (В,, Ц) and (Brl, L", j /*/c,
a e R, t > 0, have the same law.
2°) Prove that the processes (rr) and (c~'r^,) have the same law.
3°) If as usual Ta = inf {/ : B, = a) prove that the doubly-indexed processes
(LXT) and («r'L'jJ), x e R, a > 0, have the same law.
# B.12) Exercise. Prove that .TtB — .J^S~B. In other words, if you know S - В
up to time t you can recover В up to time /.
# B.13) Exercise. If В = (В1, В2) is a standard planar BM and r, is the inverse of
the local time of В' at zero, prove that X, = B* is a symmetric Cauchy process.
Compare with Exercise C.25) of Chap. III.
# B.14) Exercise. 1°) Prove that the two-dimensional process (|Br|- ^L{\B\)t) has
the same law as the processes of Theorem B.3).
2°) Conclude that the local time of |S,| (resp. S, - Bt) is equal to 2L, (resp.
25,). See also Exercise A.17).

§2. The Local Time of Brownian Motion 245
B.15) Exercise. 1") Fix / > 0. Prove that for the standard linear BM, there is a.s.
exactly one s < t such that Bs = 5,, in other words
P [3(r, s) : r < s < I and Br = B, = 5,] = 0.
[Hint: 25 is the local time at 0 of the reflected BM S - B. This result can
actually be proved by more elementary means as is hinted at in Exercise C.26) of
Chap. III.]
2°) Prove that G\ = sup{s < 1 : Bs — S\] has also the Arcsine law; thus G\
and A\ have the same law.
[Hint: Use Exercise C.20) Chapter III.]
B.16) Exercise. Let X be the standard BM reflected at 0 and 1 (see Exercise
A.14) of Chap. III).
Г) Prove that X, = 0, + Ц - Ц where 0 is a standard linear BM and L"
the symmetric local time (Exercise A.25)) of X at a.
2°) By extending Lemma B.1) to this situation prove that
Ц = sup (-A + Ц)+ , L) = sup U + Ц - \Y .
B.17) Exercise. Prove that the filtration (.?[x) of the martingale X, = /„' B\dB2s
introduced in Exercise D.13) of Chap. V, is the filtration of a BM2.
[Hint: Compute (X, X).]
B.18) Exercise. 1°) Prove that the joint law of (|fi,|, L,) has a density given by
B/л73I/2(а + b) exp (-(a + bJ/2t), a, b > 0.
Give also the law of (B,, L,).
2°) Prove that the 2-dimensional process (\B,\, L,) is a Markov process with
respect to {.7f) and find its transition function.
The reader will find a more general result in Exercise A.13) of Chap. X and
may also compare with Exercise C.17) in Chap. III.
B.19) Exercise. 1°) Prove that almost-surely the random measure v on R defined
by
Hf)= f f
Jo Jo
f(B,- Bs)ds dt
has a continuous density with respect to the Lebesgue measure. Prove that this
density is Holder continuous of order a for every a < 1.
[Hint: This last result can be proved by using the same kind of devices as in the
proof of Corollary A.8); it is also a consequence of the fact that the convolution
of two Holder continuous functions of order a is Holder continuous of order 2a.]
2°) More generally, for every Borel subset Г of [0, I]2, the measure
f(B, - Bs)ds dt

246 Chapter VI. Local Times
has a continuous density a(x, Г) with respect to the Lebesgue measure. The map
a is then a kernel on R x [0, I]2.
[Hint: Use Exercise A.15).]
3C) Prove that, if/ e Ll(R), a.s.
\m^n f f f(n(B, - Bs))dt ds = (f f(a)da\ (f {L\f db\.
B.20) Exercise. With the notation used in the proof of the Arcsine law, prove
that В is a deterministic function of <5+ and <5~, namely, there is a function / on
C(R+, RJ such that В = /(<5+, <5~).
[This exercise is solved in Chap. XIII, Proposition C.5).]
B.21) Exercise. Prove the result of Exercise A.26) on the support of dL, by
means of Proposition B.5) and the DDS theorem of Sect. 1 in Chap. V.
B.22) Exercise. Let / be a locally bounded odd function on Ш with a constant
sign on each side of 0 and such that the set [x : f(x) = 0) is of zero Lebesgue
measure. Prove that the filtration generated by M, — /0' f(Bs)dBs is that of a
Brownian motion.
[Hint: Use Exercise C.12) Chap. V.]
B.23) Exercise. Let В be the standard linear BM. Prove that f(Bt) is a (.>^
submartingale if and only if / is a convex function.
[Hint: A function / is convex if and only if / + / admits no proper local
maximum for any affine function / whatsoever.]
* B.24) Exercise. Let X be a continuous semimartingale, if it exists, such that
(*) X,=x + B,+ a(s)dL,
Jo
where В is a BM, a is a deterministic Borel function on R+ and L — L°(X).
1°) Prove that if a < 1 the law of the process L is uniquely determined by a.
[Hint: Write the expression of |X| and use Lemma B.1).]
2°) Let g,(X) — E [exp(/XX,)] and prove that
g,(X) = cxp(iXx) - у j gAWs + i^E j a(s)dLs .
As a result the law of the r.v. X, is also determined by a. Using the same device
for conditional laws, prove that all continuous semimartingales satisfying equation
(*) have the same law. In the language of Chap. IX, there is uniqueness in law
for the solution to (*). The skew BM of Exercise B.24) Chap. X is obtained in
the special case where a is constant.
3°) Prove that L^~ = /0' A -2a(s))dLs and that as a result there is no solution
X to (*) if a is a constant > 1/2.

§2. The Local Time of Brownian Motion 247
B.25) Exercise. Let X be a continuous process and La the family of local times
of BM. Prove that for each / the process
Ya =
'a = f XudV'u
Jo
is continuous.
* B.26) Exercise. 1Э) Retaining the notation of Levy's modulus of continuity in
Sect. 2 Chap. I, prove that
lim sup S,2 — Stl /h(e) <
fiO
2°) Let U I,,, where /„ = [л„, /„], be a covering of the set Z = {/ e [0, 1]
B, = 0}. Prove that
and derive therefrom that Ah(Z) > 0 (see Appendix 4 for the definition of Ah).
3°) Prove that the Hausdorff dimension of Z is > 1/2 a.s. The reverse in-
inequality is the subject of the following exercise.
* B.27) Exercise. 1°) Let у be a measure on R, С and a two constants > 0, such
that in the notation of Appendix 4, v(/) < C\I\" for every interval. If Л is a
Borel set of strictly positive v-measure, prove that the Hausdorff dimension of A
is > a.
2 ) By applying 1°) to the measure dL,, prove that the Hausdorff dimension of
Z is > 1/2 a.s. Together with the preceding exercise, this shows that the Hausdorff
dimension of Z is almost-surely equal to 1/2.
**
B.28) Exercise (Local times of BM as stochastic integrals).
1°) For / e Ск(Ш), prove that for the standard linear BM,
f f(Bs)ds= I E[f(Bs)]ds +lim [ ds f (Ps-vf)'{Bv)dBv.
Л ¦ Jo fi° Л Л
[Hint: Use the representation of Exercise C.13) Chap. V.]
2°) If we put q(x) = 2f": exp(-u2/2)du prove that
Jo
f E[f(Bs
Jo
1 f+oc
lim . / f(\')dv ...
no J^ J_^ K- -
dBv.
[Hint: Use Fubini's theorem for stochastic integrals extended to suitably inte-
grable processes.]

248 Chapter VI. Local Times
3°) Conclude that /„' f(Bs)ds = f^ f(y)L)dy where
= I 8s(y)ds - -i= f
Jo V2tt Jo
sgn(B,, - >•)<? f^M^) dBv.
\ *Jt - v )
** B.29) Exercise (Pseudo-Brownian Bridge). Г) Let h be a bounded Borel func-
function on R+ and set
M, = K (fh(s)dBs\ .
If ф is a continuous function with compact support, prove that the process
r00
Ya = / 4>{t)M,dL% аеК,
Jo
is continuous.
[Hint: Use Exercise B.25).]
2°) Let Q" be the law of the Brownian Bridge from 0 to x over the interval
[0, u] and write simply Q" for Qj] (see Exercise C.16) Chap. I for the properties
of the family (?>")). If Z is a positive predictable process prove that
рос poo
/ E[ZTt]ds= / Q"[ZU}-
Jo Jo
du
fbru
where Q"[Zu] = jZudQu.
[Hint: Compute in two different ways E [f^ fn(B,)Z,dt] where (/„) is an
approximation of the Dirac mass <5o and Z, = ф(г)М,, then let n tend to +00.]
3°) If F is a positive Borel function on C([0, 1],R) and g a positive Borel
function on Ш+ prove that
where /3" is the Brownian Bridge from 0 to 0 over the time interval [0, и].
4°) Let X be the process defined on [0. 1] by
"иг,, 0<И<1
which may be called the Pseudo-Brownian Bridge. Prove that
T 1
E[F(Xe,u<l)] = E . уг„,~ _ -/v ,
' n X
where X is the local time of ^6' at level 0 and at time 1.

§2. The Local Time of Brownian Motion 249
[Hint: Use the scaling invariance properties to transform the equality of 3°).
Then observe that 1/,/гГ is the local time of X at level 0 and time 1 and that
1 /¦'
= lim— / 1]_,.?[(Х,)
f--0 IS Jo
5°) Prove that the processes (B,; t <t\) and (BTl^,; t < ri) have the same
law.
6°) Prove that X has the law of \/2e where e is exponential with parameter
1. This is taken up in Exercise C.8) of Chap. XII. This question can be solved
independently of 5°) .
B.30) Exercise. In the notation of this section, prove that the process {g^l/2Bugl,
0 < и < 1} is a Brownian Bridge which is independent of a(g\, Bgl+U, и > 0).
[Hint: Use time-inversion and Exercise C.10) Chap. I.]
# B.31) Exercise. In the situation of Proposition C.2) Chap. V, prove that there
are infinitely many predictable processes H such that f?° Hjds < oo, and
F = E[F]+ / HsdBs.
Jo
[Hint: In the notation of this section, think about l[o.T,]-]
B.32) Exercise. Let X be a cont. loc. mart, vanishing at 0 and set
5,=supXs and X, — sgn(Xs)dX,.
.sir Л
Г) Prove that the following two properties are equivalent:
i) the processes S — X and |X| have the same law,
ii) the processes X and —X have the same law.
2°) Let X and Y be two cont. loc. mart, vanishing at zero and call /3 and у
their DDS Brownian motions. Prove that X and Y have the same law iff the 2-
dimensional processes (fi, (X, X}) and (y, {Y, Y)) have the same law. In particular,
if Y — X, then X and Y have the same law iff conditionally w.r.t. .5Q, ' , the
processes /3 and у have the same law.
3°) If ¦Уго^х'Х) = -7^\ then the laws of X and X are not equal. Let ф be a
continuous, one-to-one and onto function from R+ to [a, b[ with 0 < a < b < oo,
/3 a BM1 and A the time-change associated with /J ф(\{is\)ds, then if X, = /3a,,
the laws of X and X are not equal. Notice in addition that there is a BM denoted
by В such that X, = /0' ф(\Xs\yi|2dBs.
4°) Likewise prove that if ,F^ ' = .>*oo\ then the laws of X and X are
not equal. Change suitably the second part of 3°) to obtain an example of this
situation.

250 Chapter VI. Local Times
** B.33) Exercise (Complements to the Arcsine law). 1°) In the notation of The-
Theorem B.7), prove that for every positive Borel function F on C([0, 1], R),
E [F (Bu. и < 1) l(fl|>0)] - E ГA/О F Uu</^t, "-'
[Hint: It may be helpful to consider the quantity
where ф is a positive Borel function on R+.]
2°) Prove that for every positive Borel function / on [0, 1] x R+, one has
E [f (A + , L]) l(B|>0)] - E [(А+/Г,) / (A+/tu 1/r,)].
3°) Prove that the law of the triple T~l [Aj, Aj, L2T) is the same for all the
following random times:
i) T = t (a constant time), ii) T = or+, iii) T — xu.
B.34) Exercise. 1°) Let В be the standard BM1 and L its local time at 0; for
h > 0, prove that there is a loc. mart. M such that
log(l + h\B,\) = M, - (\/2)(M, M), + hL,.
2°) Let у be the DDS BM of -M\ set p, = y, + (\/2)t and a, = supi:Sr ps.
Using Skorokhod's lemma B.1), prove that for every t > 0,
log(l +ft|?,|)=crv, -pVi a.s.
where V, = /о'(й-' + |?,|)~2</5.
3°) Define as usual fa = inf{f : |B,| = a] and tj = inf{u : Lu > s] and prove
that
Vfj = inf [u : au - pu = log(l + h)} and Vr, = inf {u : pu = hs].
The laws of Vf and Vti are obtained at the end of Exercise C.18) Chapter VIII.
B.35) Exercise (Local times of the Brownian bridge). 1°) Let x > 0. In the
notation of this section, by considering the BM Br+r, — x, and using Theorem
B.3) prove that
P[L\ <y,Bi >b] = (l/2)P[Si >x + y, S{ - Bi > \b-x\].
Consequently, prove that, conditionally on Bx — b, the law of L\, for 0 < x < b,
does not depend on x.
2°) Using the result in Exercise B.18) and conditioning with respect to
{B\ = 0J, prove that if (/л) is the family of local times of the standard BB, then
l\ l=' (Я - 2x)+ ,
where R has the density l(r>0)'' exp (-(r2/2)). -

§3. The Three-Dimensional Bessel Process 251
§3. The Three-Dimensional Bessel Process
In Chap. XI we will make a systematic study of the one-parameter family of the
so-called Bessel processes using some notions which have yet to be introduced. In
the present section, we will make a first study of the 3-dimensional Bessel process,
which crops up quite often in the description of linear BM, using only the tools
we have introduced so far.
We first take up the study of the euclidean norm of ВМг which was begun in
Chap. V for i5 = 2 and in the preceding section for S = 1. Let us suppose that S is
an integer > 1 and let p, be the modulus of BMa. As usual, we denote by Px the
probability measure of the ВМг started at x and (.^") is the complete Brownian
filtration introduced in Sect. 2 Chap. III.
C.1) Proposition. For every S > 1, the process p,, t > 0, is a homogeneous {¦7',)-
Markov process with respect to each Ps, x e Ш.&. For S > 2, its semi-group Pts is
given on [0, oof by the densities
p*(a,b) = (a/t)(b/a)s/2Is/2-i(ab/t)exp(-(a2 + b2)/2t) for a, b > 0,
where lv is the modified Bessel function of index v, and
pf(O, b) = F(S/2JiS/2)-lrs/2bs-1 exp(-b2/2t).
Proof. Let / be a positive Borel function on [0, oo[. For s < t,
Ex [/(A) I Щ = EB, [/(|fi,_,|)] = P,-J(B,) Px - a.s.,
where f(x) = f(\x\) and P, is the semi-group of BM5.
For <5 > 2, we have
PJ(x) = Bnt)-&/2 j exp(-|* - y\2/2t) /(l.vl)dv
and using polar coordinates
PJ(x) =
Bntrs/2 f exp (-(M2 + P2)/2t) exp(-\x\pcose/t) f{p)p&
where ;; is the generic element of the unit sphere and в the angle between x and
г). It turns out that P,f(x) depends only on \x\ which proves the first part of the
result (the case S = 1 was studied in Exercise A.14) of Chap. III). Moreover,
setting Pf f(a) = P,f(x) where x is any point such that \x\ = a, we see that Pf
has a density given by
exp(-(a2+b2)/2t) / exp(-abcosO/t)o(dri)
which entails the desired result.

252 Chapter VI. Local Times
C.2) Definition. A Markov process with semi-group Pts is called a 5-dimensional
Bessel process.
Bessel processes are obviously Feller processes. We will write for short BES*
and BES*(jt) will designate a E-dimensional Bessel process started at x > 0. The
above result says that the modulus of a BM* is a realization of BES*. From the
results obtained for BM5, we thus deduce that a BES5 never reaches 0 after time
0 if 5 > 2. Moreover for S > 3, it is a transient process, that is, it converges a.s.
to infinity.
From now on, we will focus on the 3-dimensional process BES3, which we
will designate by pt. The semi-group PC has a particularly simple form which can
be seen from the expression of I\/j. We call Q, the semi-group of the linear BM
on ]0, oof killed when it hits zero. It was seen in Exercise A.15) of Chap. Ill that
Q, is given by the density
q, (x, y) — g, (x - y) - g, (x + >•). x > 0 and у > 0.
If we set h(x) = x on ]0, oo[, it is readily checked that Q,h — h. The semi-group
P,3 is what will be termed in Chap. VIII as the /j-transform of Q,, namely
x>0:
in other words, PC is given by the density x~]q,(x. y)y. For x = 0, we have
P,3/@)= f B/nt3)l/2exp(-y2/2t)y2ny)dy.
Jo
We will also need the following
C.3) Proposition. If (p,) is a BES3U) with x > 0, there is a Brownian motion
such that
p,=x+
s, + I p;
Jo
Moreover, pt ' is a local martingale (in the case x — 0, the time-set is restricted
to ]0, oo[).
Proof. We know that p, may be realized as the modulus of BM3; using the fact
that /3, — Y?\ /o' P7lB\dB\ is a BM' (p- Levy's characterization theorem) the
result follows easily from lto's formula, and the fact that p, never visits 0 after
time 0.
Remark. The first result says that p is a solution to the stochastic differential
equation dp5 = dfis + p^lds (see Chap. IX).
Another proof of the fact that p~' is a local martingale is hinted at in Exercise
B.13) of Chap. V where it is used to give an important counter-example. It will
now be put to use to prove the

§3. The Three-Dimensional Bessel Process 253
C.4) Corollary. Let P3 be the probability measure governing BES3(x) with x > 0
and Ta be the hitting time ofa>0. For 0 < a < x < b,
and P3 [Ta < oo] = a/x. Moreover, 70 = inf5>0 ps is uniformly distributed on
[О, дг].'
Proof. The local martingale p~] stopped at Ta is bounded, hence is a martingale
to which we may apply the optional stopping theorem. The proof then follows
exactly the same pattern as in Proposition C.8) of Chap. II. We then let b go to
infinity to get P^ [Ta < oo]. Finally
Ръх [70 <a] = Pi [Ta < oo] = a/x
which ends the proof.
We now turn to our first important result which complements Theorem B.3),
the notation of which we keep below, namely В is a BM'(O) and S, — sup5<, Bs.
C.5) Theorem (Pitman). The process p, = 25, - B, is a BES3@). More precisely,
ifp, is a BES3@) and J, — infs>, ps, then the processes B5, - B,, 5,) and (p,, J,)
have the same law.
Proof. Let p be a BES3@). If we put X, = 27, - р„ we shall prove that for each
/, J, — supv<, Xs. Indeed, if J, = p,, then X, = J, and for s < t, since 7, < ps,
we get X.s = 27V - ps < Js < J, = X, which proves our claim in this case; if
p, ф 7,, then p, > 7, and X, > 7, = 7ft where g, — sup[s < t : Js = p5}. Since
by the first part 7ft — supJ<ft Xs, we get the result in all cases.
We have thus proved that {p,, J,) — B7, - X,, J,) with 7, = sups<, Xs and
consequently, it remains to prove that X is a BM. To this end it is enough by
P. Levy's characterization theorem (Sect. 3 Chap. IV) to prove that X is a mar-
martingale, since plainly (X. X), = (p, p), = t.
We first notice that Js = 7, л infs<M<, pu for s < t and therefore the knowledge
of 7, and of ps, s < t, entails the knowledge of 7V, s < t. As a result .7"tx с
¦ 7^pva(Jt). On the other hand ctG,) С .7"tx since 7, = sups<, Xs and.7^p с ^х
since p, = 27, - X,. Consequently .7jx = .7,p v<jG,). Since X, < p, and
— Xt < Pt* each r.v. X, is integrable; thus, to prove that X is a martingale, it
suffices to show that for any а > 0 and s < t,
Call K the right-hand side of (*). Corollary C.4) implies that
'-' if a < z.
if «>z,
and
P3 [70 > a] = P} [Ta = oo] = j I aZ

254 Chapter VI. Local Times
¦зг, . if (z-a2z-l)/2 if a
.-Kol.Jb-.J- о if a
< z.
if и > z.
Using this and the Markov property for p, it is not difficult to prove that К =
(a -a2p;[) l(A>a).
Now, call к the left-hand side of (*); we have
k — E^lX \ 1 ¦ T'9 i^l
?3 ГF^ Iv 1 i 'is'РЛ \ i ^?"pl
о L^o Iхi '(./,>«) I -^ J Mint;-,,., р„>п) I -л J.
and using the above computation of К with t instead of л\ we obtain
since [p^i)) IS a bounded martingale. It follows that к — К which ends the
proof. D
It is interesting to observe that although 25 — В is a Markov process and is
(¦71B)-adapted, it is not a Markov process with respect to (-^s); indeed by the
early part of the preceding proof, .7^B contains some information on the future
of 2S — В after time t. This is at variance with the case of S — В studied in the
last section where .3^S~B = -7[B (see Exercise B.12)). Here .7\1S~B is strictly
contained in .WtB as is made plain by the following
C.6) Corollary. The conditional distribution of S,, hence also of S, — B,, with
respect to .7^2S~B is the uniform distribution on [0. 25, — B,].
Proof. By the preceding result it is also the conditional distribution of J, with
respect to a(ps,s < t), but because of the Markov property of p, this is the
distribution of 70 = infv>ops where p is a BES3 started at p,. Our claim follows
from Corollary C.4).
It was also shown in the last proof that
C.7) Corollary. If p is a BES3(.v), then B, = 27, - p, is a Brownian motion
started at 2J{) - x and .V^ = a (j,,.7^p).
Finally Theorems B.3) and C.5) brought together yield the
C.8) Corollary. If В is BM@) and L is its local time at 0, then \B\ + L is a
BES3@).
We will now strive to say more about Jq which is the absolute minimum of p.
C.9) Proposition. Let p be a BES3. lfT is a stopping time of the bivariate process
(p, J) such that рт = Jt, then рт+i — Рт is a BES3@) independent of{p,, t < T).

§3. The Three-Dimensional Bessel Process 255
Proof. Let us first suppose that p starts at 0. Using the notation and result of
Corollary C.7), we see that T is also a stopping time of B. Consequently by the
strong Markov property or by Exercise C.21) of Chap. IV, B, = BT+I - BT is a
BM@) independent of .F~TB. By the hypothesis made on T
BT - 2JT - рт ~ pT-
As a result, the equality J, = sups<, B, proved at the beginning of the proof of
Theorem C.5) shows that, with obvious notation,
JT+, - JT= sup Bs - BT = S,.
s<T + l
The definition of В implies that
Pt+i - Pt = 2JT+, - BT+! - pT
= 2(JT+I -JT)-(Br+, -BT)
= 2S,-B,.
Pitman's theorem then implies that pj+, — pT is a BES3@). We know moreover
that it is independent of .WTB hence of {p,, / < T).
If p starts at x > 0, by the strong Markov property, it has the same law as
р~т,+г, t > °> where p is a BES3@). It suffices to apply the above result to p; the
details are left to the reader. D
The first time at which p attains its absolute minimum
r = inf {t : p, = /о)
obviously satisfies the conditions of the above result. Therefore, since a BES3
never reaches 0, it follows that т is the only time at which p is equal to Jo. We
recall moreover that pT = Jo is uniformly distributed on [0, x] and we state
C.10) Proposition. Let p be a BES3(x) with x > 0/ the process (p,. t < r) is
equivalent to (B,. t < Ty) where В is a BM(x) and Ty is the hitting time by В of
an independent random point у uniformly distributed on [0, x].
Proof. By Corollary C.7), B, = 27, - p, is a BM started at 270 - x. For / < r,
we have J, — Jo, hence B, = 2Jo — p,\ as a result, for / < r, we have p, =
2J0 - B, = P, where p is a BMU). Moreover j$, = x - (B, - Bo) is independent
of Bo hence p, is independent of Jo = pz and г = inf {/ : ft, = Ju). Since 70 is
uniformly distributed on [0, x], the proof is complete.
The above results lead to an important decomposition of the path of BES3@).
C.11) Theorem (Williams). Pick с > 0 and the following Jour independent ele-
elements
i) a r.v. a uniformly distributed on [0. c];
ii) a BM(c) called B;

256 Chapter VI. Local Times
Hi) two BES3@) called p and p.
Put Rc = inf{/ : p, = c}; Ta — inf{/ : B, — a}. Then, the process X defined by
p,.
if / < Rc
if Rc < t
if t > Rc
Rc + Ta
is a BES3@).
Proof. If we look at a BES3@), say p, the strong Markov property (Proposition
C.5) in Chap. Ill) entails easily that the processes {p,, t < Rc} and {/0,+Я|. / > Oj
are independent and the second one is a BES3(t). Thus, the theorem follows from
the preceding results. ?
This theorem, as any path decomposition theorem, is awkward to state but is
easily described by a picture such as Figure 5 in which a is uniformly distributed
on [0, c].
-BES3@)
According to Proposition C.9), the last part can be further split up in two
independent parts at time Lc or, for that matter, at any time Ld with d > c.
Indeed, since the BES3 converges a.s. to infinity, for every с > 0, the time
LL = sup {/ > 0 : p, — c}.
where we agree that sup@) = 0, is a.s. finite. Since we have also
L, — inf {/ : p, = J, = <¦},
this is also a stopping time as considered in Proposition C.9).
In Sect. 4 of Chap. VII, the foregoing decomposition will be turned into a
decomposition of the Brownian path. We close this section with another application
to BM. We begin with a lemma which complements Proposition C.3).

§3. The Three-Dimensional Bessel Process 257
C.12) Lemma. If p is a BES3(.x), x > 0, p~x is a time-changed ВМ(л~') re-
restricted to [0. To[.
Proof. By Proposition C.3) and the DDS theorem of Sect. 1 Chap. V, we have
p-] = fi(A,) where A, = f'Qp~4ds and /Ms a BMfjt). Since p > 0 and
lim^oc p~x — 0, we have Ax — inf{/ : /?, = 0} and the result follows.
We may now state
C.13) Proposition. Let В be a BM(a), a > 0, and M — max \B,J < Tq}; then,
the following properties hold:
(i) the r.v. M has the density ax'2 on [a, oo[;
(ii) there is a.s. a unique time v < To for which Bv — M.
Furthermore, conditionally on M — m,
(Hi) the processes X] — (B,,t < v) and X2 = (Bv+I, 0 < / < To - v) are inde-
independent;
(iv) the process Xх is a BES3(a) run until it hits m;
(v) the process m — X2 is a BES3@) run until it hits m.
Proof. Using the notation of the preceding proof, we have
(ts,.t < i0) — \pc . t < Ax)
where С is the inverse of A. Thus properties i) and ii) are straightforward con-
consequences of Propositions C.9) and C.10). Property iii) follows equally from
Proposition C.9) applied at time r when the BES3(a~') process p reaches its
absolute minimum.
To prove iv) let us observe that
X[ — (p[.x. t < AT | pT = \/m).
By Proposition C.10) there is а ВМ(<зг'), say y, such that
(p;x.t<r\pT = ]/m)^(yi-xj<TUm).
As a result.
where A, = _/0' ys 4ds and С is the inverse of A. But by lto's formula.
i f -? , f -i
Y, = a — / Ys dys + I ys ' ds
Jo Jo
which entails

258 Chapter VI. Local Times
with /? another BM. As a result, the process vT1 satisfies the same stochastic
differential equation as BES3 and, by the uniqueness result for solutions of SDE's
which we will see in Sect. 3 of Chap. IX, this process is a BES3 (a). Property (iv)
now follows easily. Property (v) has a similar proof which we leave as an exercise
to the reader.
Remark. The law of M was already derived in Exercise C.12) of Chap. II.
C.14) Exercise. Extend Corollary C.6) to the stopping times of the filtration
C.15) Exercise. Let X be a BES3(O) and put L = sup{/ : X, = 1). Prove that the
law of L is the same as the law of T\ — inf{/ : B, = 1} where В is a BM@).
[Hint: Use the fact that 25, - B, is a BES3@).]
This will also follow directly from a time-reversal result in Sect. 4 Chap. VII.
** C.16) Exercise (A Markov proof of Theorem C.11)). Pick b > 0 and the fol-
following three independent elements:
(i) a r.v. у uniformly distributed on [0. b}\
(ii) a BES3@)-process p,
(iii) a BM(fr)-process B,
and define TY — inf{/ : B, = y),
X, = B, or. t. < TY), X, = p(t -TY) + y on {t > Ту).
1°) Prove that for t > 0, the conditional probability distribution P[X, e •: TY <
t | y] has a density equal to
Э
I(v>k)(/ - y)—<it{b - У,У - Y)
dy
with respect to the Lebesgue measure dy.
2°) Prove that consequently, if tQ — 0 < t\ < ti < ... < t,,, the conditional
probability distribution of the restriction of (X, XK) to {?,_, < TY < ?,} with
respeci to у has a density equal to
( 9
l(y<inf(v,)) I -—</,,-,,_, (JC/-1 - y.Xj - y)
„ - у)
with respect to the Lebesgue measure dx\ ... dxn.
3°) For 0 < с < b, prove that X conditioned by the event {у > с) is equivalent
to p + с where p is a BES3(/? - c). In particular, X is a BES3@).
4°) Use 3°) to give another proof of Theorem C.11).

§3. The Three-Dimensional Bessel Process 259
C.17) Exercise. Г) Let В be the standard linear BM and set X, = B, +t. Prove
that the process exp(—2X,) is a time-changed BMA) killed when it first hits 0.
2C) Let у — inf, X,; prove that ( — y) is exponentially distributed (hence a.s.
finite) with parameter 2 and that there exists a unique time p such that Xp = у.
The law of (—y) was already found in Exercise C.12) of Chap. II; the point here
is to derive it from the results in the present section.
C.18) Exercise. 1°) With the usual notation, put X — \B\+L. Write the canonical
decompositions of the semimartingale X in the filiations (¦^x) and (.!7\ ) and
deduce therefrom that the inclusion .У^ С -V,m is strict. This, by the way, gives
an example of a semimartingale X = M + A such that (->^"x) С (^M) strictly.
2°) Derive also the conclusion of 1") from the equality
L, = M(\BS\ + LS).
3°) Let now с ф 1 and put X = \B\ +cL. The following is designed to prove
that conversely .7~x = .i^'8' in that case.
Let D be the set of (со, t)'s such that
''™ ^ ^ Цхи~2^.а>)-Х(,-2-^<.ш)>0} = I/2-
k<n
Prove that
a) D is ../^-predictable;
b) for fixed I > 0, P[{lo : (со, t) e D}] = 1;
c) for fixed / > 0, P[{со : (w. r,(co)) e D}] — 0 where as usual r, is the inverse
of L.
Working in .Ув, prove that
f \D(s)dXs = f sgn(Bs)dB,
and derive the sought-after conclusion.
C.19) Exercise. If p is the BES3@) prove that for any t,
P Ш\ \Pt+h - p, I /y/2h log2 \Jh = 1=1.
|_''i0 J
[Hint: Use the result in Exercise A.21) Chap. II and the Markov property.]

260 Chapter VI. Local Times
§4. First Order Calculus
If M is a martingale, К a predictable process such that /0' K2d(M, M)s < сю, we
saw how stochastic integration allows to construct a new local martingale, namely
KM with increasing process K2-(M, M). We want to study the analogous problem
for the local time at zero. More precisely, if L is the local time of M at 0 and if
К is a predictable process such that \K\-L is finite, can we find a local martingale
with local time \K\ ¦ L at 0? The answer to this question will lead to a first-order
calculus (see Proposition D.5)) as opposed to the second order calculus of lto's
formula.
Throughout this section, we consider a fixed continuous semimartingale X
with local time L at 0. We use a slight variation on the notation of Sect. 2, namely
Z = {t : X, = 0} and for each t
g, = sup {s < t : Xs = 0}, d, = inf{s > t : Xs = 0}.
D.1) Lemma. If К is a locally bounded predictable process, the process Кк is
locally bounded and predictable.
Proof. If Г is a stopping time, then gj < T, so Kx is locally bounded if К
is. It is enough to prove the second property for a bounded К and by Exercise
D.20) Chap. I and the monotone class theorem, for К — lfo.rj. But in that case
КK = l[o.</r] and one easily checks that dj is a stopping time, which completes
the proof.
The following result supplies an answer to the question raised above.
D.2) Theorem, i) IfY is another cont. semimart. such that Y,i, = Ofor eveiy t > 0,
then Kgl Y, is a continuous semimartingale and more precisely
Л?| /, = Лу/о -
In particular. K/, X is a continuous semimartingale.
ii) IfY is a local martingale with local time Л at zero. K4Y is also a local
martingale and its local time at zero is equal to
f
Jo
\KKs\dA,.
In particular, if X is a local martingale, then Kg X is a local martingale with local
time at 0 equal to /0' \Kgt\dLs = /0' \Ks\dLs.
Proof By the dominated convergence theorem for stochastic integrals, the class
of predictable processes К for which the result is true is closed under pointwise
bounded convergence. Thus, by the monotone class theorem, to prove i), it is
once again enough to consider the case of К = l[o.r]- Then, because Y,It = 0 and
Kn, = hx,iT] = 1['<<1тЪ

§4. First Order Calculus 261
[ '' dY,
Jo
f 1,»,,
Jo
= K0Y0+ f l[0,dl](s)dY, = K0Y0+ [ KgidY.s
J J
f [ gi
Jo Jo
which proves our claim.
To prove ii) , we apply i) to the semimartingale |K| which clearly satisfies the
hypothesis; this yields
f
Jo
f \Kg,\sgn(Y,)dY,+ f \KXMAS
Jo Jo
f sgti(KgJs)d(Kg,Ys) + f \Kgi\dA,
Jo Jo
= \K0Y0\+
= \K0Y0\+
which is the desired result.
We may obviously apply this result to X as Xd = 0. The local time of Kg X
is thus equal to
/
Jo
/0
But the measure dLs is carried by Z and #,. = s for any s which is the limit
from the left of points of Z; the only points in Z for which g, / s are therefore
the right-end points of the intervals of Zc\ the set of which is countable. Since
L is continuous, the measure dLs has no point masses so that \Kg^\ = \KS\ for
rfL-almost all s which completes the proof. ?
If / is the difference of two convex functions, we know that f(X) is a semi-
semimartingale, and if/@) = 0, then f(X)d = 0 and consequently
['
Jo H'
In this setting, we moreover have the
D.3) Proposition. //'</> : R+ -»¦ M+ is locally bounded.
,) = f(Xn)<t>@)+
I
Jo
Proof. We apply the above formula with K, =</>(?,) and take into account that
Lg: = L, as was already observed. ?
We now apply the above results to a special class of semimartingales.
D.4) Definition. We call E the class of semimartingales X = N + V such that
the measure dV, is a.s. carried by Z = [t : X, = 0}.

262 Chapter VI. Local Times
If, for instance, M is a local martingale, then the semimartingales \M\ and M+
are in E with respectively V = L and V — \h. This will be used in the
D.5) Proposition. IfXeE, the process
X,Kg: - f KsdVs =X0K0+ f KsdNs
Jo Jo
is a local martingale. If M is a local martingale, the processes
\M,\Kg, - f KsdLs, M^Kgl - l- f KsdLb,
Jo 2 Jo
are local martingales. Finally, iftp is a locally bounded Borel function and ф(х) =
f? (p(u)du, then
Фа,)-\м,\уа,), -Фа,)-м^а,),
are local martingales.
Proof. The first two statements are straightforward consequences of Theorem D.2)
and its proof. The third one follows from the second by making K, = (p(L,) and
using the fact that /0' (p(L^)dLx = 0(L,) which is a simple consequence of time-
change formulas. ?
In Theorem B.3), we saw that the processes (S - B, S) and (|?|, L) have
the same law; if for a local martingale M we put S, = sup(<, Ms, it should not
be surprising that one can replace in the above formulas L, by 5, and \M,\ by
S, — M,, although, in this generality, the equality in law of Theorem B.3) will not
necessarily obtain (see Exercise B.32)). In fact, the semimartingale X — S — M is
in Z1, with V = S, since clearly 5 increases only on the set {S — M) = {X = 0}.
As a result, the processes
(S,~M,)Ky- KsdSs, 0E,) - E, - M,)(p(S,)
Л
where y, — sup{s < t : Ss — Ms), are local martingales.
Observe that Proposition D.3) can also be deduced from the integration by
parts formula if 0 is C1 and likewise the last formula of Proposition D.5) if 0 is
C2. For instance, if 0 e C2, then <p(S,) is a continuous semimartingale and
',) -E, - M,)ip(S,) - 0E,)-/ ip(Ss)dSs+\ tp(Ss)dMH
Jo Jo
(S, - M5)ip'(Ss)dSs.
f
Jo
But the last integral is zero, because dS is carried by {S — M = 0}. Moreover
f'Q <p(Ss)dS, = 0E,) so that

§4. First Order Calculus 263
Jo
- (S, - Mi)tp(S,
which is a local martingale. We can thus measure what we have gained by the
methods of this section which can also be used to give yet another proof of Ito's
formula (see Notes and Comments). Finally, an elementary proof of the above
result is to be found in Exercise D.16).
Ito's and Tanaka's formulas may also be used to extend the above results to
functions F(Mt, Si, {M, M),) or F(M,, L,, (M, M),) as is seen in Exercise D.9).
These results then allow to compute explicitly a great variety of laws of random
variables related to Brownian motion; here again we refer to the exercises after
giving a sample of these computations.
D.6) Proposition. If В is a standard BM and Th - inf{/ : B, = b), then for
0 <a < b and X > 0,
E [exp (-XL'l)] = A + 2X(b - a))
that is to say, LaTi has an exponential law with parameter {2(b — a)) .
Proof. By Proposition D.5), ((l/2) + k(B, -a)+) exp(-AL°) is a local martingale
for any X > 0. Stopped at 7),, it becomes a bounded martingale and therefore
E\\-+X(b-i
We now state a partial converse to Proposition D.5).
D.7) Proposition. Let (/,.*) —» F(l.x) be a real-valued Сl:'-function on Ж2+. If
M is a cont. loc. mart, and (M, M)x — oo a.s., and if F(L,. \M,\) is a local
martingale, then there exists a C2 -function f such that
Proof. If F(Lt, \M,\) is a local martingale, Ito's and Tanaka's formulae imply that
{F!(LS. \MJ) + F'X(LS. \M,\))dLs + l-F"ALs. \Ms\)d{M. M), = 0 a.s.
The occupation times formula entails that the measures dLs and d(M. M)s are
a.s. mutually singular; therefore
(F,'(Ls,0) + F;x(Ls,0))dLs = 0. F'xl(L,.\Ms\)d(M, M)S = 0 a.s.
Now since (M, M)x = oo, the local time L is the local time of a time-changed
BM and consequently Lx = oo a.s. By the change of variables formula for
Stieltjes integrals, the first equality then implies that

264 Chapter VI. Local Times
for Lebesgue-almost every /, hence for every / by the assumption of continuity
of the derivatives. By using the time-change associated with (M, M), the second
equality yields P [F^(/,, |B,|) = 0 Л-а.е.] = 1 where В is a BM and / its local
time at 0 and because of the continuity
P[f;2(/,.|1»,|) = 0 for every t] = I.
For every b > 0, it follows that
Fi':(ln,b)=0 a.s.
But by Proposition D.6) the law of lTh is absolutely continuous with respect to
the Lebesgue measure; using the continuity of F, it follows that F(-.b) = 0
for every b. As a result
and since F is continuously differentiable in / for every x, it follows that /
and g are C'-functions; furthermore the equality F/(/, 0) + F'x(l, 0) = 0 yields
g(l) — —/'(/) which entails that / is in C2 and completes the proof.
# D.8) Exercise. Let M be a uniformly integrable continuous martingale and L its
local time at 0. Set G = sup{s : Ms = 0).
1°) Prove that for any bounded predictable process К,
\J KsdLs] =
2°) Assume now first that P(MX = 0) = P(G = 0) = 0 and deduce from 1=)
that
Е[КГи\ти < oo]du =
edu).
where zu — inf {/ : L, > u). In particular,
P(LX e du) = (?
P(LK > и)
check this formula whenever M is a stopped BM, say BT, for some particular
stopping times such as T = r, T — Ta, ...
Finally, prove that in the general setting dP du-a.s.,
E [KTif; zu < oo] = E [kg\Mx\ | L^ = и] Je [iM^I | L«, = h] .
# D.9) Exercise. Let M be a local martingale, L its local time, S its supremum.
1°) Prove that the measures d(M, M) and dS are mutually singular.
2°) If F : (x.y.z) -»¦ F{x,y\z) is defined on R x R2+, and is sufficiently
smooth and if
- F;2 + Fl = 0, Fv(>', y, z) = 0 for every x, j and z,

§4. First Order Calculus 265
then F (M,, S,. (M, M),) is a local martingale. Find the corresponding sufficient
condition for F (M,, L,. (M, M),) to be a local martingale.
3°) Prove that (S, — M,J — (M, M), is a local martingale and that for any
reals a, ft, the process
aS
aS, - ~{M.M),
is a local martingale. Prove the same results when S - M is replaced by \M\ and
S by L.
4C) If В is a BM and fa = inf{f : \B,\ = a], then for a > О, Р ф 0,
E exp \-aLf - yfa\ = P W cosh aft + a si
5°) Prove an analogous formula for S and Ra = inf{f : S, — B, = a).
6°) Again for the BM and with the notation of Sect. 2, prove that
E [exp (-f, )]=,-.
* D.10) Exercise. F) Let Wbea martingale and L its local time at 0. For any p > 1
prove that ||L,||P < p\\M,\\p. For p = 1 and Мй - 0, prove that \\M,\\i = \\L,\\\.
[Hint: Localize so as to deal with bounded M and L; then apply Proposition
D.5).] For p > 1, prove that \\S,\\P < (pjp - 1)||M,||P.
2°) Show that there is no converse inequality, that is, for p > 1, there is no
universal constant Cp such that
for every M locally bounded in Lp.
D.11) Exercise. Let M be a square-integrable martingale vanishing at zero and
set s, = inf(<, Ms. Prove the following reinforcement of Doob's inequality
E[(S, -s,J]<4E[Mf].
[Hint: Use 3°) in Exercise D.9).] Prove that this inequality cannot be an equal-
equality unless M vanishes identically on [0. /].
D.12) Exercise. For the BM and b > 0 set fh = inf{/ : \B,\ - b]. Prove that Lf/
has an exponential law with parameter \jb. (In another guise, this is already in
Exercise B.10)).
D.13) Exercise. For the BM call д the law of the r.v. sj, where s, = infv<, B,.
1°) Using the analogue for л of the local martingale </>(?,) - E, - В,)ф'(Б,),
prove that д = ( - A — х)ц)' where the derivative is taken in the sense of
distributions.
Prove that consequently, ц. has the density A — xJ on ] - oo, 0[.

266 Chapter VI. Local Times
2°) Using this result, prove that the result of Exercise D.11) cannot be extended
to p ф 2, in the form
E[{S, -st)»]<CpE[\Mt\"]
with Cp = (p/p- \)p.
[Hint: Take M, = S,Af|, where Tx = inf{/ : |B,| = 1}.]
# D.14) Exercise. V) Let M be a continuous local martingale vanishing at 0, and
F a C2-function. For a > 0, prove that
а;)-; I F'(L"s)dL°s
^ Jo
is a local martingale.
2°) For the Brownian motion, compute the Laplace transform of the law of
L« where r, = M{s : L°s > /}.
D.15) Exercise. Let M be a positive continuous martingale such that Mx = 0.
Using the local martingale </>(S,) — (S, — M,)tp(St) for a suitably chosen ^), find a
new derivation of the law of the r.v. S^ conditioned on .3^, which was found in
Exercise C.12) of Chap. 11.
# D.16) Exercise. Following the hint below, give an elementary proof of the fact
that, in the notation of Proposition D.5), the process (/>(S,) — (S, - M,)ip(St) is a
local martingale.
[Hint: Assume first that M is a bounded martingale, then derive from the
equality
E[MT\a>Tii)] = Е[МТа\а>т,Л
that
E[(ST-a)+ -(ST-MT)\{sT>a)]
does not depend on the stopping time 7". This is the result for tp(x) — 1(С>„ь
extend to all functions tp by monotone class arguments.]
* D.17) Exercise. 1 ) Let X be a cont. loc. mart, and (L") the family of its local
times. For any Cx function / on R'_j_ and ct\ < ai < ... < a,,, prove that
/ (L? ,Lar С) - ? 2(X, - а,У |? (L'/' L?)
ax,
(ah-ah)...
... («,,_, -aid){X - fl,-,
is a local martingale.

§4. First Order Calculus 267
2C) For (у\ Yn) e K" and an < 1, prove that for the BM
where
ф(х,у)= 1 +^2(x-a,)+y;
n
This generalizes Proposition D.6) and gives the law of the process a —» L^
which will be further identified in the Ray-Knight theorem of Sect. 2 Chap. XI.
(See Exercise B.11) in that chapter).
* D.18) Exercise. Let ф be an increasing C'-function on R+ such that ф@) = О
and 0 < ф(х) < x for every x. Let / be а С'-function such that
and set
F(x) = I f(y)dy = f(x)(x -
Jo
1°) In the notation of this chapter, prove that
" l^ = / F""
[Hint: Using Theorem B.3), prove first that the left-hand side has the same
law as /Qr° \{f{L,)\Bt\<F(Ls))ds, then apply Proposition D.5) to f(Ls)Bs.] The same
equality holds with < in place of >.
2°) If/ is the local time at 0 of the semimartingale В — (/>(S) prove that
г
Jo
(foF-l(Ss))~ldLs.
[Hint: Write / as the limit of e ' /0 l(o<B1-0E1)<f)ufi", then follow the same
pattern as in 1°) .]
3°) Carry out the computations for ф(х) = ax, a < 1.
* D.19) Exercise. Let В be the standard linear BM and L its local time at 0.
1°) If AT is a strictly positive, locally bounded (.^"B)-predictable process, prove
that M, = KgtBu t > 0, has the same filtration as B.
2°) Prove that Nt = /„' Kg,dLs - Kgr\Bt\ has the same filtration as \B\.
3°) For p > 0, prove that the local martingale M = LP~XB is pure.
[Hint: If r, is the time-change associated with (M, M), express r, as a function
of (Ltjp) which is the local time of the DDS Brownian motion of A/.]

268 Chapter VI. Local Times
4°) Prove that, for p > 0, the local martingale Ц' - р\В,\Ц'~\ t > 0, is also
pure.
[Hint: Use Lemma B.1).]
* D.20) Exercise. (More on the class LlogL of Exercise A.16) Chap. II). We
consider a cont. loc. mart. X such that Xo = 0 and write X* = sups<, |X,| and
1°) Prove that
\X,\ log+ \X,\ - (X* - 1) - f log+ X*dL,
Jo
is a local martingale.
2°) Prove that
< {e/(e - D) (l+ sup E [\XT\ log+
where T ranges through the family of all finite stopping times and that
sup? [\XT\ log+ \XT\] < ((e + \)/e)E [X*x] + E [L*, log+ Lx].
т
3°) Prove likewise that L, log+(L,) - (L, - 1)+ - |X,|log+(L,) is a local
martingale and derive that
E [LK \og+(Lx)] < sup {((e + l)/e)E[\XT\] + E [\XT\ log+ \XT\]\.
т
4°) Conclude that the following two conditions are equivalent:
i) suprEjjXrllog+IXH] <oo;
ii) E [XZc] < oo and E [Lx log+ Lx] < oo.
# D.21) Exercise (Bachelier's equation). 1°) Let M be a cont. loc. mart, vanishing
at 0. Prove that there exists a unique strictly positive cont. loc. mart. M such that
Mo = 1 and, in the notation of this section,
s,-m, = (m, a
where s, — inf Mu.
[Hint: Try for s, = exp(—S,).]
2°) Let A be a function on K+ which is strictly decreasing, of class C1 with
non vanishing derivative and such that A@) = 1, h(oo) — 0. Given a loc. mart. M
satisfying the conditions of 1°) , prove that there exists a unique cont. loc. mart.
M such that
-h'{h-l(sl))(Sl-M,) = M,--St.
3°) Prove that the local martingale M of 2°) satisfies the Bachelier equation
dM, =dM,lh'(S,).
Give a sufficient condition on h in order that this equation have a unique solution
and then express it as a function of M.

§5. The Skorokhod Stopping Problem 269
D.22) Exercise (Improved constants in domination). p) For the BM В prove,
in the usual notation, that for к < 1,
E[(-sT,)k] < E | sup(S, - Bs)k\ < Ck,
U<t, J
where Ck is the constant defined in Exercise D.30) Chap. IV. Conclude that
(nk/sinnk) < Г(\ -к) < Ck.
[Hint: Use the results in Exercises D.12) and D.13).]
2°) Combining 1°) with Exercise D.30) Chap. IV, prove that
lim(l -k)Ck = 1.
D.23) Exercise. 1") In the usual notation, set T = inf{? : \L,B,\ = 1} and prove
that the law of LT is that of v^ where e is an exponential r.v. with parameter 1.
[Hint: Express the local time at 0 of the loc. mart. LB as a function of L and
follow the pattern of the proof of Proposition D.6).]
2C) Prove that consequently, LTBT is a bounded martingale such that the
process H of Proposition C.2) Chap. V is unbounded.
D.24) Exercise (An extension of Pitman's theorem). Retain the notation of
Proposition D.5) and the remarks thereafter and suppose that ц> is positive and
that Mo - 0.
P) Prove that
S,) = sup / yiS^
[Hint: Apply Lemma B.1).]
2°) Prove that (S, — M,)ip(St) is a time-changed reflected BM and identify its
local time at 0.
3°) Prove that (/>(S,) + (S, - M,)ip(St) is a time-changed BES3@).
§5. The Skorokhod Stopping Problem
Let д be a probability measure on R; we wish to find a stopping time T of the
standard linear BM such that the law of Вт is /x. In this generality the problem
has a trivial solution which is given in Exercise E.7); unfortunately this solution
is uninteresting in applications as T is too large, namely E[T] = oo. We will
therefore amend the problem by demanding that E[T] be finite. This however
imposes restrictions on /x. If E[T] < oo, the martingale BT is in Ml which
implies E[Bj] = 0 and furthermore (S,rJ — T л t is uniformly integrable so
that E[BJ] = E[T]. The conditions
/ x2diJL(x) < oo, I x dii(x) = 0,

270 Chapter VI. Local Times
are therefore necessary. They are also sufficient, and indeed, the problem thus
amended has several known solutions one of which we now describe. We actually
treat the case of cont. loc. martingales which by the DDS theorem is equivalent
to the case of BM.
In what follows all the probability measures д we consider will be centered,
i.e., will satisfy
/ \x\dii(x) < oo, xd/i(x) = 0.
For such а д, we define jx(x) — д([х, oo[) and
y^(jt) = jx(x)~x / t dfi{t) ifx < b = infjx : jl{x) = 0}.
J[.x.ocl
\j/^(x) = x if x > b.
The functions д and x/r,, are left-continuous; \j/^ is increasing and converges to b
as x —> b. Moreover, г}гц(х) > x on ] - oo, b[ and b — inf{x : i/r/((x) = x}.
E.1) Lemma. For every x e] — oo, b[ and every a < 0,
— ~r exp |
ix(a) \jftl(x)-x
Proof. By definition
" X У[л.
and taking regularizations to the right,
Let us call Д (resp. v) the measure associated (Sect. 4 Chap. 0) with the right-
continuous function of finite variation д(х+) (resp. i/rM(x + )); we have p. = — д
and using the integration by parts formula of Sect. 4 Chap. 0, the above equality
reads
(+) jx(x)dv(x) + (i/lt (x+) - x) djx(x) = 0,
this equality being valid in ] — oo. b[ where i/f,(x) > x.
By the reasoning in Proposition D.7) Chap. 0, there exists only one locally
bounded solution Д to (+) with a given value in a given point. But for a < 0, the
function
is a solution to (+). This is seen by writing
ф(х+) (ф,Лх+) -x) = exp (- f —^
and applying again the integration by parts formula which yields
(Irll(x+) - x) с1ф(х) + <l>(x)dv{x) = 0.
The proof is then easily completed.

§5. The Skorokhod Stopping Problem 271
The preceding lemma shows that the map /i —»• i/^ is one-to-one; the next
lemma is a step in the direction of showing that it is onto, namely, that every
function possessing the properties listed above Lemma E.1) is equal to фц for
some /i.
E.2) Lemma. Let ф be a left-continuous increasing function and a < 0 < b two
numbers such that \//(x) = 0 for x < a, i//(x) > x for a < x < b, and \jr(x) = x
for x > b. Then, there is a unique centered probability measure /i with support in
[a, b] and such that i/^ = \J/.
Proof. We set p(x) = 1 for x < a,
-f ('/Ф) -s)~lds 1 for a < x < b.
Ja /
This function is left-continuous on ] — oo, b[\ furthermore, it is decreasing. Indeed,
this is easy to see if \j> is C1; if not, we use а С°° function j > 0 with support
in ]0. 1] and such that / j(y)dy — 1, and we set г//„(х) = n f f(x + y)j(ny)dy
as in the proof of Theorem A.1). Then \j/n > \J/ and limn ф„(х) — ф(х+),
whence the property of Д follows. We complete the definition of Д by setting
ji(b) = \\mx^b..x<biJ-(x), and Д(х) = 0 for x > b. There is then a unique
probability measure /i such that ?L(x) — д([х, oo[). By differentiating the equality
-x) = -aexpl - / {\jr(s) - s)~X ds\
\ Ja /
we get
( = -p.(x)d\//(x)
which may be written d(\jjjl) — x dp.. As a result
if(x)P(x) = j t dii(t).
J[\.oo[
Taking x < a this equality shows that /i is centered and the proof is complete.
We will also need the following
E.3) Lemma. If f x2dfi(x) < +oo, then
/ x2dii(x) = / i/fi(x)p(x)dx.
Proof. By eq(*) we have
/ x2dli(x) = - dx
and integrating by parts
Jx2dii(x) = J pi(x)irlt(x)dx - [x

272 Chapter VI. Local Times
But
lim xfx(x+)i/,Ax+) — lim x I I d/x(t) < lim / t2d/u(t) = 0
x^x x~*x J].x.x[ x^xJ]x.^l
whereas, because /i is centered,
lim (-.х)A(х+)фц(х+) = lim x / / dn(l)
¦^~x x^-x J]-x.x]
< lim
From now on we consider a cont. loc. mart. M such that MQ = 0 and
{M, М)ж = oo. We set Tx = infj/ : M, — x] and S, = supJ<r Ms. The main
result of this section is the following
E.4) Theorem. If /л is a centered probability measure, the stopping time
7д =inf{/>O:S, >фц(М,)}
is a.s. finite, the law of Mj is equal to /i and, moreover,
i) MT'1 is a uniformly integrable martingale,
ii) E[{M,M)Tu]=fx2d[x(x).
We illustrate the theorem with the next
Fig. 6.
Proof of the Theorem. The time 7^ is a.s. finite because for any x > 0 we have
Тм <inf{f >TX: M, = ^(дг)}
and this is finite since (M, M)x = oo. To complete the proof of the theorem we
use an approximation procedure which is broken into several steps.

§5. The Skorokhod Stopping Problem 273
E.5) Proposition. Let (/i,,) be a sequence of centered probability measures such
that (i/,,,,) increasespointwise to i/^/ then
i) (Mn) converges weakly to /л;
ii) lim,, / \x\dfxn(x) = f \x\dii(x);
Hi) (Mrm) converges a.s. to Mj:.
Proof. The sequence (fin) is weakly relatively compact because for any К > 0,
lin([K,oo[) < — tdfin(l)<— t dfi,,(t)
л J{K.-yo{ Л J[0.oc{
= Т7^„(О)Д»(О)< -^@),
л л
and, recalling that /i is centered,
— (-t)diiAt) <— / (-t)diin(t)
Л J]-oo.-K] K J]-oc.O]
Л l\
Moreover, since \j/IXn increases to i/^ it is easily seen that bn = inf{/ :
\j/IJn(t) — t\ increases to b. From Lemma E.1) it follows that Д„(х)/Д„@) con-
converges to Д(х)/Д@) on ] — oo, b[ and Дя = Д = 0 on ]b, oo[. Let (/i,,() be
a subsequence converging weakly to a probability measure v. If x is a point of
continuity of v, then
НтДЯ((х) = v(x).
If we choose x < b, it follows that Д,„@) has a limit /, otherwise Д„, (х)/Д„(@)
could not have a limit. Moreover
С(х)// = Д(х)/Д@).
By taking a sequence х„ of continuity points of v increasing to zero, it follows
that / = Д@) and v = Д which proves i).
Using the lower semi-continuity of /i —> f,0oc,t d/i(t), we have
' ^„@
[O.cc[ " ^1().эс[
Hm t dfi,,(t) < lim /
/i J[0.oc[ " JlO
= НшД„@)^„@) = Д@)^@)= f
Ju
using the fact that \in and д are centered we also have
lim / t dii,,(t) = / t dii(t)
" J]-oo.0] J]-oo.O]
t dfi(t);
[O.cx)[
which establishes ii).

274 Chapter VI. Local Times
Finally, the sequence Т„ — ТЦп increases to a stopping time R < Тц. But for
p < и, we have
ФиРШт.) < Фц.(Мт.) < ST. SSr;
passing to the limit, we get
¦фЦрШк) <lim фц„(М1:) < SR
hence i/,j(MR) < SR and R = Тц. By the continuity of M the proof is complete.
E.6) Lemma. There exists a sequence of centered probability measures \xn such
that
i) fin has a compact support contained in an interval [a,,, b,,];
ii) фЦп is continuous on К and strictly increasing on [а„, оо[;
Hi) the sequence (ф1Хл) increases to ij/,, everywhere on K.
Proof. Let a — sup{x : ij/^ix) — 0), and (а„) resp: (bn) a sequence decreasing
resp: increasing strictly to a resp: b. The number
<5„ = inf(i/f (x) - x; — oo < x < bn\
is finite and > 0. Let j be the function in the proof of Theorem A.1), and for
each n, pick an integer к„ such that
/
For —oo < x < bn, set
/ , (X) \ f
Vn(x) — l(x -а„+1I(а„+1<л<«„ + l[o,,.oo[) / Фц(х + y)knj(
The function ф„ enjoys the following properties:
i) Фп < Фц and ф„ is continuous on ] - oo, Ь„],
ii) ф„ > 0 on ]an+i,bn],
iii) ф„(х) > x on ] - oo, bn],
(indeed, for x e [0, b,,],
ф„(х)-х=к„ I (\lrfl(x + y)-(x + >•)) j (k,,y) dy + kn / у j(kny)dy
which is > 0 by the choice of к„).
We now define ф„ on ]Ь„, oof in the following manner. Since bn < ф„{Ь„) <
Фц(Ь„), we let фп be affine on [Ь„, ф^ф»)] and set фп(х) = x for x > дF„) in
such a way that ф„ is continuous. Finally, we set ф„ = ф\ v ф2 v • • • v ф„; by
Lemma E.2), the sequence (ф„) enjoys the properties of the statement.

§5. The Skorokhod Stopping Problem 275
End of the Proof of Theorem E.4). By Proposition E.5) and Lemma E.6) it is
enough to prove the first sentence when the support of fi is contained in the
compact interval [a.b] and i/f^ is continuous and strictly increasing on [a, oof.
Since фц(х) = 0 for x < a we have Tfl < Ta and since \j/(x) — x for x > b we
also have T/x < Tb, hence MT" is bounded.
Let у be the inverse of the restriction of i/^ to [a, oo[; for ф € Ск we set
g = фоу and G(x) = /J)V g(u)du. By the remarks following Proposition D.5) the
process X, = G(S,) - (S, - M,)g(St) is a local martingale. The functions ф and
G being bounded, XT" is a bounded martingale and consequently
E[G{STii)-(STl,-MTii)g{STi,)]=0.
By the definitions of g and 7^ this may be written
Г гФ,ЛМТ/1) -i
? I У 0 о y(ii)rfii - (^ (AfrJ - МТц) ф (МТн)\ - 0.
If у is the law of MTi we have
| v(dx) I 0(u)rf^(u) + I (х- ^^х^ф^^х) = О,
and after integrating by parts
х) + (x - ^(x))dv(x)] = 0.
/
Since ф is arbitrary in С к it follows from Lemma E.1) and its proof that v — /л.
To prove i) choose a sequence (ф„) according to Lemma E.6). For each n,
the process Afr"» is a bounded martingale. Moreover \Мт„„\ converges a.s. to
\MT)i | and by Proposition E.5) ii), E [\МТIп |] to E [\MTji |]; it follows that \MT)in \
converges to \MTJ in Z.1. The proof of i) is then easily completed.
It remains to prove ii). When \x has compact support, A/r'< is bounded and by
Proposition A.23) Chap. IV, E [{M, M)Tit] = fx2dn(x). To get the general case
we use again an approximating sequence. Set, with A(x) = x,
By Lemma E.2), ф„ corresponds to a measure д„ and by Lemma E.3) if
fx2dii,,(x) < oo,
/ x2d[i,,(x) = / \1/ц (х)Д„
J J-n
By Lemma E.1), for —n < x < n, we have Д„ = С„Д(х) where lim,, C,, — 1.
Therefore
x2dix,,(x) = С„ I фц(х)р.(х)с1х + I ф|x{x)^^L„(x)dx.
J—n J—n

276 Chapter VI. Local Times
We will prove that the last integral on the right, say /„, goes to 0 as и tends to
infinity. Indeed, Д„ is also constant on [и, г//,Ап)] and is equal to С„Д(и). Hence
h = С„^(и)Д(и)(^.(и)-и).
If X is a r.v. with law д, then:
In/Cn = E[X\^n)]E[(X~n)\(X>n)]/P(X>n)
< E[X2]U2E[((X-n)+J]l/2.
which proves our claim.
As a result, lim,, f x2d/j.n(x) < oo. By the proof of Proposition E.5) the
sequence {Tlh} increases to T^ so that
E[{M, М)т„] = Vim E[{M, M)Tiin] = \ x2d[x{x).
# E.7) Exercise. Let В be the standard linear BM.
1°) For any probability measure д on M prove that there is a ,У[р measurable
r.v. say Z, such that Z(P) = д.
2°) Define a .^-stopping time T by
T = inf{t > 1 : B, = Z).
Prove that the law of Bj is \x and that E[T] — oo.
* E.8) Exercise (A uniqueness result). 1°) Let g be a continuous strictly increas-
increasing function such that \imx^.0Og(x) — 0, g(x) > x and g(x) = x for all
x > inf(w : g(u) - u}. If for T = inf {/ : 5, > g(M,)} the process MT is a
uniformly integrable martingale and if MT has law д, prove that
/ t
In particular if д is centered, then g = i/r,,.
[Hint: Use the first part of the proof of Theorem E.4).]
2°) Extend the result to the case where g is merely left-continuous and in-
increasing.
E.9) Exercise. In the situation of Theorem E.4) prove that the law of STi is given
Ьу
P [ST > x] = exp ( - / ———) ,
where y(s) = inf {г : ^@ > •$}•

Notes and Comments 277
E.10) Exercise. Let В be the standard linear BM and fi a centered probability
measure. Prove that there is an increasing sequence of finite stopping times Т„
such that the random variables Вт„^ — Вт„ are independent, identically distributed
with law \x and E [7"n+i — Г„] = f x2dn(x).
E.11) Exercise. Prove that the time Tt, of Theorem E.4) has the following min-
minimality property: if R is a stopping time such that R < Тц and MR — MTpt, then
Notes and Comments
Sect. 1. The concept and construction of local time in the case of Brownian
motion are due to P. Levy [2]. The theory expanded in at least three directions.
The first to appear was the theory of local times for Markov processes which is
described in Blumenthal-Getoor [1] (see also Sharpe [1]) and will be taken up in
the Notes and Comments of Chap. X. A second approach is that of occupation
densities (Geman and Horowitz [1]). The point there is to show that the measure
A —> Jq 1л(Х,)Л is absolutely continuous with respect to a given deterministic
measure which usually is the Lebesgue measure on R. This is often done by Fourier
transform methods and generalizes in the theory of intersection local times which
has known much progress in recent years and for which the reader may consult
Geman et al. [1], Rosen [1], Le Gall [4] and Yor [18]; the Markov view-point on
this question being thoroughly developed in Dynkin [2]. These two approaches to
local times are fleetingly alluded to in some exercises e.g. Exercise B.19).
The third and possibly most useful line of attack stems from the desire to
enlarge the scope of Ito's formula; this is the semimartingale point of view which
first appeared in Meyer [5] after earlier results of Tanaka [1] for Brownian motion
and Millar [2] for processes with independent increments. The case of continuous
semimartingales is the subject of this section. The reader can find another exposi-
exposition based on the general theory of processes in Azema-Yor [1] and the extension
to local times of regenerative sets in Dellacherie-Meyer [1] vol 4.
Theorem A.7) which extends or parallels early results of Trotter [1], Boylan
[1] and Ray [1] is taken from Yor [4] and Theorem A.12) from Bouleau-Yor [1]
as well as Exercise A.29). The approximation results of Corollary A.9), Theorem
A.10) and Exercise A.20) as well as some others to be found in Chap. XII, were, in
the case of Brownian motion, originally due to Levy (see Ito-McKean [1] and for
semimartingales see El Karoui [1]). Ouknine-Rutkowski [1] give many interesting
"algebraic" formulae for the computation of local times, some of which we have
turned into exercises.
Exercise A.21) is from Weinryb [1], Exercise A.20) from McGill et al. [1] and
Exercise A.26) from Pratelli [1]. Exercise A.17) is due to Yoeurp [1], Exercises
A.14) and A.22) are respectively from Yor [5] and [12]. Exercise A.29) is from
Biane-Yor [1]; it extends a result which is in Ito-McKean [1] (Problem 1, p. 72).
Principal values of Brownian local times have been studied in depth by Yamada

278 Chapter VI. Local Times
([2], [3], [4], [6]) and also by Bertoin [2] to whom Exercise A.30) is due; they
have been investigated for physical purposes by Ezawa et al. ([1], [2], [3]).
Sect. 2. The results of the first half of this section are due to Levy but the proofs
are totally different. Actually Levy's study of Brownian local time was based on
the equivalence theorem B.3), whereas we go the other way round, thanks to
Lemma B.1) which is due to Skorokhod [2] (see also El Karoui and Chaleyat-
Maurel [1]). Among other things, Theorem B.3) shows that the Brownian local
time is not after all such an exotic object since it is nothing else than the supremum
process of another BM.
Corollary B.8) gives a precise labeling of the excursions of BM away from
zero which will be essential in Chap. XII.
The first proof of the Arcsine law appears in Levy ([4], [5]). The proof pre-
presented here is found in Pitman-Yor [5] and Karatzas-Shreve [1] but the original
ideas are due to Williams [1] and McKean [3]. There are other proofs of the Arc-
sine law, especially by the time-honoured Feynman-Kac's approach which may
be found in Ito-McKean [1]. Another proof relying on excursion theory is found
in Barlow-Pitman-Yor [1] (see Exercise B.17) Chap. XII).
Exercise B.13) is due to Spitzer [1] and Exercise B.15) is in Ito-McKean [1].
The equality in law of Exercise B.15), 2C) is a particular case of a general result
on Levy processes (see Bertoin [7]) which is an extension to continuous time of
a combinatorial result of Sparre Andersen (see Feller [4]).
Exercise B.22) is taken from Lane [1] and Exercise B.24) from Weinryb [I].
Exercise B.27) is taken from Le Gall [3]; another method as well as Exercise B.26)
may be found in Ito-McKean [1]; the results are originally due to Besicovitch and
Taylor [1] and Taylor [1]. Exercise B.28) is from Yor [18] and Exercise B.29)
from Biane et al. [1].
Pushing the ideas of Exercise B.32) a little further we are led to the following
open
Question 1. Is the map /J —»- /3 ergodic?
Dubins and Smorodinsky [1] have given a positive answer to a discrete ana-
analogue of Question 1 for the standard random walk on Z. Furthermore Dubins et
al. [1] A993) give another interesting question equivalent to Question I along the
lines of Exercise B.32). Let us note that Question I may be extended to functions
of modulus 1 other than sgn and also to a ^-dimensional setting using the result
in Exercise C.22) Chapter IV.
Exercise B.33) presents one of many absolute continuity relationships (see,
e.g., Yor [25]) which follow from scaling invariance properties; for a different
example, see Exercise B.29) relating the Brownian and Pseudo-Brownian bridges.
The method used in Exercise B.33) may be extended to yield Petit's generalization
of Levy's Arcsine law (see Petit [1] and Carmona et al. [1]).
Sect. 3. Most of the results of this section are from Pitman [1], but we have
borrowed our proof of Theorem C.5) from Ikeda-Watanabe [2]. The original proof
of Pitman uses a limiting procedure from the discrete time case to the continuous

Notes and Comments 279
time case. This can be used successfully in many contexts as for instance in Le
Gall [5] or to prove some of the results in Sect. 3 Chap. VII as in Breiman [1].
For other proofs of Theorem C.5) see Pitman-Rogers [1], Rogers [2] and Jeulin
[1], as well as Exercise D.15) in Chapter Vll. A simple proof has been given by
lmhof [3].
Theorem C.11) is due to Williams [3] as well as several exercises. Exercise
C.18) is taken from Emery-Perkins [1].
Sect. 4. The better part of this section comes from Azema-Yor [1] and Yor [8].
Earlier work may be found in Azema [2] and extensions to random closed sets in
Azema [3]. As mentioned below Proposition D.5) another proof of Ito's formula
may be based on the results of this section (see Azema-Yor [1]) and thus it is
possible to give a different exposition of many of the results in Chaps. IV and VI.
Exercise D.9) is due in part to Kennedy [1]. Exercise D.11) is from Pitman
[2] and is continued in Exercise D.13).
Exercise D.17) is in Ito-McKean [I] (see also Azema-Yor [2]) and Exercise
D.20) in Brassard and Chevalier [I]. Exercise D.21) is due to L. Carrara (private
communication). Exercise D.25) comes from Pitman [7].
Let us mention the
Question 2. Is it possible to relax the hypothesis on F in Proposition D.7)?
Sect. 5. The problem dealt with in this section goes back to Skorokhod [2] and
has received a great many solutions such as in Dubins [3], Root [1], Chacon-Walsh
[1] to mention a few. In discrete time the subject has been investigated by Rost
(see Revuz [3]) and has close connections with Ergodic theory.
The solution presented in this section is taken from Azema-Yor [2] with a
proof which was simplified by Pierre [1] (see also Meilijson ([1] and [2]) and
Zaremba [1]). We thank D. Lamberton who helped us improve some technical
points concerning the approximation procedure. Further simplification is to be
found in Vallois [1]. Another proof given by Rogers [1] is based on Excursion
theory and is outlined in Exercise D.14) of Chap. XII. This construction has
specific extremal properties described in Exercise E.11) due to Meilijson (see also
Perkins [7] and Dubins-Gilat [1]). A more complete list of references and newer
developments on this subject are found in Kertz-Rosler [1] and, e.g., Hobson [1].

Chapter VII. Generators and Time Reversal
In this chapter, we take up the study of Markov processes. We assume that the
reader has read Sect. 1 and 2 in Chap. III.
§1. Infinitesimal Generators
The importance of the theory of Markov processes is due to several facts. On the
one hand, Markov processes provide models for many a natural phenomenon; that
the present contains all the information needed on the past to make a prediction
on the future is a natural, if somewhat overly simplifying idea and it can at least
often be taken as a first approximation. On the other hand, Markov processes arise
naturally in connection with mathematical and physical theories.
However, the usefulness of the theory will be limited by the number of pro-
processes that can be constructed and studied. We have seen how to construct a
Markov process starting from a t.f, but the snag is that there aren't many t.f.'s
which are explicitly known; moreover, in most phenomena which can be modeled
by a Markov process, what is grasped by intuition is not the t.f. but the way in
which the process moves from point to point. For these reasons, the following
notions are very important.
A.1) Definition. Let X be a Feller process; a function f in Co is said to belong
to the domain (/a of the infinitesimal generator of X if the limit
A/= lim-(/>,/-/)
iio t
exists in Co. The operator A : C/A —>¦ Co thus defined is called the infinitesimal
generator of the process X or of the semi-group P,.
By the very definition of a Markov process with semi-group (P,), if / is a
bounded Borel function
E [f(X,+h) - f(X,) | .Щ = Phf(X,) -
As a result, if / € С/д, we may write
E [f (Xl+h) - f(Xt) | .Щ = hAf(X,) + o(h).

282 Chapter VII. Generators and Time Reversal
Thus A appears as a means of describing how the process moves from point to
point in an infinitesimally small time interval.
We now give a few properties of A.
A.2) Proposition. Iff e C/A, then
i) P,fe (/A for every t;
ii) the function t —>¦ P,f is strongly differentiable in Co and
— P,f = AP,f = P,Af\
at
Hi) Ptf-f = /„' Ps Af ds = /„' APJ ds.
Proof. For fixed t, we have, using the semi-group property,
lim - [Ps(P,f) - Р.П = Hm P, Г- (PJ - /I = P,Af
л—«О S s^O \_S J
which proves i) and AP,f — P,Af. Also, / —*¦ P,f has a right-hand derivative
which is equal to P,Af.
Consider now the function t ->¦ /„' PsAf ds. This function is differentiable
and its derivative is equal to P,Af. Since two continuous functions which have
the same right derivatives differ by a constant, we have Ptf = /0' PsAf ds + g
for some g, which completes the proof of ii); by making t — 0, it follows that
g — f which proves iii). ?
Remark. The equation j^Ptf — P,Af may be written in a formal way
— P,{x,-) = A*P,(x,-)
at
where A* is the formal adjoint of A for the duality between functions and mea-
measures. It is then called the forward or Fokker-Planck equation. The reason for
the word forward is that the equation is obtained by perturbing the final posi-
position, namely, P,Af is the limit of P, (j (Pef — /)) as e ->¦ 0. Likewise, the
equation ^Ptf — AP,f which is obtained by perturbing the initial position i.e.
AP,f = !imE^o ~(Ре - I)Ptf, is called the backward equation. These names are
especially apt in the non-homogeneous case where the forward (resp. backward)
equation is obtained by differentiating PSJ with respect to t (resp. s).
A.3) Proposition. The space (/A is dense in Co and A is a closed operator.
Proof. Set Ahf = i (Phf - f) and BJ = \ /„* Pjdt. The operators Ah and
Bs are bounded on Co and moreover
, = BsAh =A,Bh = BhAs.
For every s > 0 and / € Co,

§1. Infinitesimal Generators 283
lim AhBJ = lim As(B,,f) = AJ:
therefore Bs/ € i/A and since lirn^o BJ = f, C/A is dense in Co.
Let now (fu) be a sequence in (/A, converging to / and suppose that (Af,,)
converges to g. Then
Bsg = \imBsAfn =\im B,[ Mm A,J,,
n ч \ h
= limlim/lj (Bhfn) = НтЛ.5/„ = AJ.
n h n
It follows that / e C/A and Af = Нтл^„ Asf — g which proves that Л is a
closed operator. ?
The resolvent Up, which was defined in Sect. 2 Chap. Ill, is the resolvent of
the operator A as is shown in the next
A.4) Proposition. For every p > 0, the map f ->¦ pf — Af from i/A to Co is
one-to-one and onto and its inverse is Up.
Proof. If / € ?/д, then
UP(pf-Af) = [ e->"P,(pf - Af)dt
Jo
integrating by parts in the last integral, one gets Up(pf — Af) — f. Conversely,
if / € Co, then, with the notation of the last proposition
roc /Phf-
lim A,,Upf = lim UpA,, f = lim / e~ptP, ,
/<—о ' л-.о ' ' л-^oJo \ h
which is easily seen to be equal to pUpf — f. As a result, (pi — A)Upf = f
and the proof is complete. ?
The last three propositions are actually valid for any strongly continuous semi-
semigroup of contractions on a Banach space. Our next result is more specific.
A.5) Proposition. The generator A of a Feller semi-group satisfies the following
positive maximum principle: if f € (/A, and if x0 is- such that 0 < f(x0) =
sup{/(.v). x € E), then
< 0.
Proof. We have Af(xQ) - lim,10 ) (P,f(x0) - /(*„)) and
Ptf(x0) - f(x0) < f(xo)(P,(xo. E) - 1) < 0.

284 Chapter VII. Generators and Time Reversal
The probabilistic significance of generators which was explained below Def-
Definition A.1) is also embodied in the following proposition where X is a Feller
process with transition function (Pt).
A.6) Proposition. If f e (/A, then the process
M/= /(*,)-/(*„)- f Af(Xs)ds
Jo
is a (•i^"°. Pv)-martingale for every v. If, in particular, Af — 0, then f(X,) is a
martingale.
Proof. Since / and Af are bounded, M, is integrable for each t. Moreover
?,. [m/ I ./rf\ = Ml + Ev \f(X,) - f(Xs) - j Af(Xu)du |.^°1.
By the Markov property, the conditional expectation on the right is equal to
EXi Г/ (X,-,) - f(X0) - I Af{Xu)du 1 .
But for any у е Е,
Ey Г / (*,_,) - f(X0) - [ Af{Xu)du\
= P,-sf(y) ~ /(>) - / PuAf(y)du
Jo
which we know to be zero by Proposition A.2). This completes the proof. We
observe that in lieu of (.J^"°), we could use any filtration (-'?,) with respect to
which X is a Markov process. ?
Remark. This proposition may be seen as a special case of Exercise A.8) in
Chap. X. We may also observe that, if / € (/a, then f(X,) is a semimartingale;
in the case of BM, a converse will be found in Exercise B.23) of Chap. X.
Conversely, we have the
A.7) Proposition. If f € Co and. if there exists a function g e Co. such that
f(Xt)-f(XQ)- [ g(Xs)ds
Jo
is a (.7^, Px)-martingale for every x, then f € ГУА and Af = g.

§1. Infinitesimal Generators 285
Proof. For every x we have, upon integrating,
P,f(x) - f(x) - f Psg(x)ds = 0,
Jo
hence
-tiP,f-f)-g
1 /"' 1 /"'
- / (Psg-g)ds < - / \\Ps
I J{) ' Jo
Psg-g\\ds
t
which goes to zero as t goes to zero. ?
The two foregoing results lead to the following
A.8) Definition. If X is a Markov process, a Borel function f is said to belong
to the domain Вд of the extended infinitesimal generator if there exists a Borel
function g such that, a.s., _/"„' \g(Xs)\ds < +00 for every t, and
f{X,)- f{X0)- / g(Xs)ds
Jo
is a (,'Ti, Px )-right-continuous martingale for every x.
Of course Вд D %; moreover we still write g — Af and call the "operator"
A thus defined the extended infinitesimal generator. This definition makes also
perfect sense for Markov processes which are not Feller processes. Actually, most
of the above theory can be extended to this more general case (see Exercise A.16))
and the probabilistic significance is the same. Let us observe however that g may
be altered on a set of potential zero (Exercise B.25) Chap. Ill) without altering
the martingale property so that the map / -*¦ g is actually multi-valued and only
"almost" linear.
The remainder of this section is devoted to a few fundamental examples. Some
of the points we will cover are not technically needed in the sequel but are useful
for a better understanding of some of the topics we will treat.
There are actually few cases where (/\ and A are completely known and
one has generally to be content with subspaces of (/A. We start with the case
of independent increment processes for which we use the notation of Sect. 4 in
Chap. 111. Let .У be the Schwartz space of infinitely differentiable functions /
on the line such that Нт|Д|_ос/(*)(дг)/>(дг) = 0 for any polynomial P and any
integer к. The Fourier transform is a one-to-one map from .У onto itself.
A.9) Proposition. Let X be a real-valued process with stationary independent in-
increments; the space .'/ is contained in (/a and for / e .У
у
A fix) = Pf'(x) + у f"(x) + J \f(x + y)- f(x) - Y^/'U)j v(dy).

286 Chapter VII. Generators and Time Reversal
Proof. We first observe that \\j/\ increases at most like \u\2 at infinity. Indeed
1-й]1
and
+x
IUX
< 2v ([-I, 1Г)
+x
v(dx)
:
J-1
\+x2
e'"x — 1 — iiix] v(dx):
it remains to observe that the last integrand is majorized by с|х|2|г<|2 for a con-
constant с
Let then / be in .У; there exists a unique g € .У such that /(y) =
f e'yi'g(v)dv. If we set g.v(u) = e'vl'^(u), we have P,f(x) — (/(,, gv) as is proved
by the following string of equalities:
|М/,Ял/ - je 8 i V, v v - J e g l
= / д,(</у) / e'ix+-ug(v)dv = / /(.r + y)n,(dy) = P,f(x).
As a result
,f(x)-Je яЛ»)и- -Я» ,&г + 2 (Л-r)
where, because <?s* is the Fourier transform of a probability measure,
\H(t,X)\< sup |
0<s</
by the above remark, (|^|, \gx\) and (|^|2. l^vl) are finite so that j(P,/(jr) —
/(jr)) converges uniformly to {^/,gx). As /'()') — ' j vgy(v)dv and /"(.V) =
i2 f v2gy(v)dv, we get for A the announced formula. ?
The three following particular cases are fundamental. To some extent, they
provide the "building blocks" of large classes of Markov processes.
If A" is the linear BM, then obviously A/(a) = \f"(x) for every / € .V, and
if X = а В where В is the linear BM, then Af(x) = ^f"(x), /€•*/. In this
case we can actually characterize the space С/л. We call C5 the space of twice
continuously differentiable functions / on Rd(d > 1) such that / and its first and
second order derivatives are in Co.
A.10) Proposition. For the linear BM, the space C/A is exactly equal to the space
Cl and Af(x) = jf" on this space.

§1. Infinitesimal Generators 287
Proof From Proposition A.4), we know that C/A = Up(Cq) for any p > 0, and
that AUpf = pUpf — f. We leave as an exercise to the reader the task of
showing, by means of the explicit expression of Up computed in Exercise B.23)
of Chap. Ill, that if / e Co then Upf e C2 and pUpf - f = {(Upf)".
If, conversely, g is in C2 and we define a function / by
f = Pg- 2#"
the function g - Upf satisfies the differential equation y" — jy = 0 whose only
bounded solution is the zero function. It follows that g = Upf, hence g e CSA
and Ag = \g". a
The other particular cases are: i) the translation at speed /3 for which C/A is the
space of absolutely continuous functions in Co such that the derivative is in Co
and Af(x) = fif'(x), ii) the Poisson process with parameter X for which C/A — Co
(see Exercise A.14)) and
Af(x) = X(f(x+ })- f(x)).
In all these cases, we can describe the whole space C/A, but this is a rather unusual
situation, and, as a rule, one can only describe subspaces of C/A.
We turn to the case of BM''.
A.11) Proposition. For d > 2, the infinitesimal generator of BMcl is equal to ^A
on the space C2.
Proof. For / e Co, we may write
P,f(x) = Btt)-' f e
Jrj
If / € Cq, using Taylor's formula, we get
dz.
= f(x) + -tAf(x) + Bnyd/2'-J(t,x)
where
with в some point on the segment [x. x + z-Jt]. Set
F(x. z. t) = max
For any R > 0, we have

288 Chapter VII. Generators and Time Reversal
\J(t,x)\ < / F(x,z,t)e
J\;\<R
As t goes to zero, the uniform continuity of second partial derivatives entails that
the first half of the sum above goes to zero uniformly in x\ consequently
lim sup \J(t. x)\ < 2max
d2f
J\;\>R \ij j
By taking R large, we may make this last expression arbitrarily small which
implies that
||
for every / e C02. ?
Remarks. 1) At variance with the case d — 1, for d > 1, the space С„ is not equal
to (/a.. Using the closedness of the operator A, one can show without too much
difficulty that (SA is the subspace (of Co) of functions / such that Af taken in
the sense of distributions is in Co and A is then equal to \Af.
2) In the case of BM, it follows from Proposition A.2) that for /eC02
— P,f= -AP,f= -P,Af.
dt и 2 и 2 i J
Actually, this can be seen directly since elementary computations prove that
а l з2
at 2 bxl
and the similar formula in dimension d. It turns out that the equality j;Pif —
\AP,f is valid for any bounded Borel function / and t > 0. In the language
of PDE's, g, and its multidimensional analogues are fundamental solutions of the
heat equation ^ + \Af — 0.
If В is a BM''@) and a a d x d-matrix, one defines a Revalued Markov
process X by stipulating that if Xo = x a.s., then X, = x + aB,. We then have
the
A.12) Corollary. The infinitesimal generator of X is given on Сц by
2^
where у = aal, with a1 being the transpose of a.

§1. infinitesimal Generators 289
Proof. We have to find the limit, for / € Cq, of
-E[f(x+aB,)- f(x)]
where E is the expectation associated with B; it is plainly equal to Ag@)/2, where
= f(x+ay), whence the result follows by straightforward computations. ?
Remark. The matrix у has a straightforward interpretation, namely, ty is the
со variance matrix of X,.
Going back to Proposition A.9), we now see that, heuristically speaking, it
tells us that a process with stationary independent increments is a mixture of a
translation term, a diffusion term corresponding to y/" and a jump term, the
jumps being described by the Levy measure v. The same description is valid for
a general Markov process in Ш'1 as long as (/A D C2K\ but since these processes
are no longer, as was the case with independent increments, translation-invariant
in space, the translation, diffusion and jump terms will vary with the position of
the process in space.
A.13) Theorem. If P, is a Feller semi-group on M.d and C^? С C/A, then
i) C\ С С/А;
ii) For every relatively compact open set U, there exist functions a,j, b,, с on U
and a kernel N such that for f € C\ and x € U
OXj ¦ . OX; OX i
i i.j J
[
md\{
N(x.dy)
i '
where N(x, ¦) is a Radon measure on M.d\{x], the matrix a(x) — \\ajj(x)\\ is
symmetric and non-negative, с is < 0. Moreover, a and с do not depend on U.
Fuller information can be given about the different terms involved in the de-
description of A, but we shall not go into this and neither shall we prove this result
(see however Exercise A.19)) which lies outside of out main concerns. We only
want to retain the idea that a process with the above infinitesimal generator will
move "infinitesimally" from a position x by adding a translation of vector b(x),
a gaussian process with covariance a(x) and jumps given by N(x, •); the term
c(x)f(x) corresponds to the possibility for the process of being "killed" (see Ex-
Exercise A.26)). If the process has continuous paths, then its infinitesimal generator
is given on C\ by
Af(x)=c{x)f{x) + Ybi(x)-~ (x) + Yau(x)——-(.r)
where the matrix a(x) is symmetric and non-negative. Such an operator is said to
be a semi-elliptic second order differential operator.

290 Chapter Vii. Generators and Time Reversal
As was noted at the beginning of this section, a major problem is to go the other
way round, that is, given an operator satisfying the positive maximum principle, to
construct a Feller process whose generator is an extension of the given operator.
Let us consider a semi-elliptic second order differential operator 1]а,/(х)дД^-
without terms of order 0 or 1. If the generator Л of a Feller process X is equal
to this operator on Cj,, we may say, referring to Corollary A.12), that between
times t and t + h, the process X moves like cr(x)B where cr(x) is a square root
of a(x), i.e., a(x) = <j(x)<j'(x), and В a BMrf; in symbols
Xt+h = X, +a(X,) (B,+h -B,) + 0(A).
The idea to construct such a process is then to see it as an integral with respect
to B,
X,= [ a(Xs)dBs,
Jo
or, to use a terminology soon to be introduced, as a solution to the stochastic dif-
differential equation dX = a(X)dB. As the paths of В are not of bounded variation,
the integral above is meaningless in the Stieltjes-Lebesgue sense and this was one
of the main motivations for the introduction of stochastic integrals. These ideas
will be developed in the following section and in Chap. IX.
A.14) Exercise. 1°) Let тс be a transition probability on E such that тс (C0(E)) С
C0(E) and / be the identity on C0(E). Prove that P, = ехр(/(тг - /)) is a Feller
semi-group such that ' = C0(E) and A — тс — I. Describe heuristically the
behavior of the corresponding process of which the Poisson process is a particular
case.
[Hint: See the last example in Exercise A.8), Chap. III.]
2°) More generally, if Л is a bounded operator in a Banach space, then T, —
exp(M) is a uniformly continuous semi-group (i.e. lim,j,0 II7^+* — 71,11 = 0) of
bounded operators (not necessarily contractions) with infinitesimal generator A.
3C) Prove that actually the three following conditions are equivalent
i) (T,) is uniformly continuous;
ii) (/ is the whole space;
iii) Л is a bounded operator.
If these conditions are in force, then T, = exp(M).
[Hint: Use the closed graph and Banach-Steinhaus theorems.]
A.15) Exercise. A strongly continuous resolvent on Co is a family (?Д), к > 0,
of kernels such that
i) \\Wx\\ < 1 for every X > 0;
ii) Uk - UIX = (/л - Х)ики„ = (д - X)U^Uk for every pair (Я, д);
iii) for every / € Co, lim^*, \\Wkf - f\\ = 0.
It was shown in Sect. 2 Chap. Ill that the resolvent of a Feller semi-group is a
strongly continuous resolvent.

§1. Infinitesimal Generators 291
Г) If (t/л), A > 0, is a strongly continuous resolvent, prove that each operator
Uj, is one-to-one and that if the operator A is defined by XI — A — U-~{, then A
does not depend on X. If (U>.) is the resolvent of a Feller semi-group, A is the
corresponding generator.
2°) Prove that / e C/A if and only if lim^^ X(XUAf - f) exists and the limit
is then equal to Af.
A.16) Exercise. For a homogeneous transition function P,, define Bo as the set
of bounded Borel functions / such that lim,_*0 II Ptf — /II = 0, where ||/|| =
supv |/(x)|. Define ?/д as the set of those functions / for which there exists a
function Af such that
lim
t->0
-(PJ-f)-Af
= 0.
Prove that C/A с So and extend the results of the present section to this general
situation by letting So play the role held by Co in the text.
A.17) Exercise. If (P,) is a Feller semi-group and / a function in Q such that
Г1 (P,f - /) is uniformly bounded and converges pointwise to a function g of
Co, then / e C/A and Af = g.
[Hint: Prove that / e Uk(C0).]
A.18) Exercise. Let P, and Q, be two Feller semi-groups on the same space with
infinitesimal generators A and S. If C/A с С/в and В — A on C/A, prove that
P, = Qt. Consequently, the map P, -*¦ A \s one-to-one and no strict continuation
of an infinitesimal generator can be an infinitesimal generator.
[Hint: For / € (/a, differentiate the function 5 ->¦ QsP,~sf.]
A.19) Exercise. Let Л be a linear map from C°°(R'') into C(№), satisfying the
positive maximum principle and such that A1 = 0. We assume moreover that A is
a local operator, namely, if / = 0 on some neighborhood of x, then Af(x) = 0.
Iе) Prove that A satisfies the local maximum principle: if / has a local maxi-
maximum at .v, then Af(x) < 0.
2C) If, for some x, the function / is such that |/(v) - f(x)\ = 0(| v - .v|2) as
v -> x, prove that Af(x) — 0.
[Hint: Apply A to the function /(y) + a\у - x\2 for suitably chosen a.]
3") Call x' the coordinate mappings and set bj(x) = Ax'{x), a,,(x) =
A(x'X1)(x)-bi{x)x)(x)-bj(x)x'(x). Prove that for every x, the matrix (аи(х))
is non-negative.
[Hint: Use the functions f(y) = jy,L\ % (x'(.v) - /'U))| where в e R'1.]
4°) Prove that
Afix) = \^Гаи{хIЩг(х) + У]Ь;(х)^{х).
2 *7-f dxidxj ^ dxt
[Hint: Use Taylor's formula.]

292 Chapter VII. Generators and Time Reversal
# A.20) Exercise. Let у be the Gaussian measure in W1 with mean 0 and covariance
matrix lj. Let / be a C^-function in RJ, bounded as well as its derivatives and
В the BM^O).
1) (Chernoff s inequality). Prove that
]= I 4(Px-,f)(B,)dB,
Jo
and derive that
f (f- f fdyJdy<J\Vf\2dy.
2C) Suppose further that ||V/|| < 1 and prove that there exists a BM1 say X
and a r.v. г < 1 such that /(?,) - E[f(B{)] = XT.
[Hint: ||V(Pi-,/)ll < I-]
3°) Prove that for every и > 0,
el: f(x) > f fdy+u}) < yj - f exp(~x2/2)dx.
4C) Extend the result of Г) to functions / in L2(y) such that V/ is in L2(y)
and prove that the equality obtains if and only if / is an affine function.
* A.21) Exercise. In the case of BM1 let F be a r.v. of L2(.F^). Assume that there
is a function ф on Q xR+ x?2 such that for each /, one has F(a>) = ф(со, t, в, (to))
(see Exercise C.19) Chap. HI) and (x, I) -* Ф(ш.х.1) = Ех[ф(ш. t. •)] is for a.e.
w a function of C2;,'. Prove then that the representation of F given in Sect. 3
Chap. V is equal to
E[F]+ f <P[(to,B,(to),s)dB,(to).
Jo
Give examples of variables F for which the above conditions are satisfied.
[Hint: Use Exercise C.12) Chap. IV.]
A.22) Exercise. Prove that the infinitesimal generator of the BM killed at 0 is
equal to the operator j^i on C|(]0, oc[).
# A.23) Exercise (Skew Brownian motion). Prove that the infinitesimal generator
of the semi-group defined in Exercise A.16) of Chap. Ill is equal to jj^ on the
space {/ e Co : /" exists in R\{0}, /"@-) = /"@+) and A - a)f'@-) =
# A.24) Exercise. If X is a homogeneous Markov process with generator A, prove
that the generator of the space-time process associated with X (Exercise A.10) of
Chap. Ill) is equal to ^ + A on a suitable space of functions on R+ x E.

§2. Diffusions and Ito Processes 293
A.25) Exercise. 1J) Let X be a Feller process and U the potential kernel of
Exercise B.29) Chap. III. If / e Co is such that Uf e Q, then Uf e C/A and
-AUf = f. Thus, the potential kernel appears as providing an inverse for A.
[Hint: Use Exercise A.17).]
2°) Check that for BM3 the conditions of 1'"') are satisfied for every / e
CK- In the language of PDE's, — \j\x\ is a fundamental solution for A, that is:
— /4(l/|jr|) = <50 in the sense of distributions.
A.26) Exercise. Let X be a Feller process and с a positive Borel function.
Iе) Prove that one defines a homogeneous transition function Q, by setting
- j c(X
s)dA 1.
Although we are not going into this, Q, corresponds to the curtailment or "killing"
of the trajectories of X performed at the "rate" c(X).
2°) If / is in the domain of the generator of X and, if с is continuous, prove
that
(Q,//) A//
/10
pointwise. The reader is invited to look at Proposition C.10) in the following
chapter.
A.27) Exercise. Iе) Let Z be a strictly positive r.v. and define a family of kernels
on R+ by
Ptf(x) = E[f((tZ)vx)].
Prove that (P,) is a semi-group iff Z ' is an exponential r.v., i.e. Z~' = Xe,
к > 0.
2°) If this is the case, write down the analytical form of (P,), then prove that
it is a Feller semi-group, that C/A — Co(K+) and that
\f{x) = k f (f{y)-f(x))y-2dy.
J x
§2. Diffusions and Ito Processes
In the foregoing section, we have seen, in a heuristic way, that some Markov
processes ought to be solutions to "stochastic differential equations". We now take
this up and put it in a rigorous and systematic form thus preparing for the discussion
in Chap. IX and establishing a bridge between the theory of the infinitesimal
generator and stochastic calculus.
In the sequel, a and b will denote a matrix field and a vector field on M.'1
subject to the conditions
i) the maps x -*¦ a(x) and x -*¦ b(x) are Borel measurable and locally bounded,

294 Chapter VII. Generators and Time Reversal
ii) for each x, the matrix a(x) is symmetric and non-negative i.e. for any X e
With such a pair (a, b), we associate the second order differential operator
()>]^()
In Sect. 1, we have mentioned that some Markov processes have infinitesimal
generators which are extensions of such operators. It is an important problem to
know if conversely, given such an operator L, we can find a Markov process
whose generator coincides with L on C\.
1
B.1) Definition. A Markov process X = (Q. .W, .^", X,, Px) with state space
is said to be a diffusion process with generator L if
i) it has continuous paths,
ii) for any x e №.J and any f e Cjf,
We further say that X has covariance or diffusion coefficient a and drift b.
This is justified by the considerations in Sect. 1.
Let us stress that the hypothesis of continuity of paths includes that f = 00 a.s.
As a result, if {K,,} is an increasing sequence of compact sets such that Kn с К'^+[
and [Jn Kn — Rd, then setting а„ = Тщ, we have limn an = +00. Furthermore,
the necessity of the non-negativity of a follows from Theorem A.13) and Exercise
A.19) but is also easily explained by Exercise B.8).
Observe also that one could let a and b depend on the time s, and get for each
s a second-order differential operator Ls equal to
9
2 j-*
The notion of diffusion would have to be extended to that of поп homogeneous
diffusion. In that case, one would have probability measures Л., corresponding
to the process started at x at time s and demand that for any / e C?\ v < t.
,)] = fix) + ?.<., \f Luf(X,,)du\ .
In the sequel, we will deal mainly with homogeneous diffusions and write for
/ € C\
M[ = f(Xt) - f(X0) - f Lf(Xs)ds.
Jo
The process Mf is continuous; it is moreover locally bounded, since, by the
hypothesis made on a and b, it is clearly bounded on [0. а„ л и]. Likewise, if
/ e C?\ M1 is bounded on every interval [0, t] and the integrals in ii) are finite.

§2. Diffusions and Ito Processes 295
B.2) Proposition. The property ii) above is equivalent to each of the following:
Hi) for any f ? C^, Mf is a martingale for any Px:
iv) for any f ? С2, М1 is a local martingale for any /\.
Proof. If iii) holds, then, since Mq = 0,
P,f(x) - f(x) - Ex \j LfiX^ds] = E,[M,'] = 0,
and ii) holds.
Conversely, if ii) holds, then Ex, [MJ ] — 0 for every s and t. By the Markov
property, we consequently have
Ex [m/ 1.%] = M{ + Ex \f(X.) - f(Xs
s) - f Lf(Xu)du\.%
which shows that ii) implies iii).
If M* is a local martingale and is bounded on [0, /] for each t, then it is a
martingale; thus, iv) implies iii).
To prove that iii) implies iv), let us begin with / in C\. There is a compact
set H and a sequence {fp\ of functions in C^ vanishing on H' and such that {fp}
converges uniformly to / on H as well as the first and second order derivatives.
For every /, the process M,1' - MJ is bounded on [0. /] by a constant cp which
goes to zero as p ->¦ oo. By passing to the limit in the right-hand side of the
inequality
Ex
- M
we see that MJ is a martingale.
Let now / be in C2; we may find a sequence {g,,} of functions in C\ such
that g,, = / on Kn. The processes M1 and Mx" coincide up to time a,,. Since
Mx" is a martingale by what we have just seen, the proof is complete.
Remarks. Г) The local martingales M/ are local martingales with respect to
the uncompleted ст-fields .P^° = a(X,,s < t) and with respect to the usual
augmentation of (.^"°).
2 ) Proposition B.2) says that any function in C\ is in the domain of the
extended infinitesimal generator of X. If X is Feller, by arguing as in Proposition
A.7), we see that C2K С С/a and A = L on C\. In the same vein, if Lf = 0. then
f(Xt) is a local martingale, which generalizes what is known for BM (Proposition
C.4) Chap. IV). By making f(x) = x, we also see that X is a local martingale if
and only if L has no first order terms.
If we think of the canonical version of a diffusion where the probability space
is W = C(R+, Rd) and X is the coordinate process, the above result leads to the

296 Chapter VII. Generators and Time Reversal
B.3) Definition. A probability measure P on W is a solution to the Martingale
problem n(x.a.b) if
ii) for any f e C^, the process
M[ = f(Xt) - f(X0) - f Lf(Xs)ds
Jo
is a P-martingale with respect to the filtration (a(Xs. s < t)) — (¦^
The idea is that if (Q, Xt, Px) is a diffusion with generator L, then X(PX) is a
solution to the martingale problem л(х, a, b). Therefore, if one wants to construct
a diffusion with generator L, we can try in a first step to solve the corresponding
martingale problem; then in a second step if we have a solution for each x, to
see if these solutions relate in such a way that the canonical process is a diffusion
with L as its generator. This will be discussed in Chap. IX.
For the time being, we prove that the conditions in Proposition B.2) are equiv-
equivalent to another set of conditions. We do it in a slightly more general setting which
covers the case of non-homogeneous diffusions. Let a and b be two progressively
measurable, locally bounded processes taking values in the spaces of non-negative
symmetric d x ^/-matrices and d-vectors. For / e C2(Rd), we set
Ls(co)f(x) = - > au(s, <и)т—7-U) + > bis, ш)-^-(дг).
2 *r-? dxidxi *—' dx:
i.j ' J i '
B.4) Proposition. Let X be a continuous, adapted, Rd-valued process; the three
following statements are equivalent:
i) for any f e C2, the process MJ = f(X,) - /(Xo) - /0' LJ(Xs)ds is a local
martingale;
ii) foranyO e Rd, the process M? = {в, X,-Xo-/O' b(s)ds) is a local martingale
and
[Мв,Мв), = / (9,a(s)e)ds:
Jo
Hi) for anv в е
X," = exp ({в, X, - Xo - f b(s)ds) - 1 f (9, a{s)9)ds\
is a local martingale.
Proof, i) => ii). For f(x) = (9,x), we get Mf — Me which is thus a local
martingale. Making f(x) = (в, хJ in i), we get that
Hf = (в, X,J - (в, Х0J -2 f (9,Xs)(e,b(s))ds- f (9,a(s)9)ds
Jo * Jo

§2. Diffusions and Ito Processes 297
is a local martingale. Writing X — Y \f X - Y is a local martingale, we have,
since 2@, Xo)Mf is a local martingale, that
(AffJ- / F,a(sN)ds ~ (Л/? + @, Xo)J- / F.a(sN}ds.
Jo Jo
Setting A, — @. /0' b(s)ds), we further have
(AffJ- / (9.a{s)9)ds ~ (Л/,н + @, Xo)) - @, X0>2 - / (9.a(sH)ds - H*
Jo Jo
= (@, X,) - A,J - @, X,J+2 f (9, Xs)dAs
Jo
= -2(9. X,)A, + A2+2 / (9,X,)dA,.
Jo
As @, X,) = М? + (в, Xo)+A, is a semimartingale, we may apply the integration
by parts formula to the effect that
(M"f - / {9.a(s)B)ds ~ A2 - 2 / A,-t/A, = 0
Jo Jo
which completes the proof.
ii) => iii). By Proposition C.4) in Chap. IV, there is nothing to prove as
iii) => i). We assume that iii) holds and first prove i) for /(>•) = exp(@. v>).
The process
V, = exp @, I b(s)ds) + - f @. a(sN)ds
is of bounded variation; integrating by parts, we obtain that the process
f ( 1 \
. X, -Xo>)- / exp({6». X, - Xo>) ( (9.b(s) +-a(sH) ) ds
Jo V 2 /
is a local martingale. Since L,f(x) = exp(@. X.,)) ((Q.b(s) + \а(и)в}), we have
proved that
f(X0)(f(X,)-j LJ(Xs)ds\
is a local martingale. The class of local martingales being invariant under multi-
multiplication by, or addition of, .^-measurable variables, our claim is proved in that
case. To get the general case, it is enough to observe that exponentials are dense
in C2 for the topology of uniform convergence on compact sets of the functions
and their two first order derivatives. ?

298 Chapter VII. Generators and Time Reversal
Remarks. 1°) Taking Proposition B.2) into account, the implication ii) => i) above
is a generalization of P. Levy's characterization theorem (Theorem C.6) of Chapter
IV) as is seen by making a = Id and b = 0.
2°) Another equivalent condition is given in Exercise B.11).
B.5) Definition. A process X which satisfies the conditions of Proposition B.4) is
called an Ito process with covariance or diffusion coefficient a and drift b.
Obviously Ito processes, hence diffusions, are continuous semimartingales. We
now show that they coincide with the solutions of some "stochastic differential
equations" which we shall introduce and study in Chap. IX. We will use the
following notation. If X = (X1 Xd) is a vector semimartingale and К — (K/j)
a process taking its values in the space of r x uf-matrices, such that each K,j is
progressive and locally bounded we will write К ¦ X or /0' K<dXs for the r-
dimensional process whose i-th component is equal to ?• /„' Kjj(s)dXj(s)-
B.6) Proposition. Let ft be an (.7^)-BMr defined on a probability space
(i2, .7". .7[, P) and a fresp. b) a locally bounded predictable process with val-
values in the d x r-matrices (resp. W1); if the adapted continuous process X satisfies
the equation
(*) X, = Xo + f cr(s)d& + [ b(s)ds.
it is an Ito process with covariance aa' and drift b. If in particular, o(s) = cr(X5)
and b(s) — b(X^) for two fields a and b defined on Rd, and if Xq = x a.s., then
X(P) is a solution to the martingale problem я(х, aa'. b).
Proof. A straightforward application of Ito formula shows that condition i) of
Proposition B.4) holds. D
We now want to prove a converse to this proposition, namely that given an Ito
process X, and in particular a diffusion, there exists a Brownian motion C such
that X satisfies (*) for suitable a and b. The snag is that the space on which X
is defined may be too poor to carry a BM; this is for instance the case if X is
the translation on the real line. We will therefore have to enlarge the probability
space unless we make an assumption of non degeneracy on the covariance a.
B.7) Theorem. //' X is an ltd process with covariance a and drift b. there ex-
exist a predictable process a and a Brownian motion В on an enlargement of the
probability space such that
\ o(s)dBs+ / b(.s)ds.
Jo Jo
X, = Xo +
h Jo
Proof. By ii) of Proposition B.4), the continuous vector local martingale M, —
X, — Xo — /„' b(s)ds satisfies (M1. MJ), = /0' atj(s)ds. The result follows imme-
immediately from Proposition C.8) in Chap. V. D

§2. Diffusions and Ito Processes 299
Remark. By the remark after Proposition C.8) in Chap. V, we see that if a is
dP®dt a.e. strictly non-negative, then a and В may be chosen such that a — a a'.
If in particular a{s) — c(Xs) where с is a measurable field of symmetric strictly
non-negative matrices on Rrf, we can pick a measurable field у of matrices such
that yy' = с and take a(s) = y(Xs).
# B.8) Exercise. Г) In the situation of Proposition B.2) and for f.geC2 prove
that
(Mf,M*),= f (L(fg)-fLg-gLf)(X,)ds= f Df,aVg)(Xs)ds.
Jo Jo
[This exercise is solved in Sect. 3 Chap. VIII].
2°) Deduce from Iе) the necessity for the matrices a(x) to be non-negative.
# B.9) Exercise. In the situation of Proposition B.2), prove that if / is a strictly
positive C2-function, then
(/(X,)//(X0)) exp (- j^ (Lf/f)(Xs)di
is a Pv-local martingale for every x.
B.10) Exercise. If X is a ^/-dimensional Ito process with covariance a and drift
0, vanishing at 0, prove that for 2 < p < oo, there is a constant С depending only
on p and d such that
Г 1 Г/ Г'
E\ sup \X,\P\ <CE I Trace
l_O<s<f J \Jo
a(s)ds
B.11) Exercise. Prove that the conditions in Proposition B.4) are also equivalent
to
iv) for any / on [0, oo[xR'/ which is once (twice) differentiable in the first
(second) variable, the process
- f
is a local martingale. Compare with Exercise A24).
B.12) Exercise. In the situation of Proposition B.4), suppose that a and b do not
depend on w. If и is a function on [0, oofxR'', which is sufficiently differentiable
and such that |y — L,u + g in ]0, oo[xRrf, prove that
u(t -s,Xs)+ I g(t -r,Xr)dr
Jo
is a local martingale on [0, /[.

300 Chapter VII. Generators and Time Reversal
§3. Linear Continuous Markov Processes
Beside the linear BM itself, many Markov processes, with continuous paths, de-
defined on subsets of R such as the BES3 or the reflected BM have cropped up in our
study. The particular case of Bessel processes will be studied in Chap. XI. This is
the reason why, in this section, we make a systematic study of this situation and
compute the corresponding generators.
We will therefore deal with a Markov process X whose state space E is
an interval (l.r) of R which may be closed, open or semi-open, bounded or
unbounded. The death-time is as usual denoted by ?. We assume, throughout the
section, that the following assumptions are in force:
i) the paths of X are continuous on [0, f [;
ii) X enjoys the strong Markov property;
iii) if ? < oo with strictly positive probability then at least one of the points / and
r does not belong to E and lim,u X, ? E a.s. on {? < oo}; in other words X
can be "killed" only at the end-points of E which do not belong to E.
Property i) entails that the process started at x cannot hit a point у without
hitting all the points located between x and y. The hitting time of the one-point
set {x} is denoted by Tx\ we have
Tx = inf{/ > 0 : X, = x]
where as usual inf@) = +oo. Naturally, Я>, = x on {TX < oo}.
Finally, we will make one more assumption, namely, that X is regular, for
с
any x € E =]l,r[ and у e E, Px[Ty < oo] > 0. This last hypothesis means
that E cannot be decomposed into smaller sets from which X could not exit (see
Exercise C.22)).
From now on, we work with the foregoing set of hypotheses.
For any interval / — ]a, b[ such that [a. b] с Е, we denote by 07 the exit time
of /. For x e /, we have 07 = Ta л Th P,-a.s. and for x ? /, 07 = 0 /\-a.s. We
also put mi(x) = Ex[o[].
C.1) Proposition. If I is bounded, the function mr is bounded on I. In particular
at is almost-surely finite.
Proof. Let у be a fixed point in /. Because of the regularity of X, we may pick
a < 1 and / > 0 such that
max (P,[Ta > 1], Py[Th > 1]) = a.
If now у < x < b, then
Pxl<*i > t] < Px[Th >t]< Py[Tb >t]<a;
the same reasoning applies to a < x < у and consequently

§3. Linear Continuous Markov Processes 301
sup Px[oi > /] < a < 1.
xel
Now, since O] — и +aj ови on {cr/ > «), we have
/>v[ct, > nt] = Px[{a, > (n- \)t)n((n- \)t+a,o9in-Ut > nt)],
and using the Markov property
Px[cr, > nt] = Ex [\{ai>(n-\)t)EXu,_Ul[\«j,>t)i\ ¦
On {a/ > (и - 1)/}, we have X(n_,)r e / P,-a.s. and therefore
Px[a, > nt] < aP[oj > (n - \)t].
It follows inductively that Px[<7/ > nt] < a" for every x e I, and therefore
30
\-l
sup Ex[oi] < sup N tPx[(Ti > nt] < t(\ — a) ,
x€l xel „_q
which is the desired result. ?
For a and b in E and l<a<x<b<r, the probability Px[Tb < Ta] is the
probability that the process started at x exits ]a. b[ by its right-end. Because of
the preceding proposition, we have
P.x[Ta < Ть] + P\[Tb <Tu] — \.
C.2) Proposition. There exists a continuous, strictly increasing function s on E
such that for any a, b, x in E with l<a<x<b<r
P.x[Tb < Ta] = (s(x) - s(a))/(s(b) - s(a)).
Ifs is another function with the same properties, then s = as + ft with a > 0 and
Proof. Suppose first that E is the closed bounded interval [/. r]. The event {Tr <
Ti} is equal to the disjoint union
[Tr < T,.Ta < Th}U{Tr <T,,Tb < Ta).
Now T, = Ta + T, о вТа and Tr = Ta + Tr о вт„ on the set {Ta < Th). Thus
Px [T, < ТГ, Ta < Tb] = Ex [\{Tu<Th)hT,<T,) О вТа]
and since [Ta < Th] e .Щ-а, the strong Markov property yields
"x I'r < 'I, 'a < 'hi — tLx Yl{T,,<Tb)LXTl, \_l{Tr<T,)\\
and because XTa = a a.s.,
Px [Tr < T,; Ta < Th] = Px [Ta < Th] Pa [Tr < T,].

302 Chapter Vll. Generators and Time Reversal
We finally get
Px [Tr < T,] = Px [Ta < 7i] Pa [Tr < T,] + Px [Tb < Т„] Ph [Tr < T,].
Setting s(x) — Py[Tr < 7}] and solving for Рх[Ть < Та] we get the formula in
the statement.
To prove that s is strictly increasing, suppose there exists x < v such that
s(x) = .v(v). Then, for any b > y, the formula just proved yields Py[Ti, < Tx] = 0
which contradicts the regularity of the process.
Suppose now that E is an arbitrary interval. If/2 < /1 < r\ < ri are four points
in E, we may apply the foregoing discussion to [/,, r\ ] and [/2, r2]; the functionss\
and S2 thus defined obviously coincide on ]/], r\ [ up to an affine transformation. As
a result, a function .v may be defined which satisfies the equality in the statement
for any three points in E. It remains to prove that it is continuous.
If a < x and {a,,} is a sequence of real numbers smaller than x and decreasing
to a, then Та„ \ Ta Pv-a.s.; indeed, because of the continuity of paths, X\\mii т11п = а
Pc-a.s., so that lim,, Та„ > Та, and the reverse inequality is obvious. Consequently,
[Ta < T,,} = lim,,{7;,,, < Tb] PA-a.s. since obviously Px[Ta = Tb] = 0. It follows
that s is right-continuous in a and the left-continuity is shown in exactly the same
way. The proof is complete. ?
C.3) Definition. The function s of the preceding result is called the scale function
ofX.
We speak of the scale function although it is defined up to an affine transfor-
transformation. If s(x) may be taken equal to x, the process is said to be on its natural
scale. A process on its natural scale has as much tendency to move to the right
as to the left as is shown in Exercise C.15). The linear BM, the reflected and
absorbed linear BM's are on their natural scale as was proved in Proposition C.8)
of Chap. 11. Finally, a simple transformation of the state space turns X into a
process on its natural scale.
C.4) Proposition. The process X, = s(X,) satisfies the hypotheses of this section
and is on natural scale.
Proof. Straightforward. ?
The scale function was also computed for BES1 in Chap. VI. In that case,
since 0 is not reached from the other points, to stay in the setting of this section,
we must look upon BES3 as defined only on ]0. oc[. The point 0 will be an
entrance boundary as defined in Definition C.9). In this setting, it was proved in
Chap. VI that s(x) = —\/x (see also Exercise C.20) and the generalizations to
other Bessel processes in Chap. XI). The proof used the fact that the process X
of Proposition C.4) is a local martingale; we now extend this to the general case.
We put R = ? Л 7/ лТг. Let us observe that if f is finite with positive probability
and lim,|.c X, = I (say), then lim^—/л(дг) is finite and we will by extension write
.v(/) for this limit. Accordingly, we will say that 5(*Xf) = s(l) in that case.

§3. Linear Continuous Markov Processes 303
C.5) Proposition. A locally bounded Borel junction f is a scale junction if and
only if f(X,)R is a local martingale. In particular, X is on its natural scale if and
only if XK is a local martingale.
Proof If f(X,)H is a local martingale, then fora < x < b. the process /(X,)'"'~'Th
is a bounded {.У,, Pv)-martingale and the optional stopping theorem yields
/(дг) = f(a)Px [Ta < T,,] + f(b)Px [Ti, < Ta].
On the other hand, as already observed,
1 = P, [Ta < 7ft] + P, [Т„ < Ta] .
Solving this system of linear equations in РХ[Т„ < Ть] shows that / is a scale
function.
Conversely, by the reasoning in the proof of Proposition C.2), if / is a scale
function, then it is continuous. As a result, for [a. b] С Е and e > 0, we may find
a finite increasing sequence A = (ak)k=0 K of numbers such that a0 = a, aK = b
and \f(ak+\) - f(ak)\ < e for each к < К - 1. We will write 5 for Ta л Tb.
We define a sequence of stopping times Tn by 7b = 0, T\ = Ta and
Т„ = inf (t > Tn-\ : X, e A\ {Х7„_,}) л 5.
Clearly, {7",,} increases to 5 and if / is a scale function, the strong Markov property
implies, for x e]a. b[,
Ex [f(XsT) | .Уг„_,] = Ex [f(XsTj) o^, | .^,.,]
= EXl { [f(XsTi)} = f(XsTn ,) P,-a.s.;
in other words, {j (Xj ). .Syn} is a bounded Px-martingale.
For / > 0, let N = inf{« > 1 : T,, > t) where as usual inf@) — oo. The r.v. N
is a stopping time with respect to (-^„), so that, by the optional stopping theorem
f(x) = EAf(Xsn)].
But. on the set {/ < 5}, N is finite and Tn-\ < t < TN < S which by the choice
of A implies that
Since s is arbitrary, it follows that f(x) — ?Л[/(Х;9)] and another application of
the Markov property shows that f(Xf) is a martingale. The proof is now easily
completed.
Remarks. We have thus proved that, up to an affine transformation, there is at
most one locally bounded Borel function / such that j(X,) is a local martingale.
This was stated for BM in Exercise C.13) of Chap. II. If R = oo, we see that s
belongs to the domain of the extended infinitesimal generator A and that As = 0,
a fact which agrees well with Theorem C.12) below.

304 Chapter VII. Generators and Time Reversal
We now introduce another notion, linked to the speed at which X runs through
its paths. We will see in Chap. X how to time-change a BM so as to preserve the
Markov property. Such a time-changed BM is on its natural scale and the converse
may also be shown. Thus, the transformation of Proposition C.4) by means of the
scale function turns the process into a time-changed BM. The time-change which
will further turn it into a BM may be found through the speed-measure which we
are about to define now. These questions will be taken up in Sect. 2 of Chap. X.
Let now J =]c, d[ be an open subinterval of /. By the strong Markov property
and the definition of s, one easily sees that for a < с < x < d < b,
(*) nti(x) = Ex [cry +<7/ oeaj]
s(d)-s(x) s(x)~s(c)
= ntj(x) + m(c) +
s(d)-s(c) s(d)~s(c)
Since ntj(x) > 0, it follows that mi is an s-concave function. Taking our cue
from Sect. 3 in the Appendix and Exercise B.8) in Chap. VI, we define a function
G/ on E x E by
G,(x.y) =
(.s(x)~s(a))(s(b)-s(y))
— if a < x < у < b,
s(b)-s(a) ~ -'-
(s(y)~s(a))(s(b)-s(x))
— if a < v < x < b,
s(b)-s(a)
0 otherwise.
C.6) Theorem. There is a unique Radon measure m on the interior E of E such
that for any open subinterval I —]a, b[ with [a, b] С Е,
m,(x) = / Gi{x,y)m(dy)
for any x in I.
Proof. By Sect. 3 in the Appendix, for any /, there is a measure v7 on / for which
m,(x) = J Gdx,y)v,(dy).
Thus, we need only prove that if J с / as in (*), vj coincides with the restriction
of V[ to J. But, this is a simple consequence of the definition of i>/; indeed, if we
take the ^-derivatives in (*), we see that the derivatives of mt and mj differ by a
constant and, consequently, the associated measures are equal. ?
C.7) Definition. The measure m is called the speed measure of the process X.
For example, using the result in Exercise C.11) of Chap. II, one easily checks
that, if we take *(дг) = дг, the speed measure of Brownian motion is twice the
Lebesgue measure. The reader will also observe that, under m, every open sub-
subinterval of E has a strictly positive measure.

§3. Linear Continuous Markov Processes 305
We will see, in Theorem C.12), that the knowledge of the scale function and
of the speed measure entails the knowledge of the infinitesimal generator. It is
almost equivalent to say that they determine the potential operator of X killed
when it exits an interval, which is the content of the
C.8) Corollary. For any 1 =]a, b[, x e 1 and f e ./?(R)+,
<• Та Л Th
f(X,)ds\ = J
f(Xs)ds\= I G,{x.y)f(y)m(dy).
Proof. Pick с such that a < с < b. The function v(x) = Ек \/^лТ" \]c,b[(Xu)du\
is .v-concave on ]a, b[ and v(a) — v(b) — 0; therefore
v(x) = - / G,(x,y)v"(dy)
where v" is the measure which is associated with the s-derivative of v (see Ap-
Appendix 3). On the other hand, by the Markov property,
v(x) = Ex[TcATh]+S(^i(X) v(c) on ]c. b[,
s(b) -s(c)
s(x) - s(a)
v(x) — v(c) on ]a,c).
s(c) -s(a)
Since the function E [Tc л 7i] is equal to / G](.,(,[(., y)m(dy), the measure associ-
associated to its second i-derivative is equal to — l]t-,ь[гп\ the measures associated with
the other terms are obviously 0 since their ^-derivatives are constant. Therefore
v(x)= / G,(x,y)\V4(y)m{dy).
J
which is the result stated in the case where / = \\c,b[- The proof is completed by
means of the monotone class theorem and the usual extensions arguments. D
From now on, we will specialize to the case of importance for us, namely
when E is either ]0, oo[ or [0, oof and we will investigate the behavior of the
process at the boundary {0}. The reader may easily carry over the notions and
results to the other cases, in particular to that of a compact subinterval. If 0 ? E
and if ? = oo a.s., we introduce the following classification.
C.9) Definition. IfE =]0, oo[, the point 0 is said to be a natural boundary if for
all t > 0 and у > 0,
lim PX[TX < Л = 0.
// is called an entrance boundary if there are t > 0 and у > 0 such that
lim/^[Г,. < t] > 0.

306 Chapter VII. Generators and Time Reversal
An example where 0 is an entrance boundary is given by the BES3 process de-
described in Sect. 3 of Chap. VI (see more generally Bessel processes of dimension
> 2 in Chap. XI). Indeed, the limit in Definition C.9) is monotone, hence is equal,
in the case of BES3 to the probability that BES3@) has reached у before time /
which is strictly positive. In this case, we see that the term "entrance boundary"
is very apt as BES3 is a process on [0, oc[ (see Sect. 3 Chap. VI) which does not
come back to zero after it has left it.
At this juncture, we will further illustrate the previous results by computing
the speed measure of BES3. Since the scale function of BES3 is equal to (-\/x),
by passing to the limit in Corollary C.8), we find that the potential operator of
BES3 is given, for x > 0, by
U f(x) = Ex \j f(Xs)ds\ = f u(x, y)f(y)m(dy)
where m is the speed measure and u(x, y) = inf(l/x, \/y). To compute the
potential kernel t/@, •) of BES3@), we may pass to the limit; indeed for e > 0,
if / vanishes on [0, e], the strong Markov property shows that
y-'f(y)m{dy).
Passing to the limit yields that U@, •) is the measure with density y~l l(v>0) with
respect to m. On the other hand, since the modulus of BM3@) is a BES3@), (/@, ¦)
may be computed from the potential kernel of BM3 (Exercise B.29) Chap. Ill); it
follows, using polar coordinates, that
/ if(p)p2
1 Г f(\x\\ 1 Г2 C f
Uf@) = — ^-dx = — / / pif(p)p2cos<l>ded<l>dp
2n JRi \x\ 2л- Jo J^/jJo
= 2 [ f{p)pdp.
Jo
Comparing the two formulas for Uf{0) shows that the speed measure for BES3
is the measure with density 2y2 l(v>0) with respect to the Lebesgue measure.
We now turn to the case E — [0, oc[. In that situation, 5@) is a finite number
and we then always make .s@) — 0. The hypothesis of regularity implies that 0 is
visited with positive probability; on the other hand, since regularity again bars the
possibility that 0 be absorbing, by the remarks below Proposition B.19) Chap. Ill,
the process started at 0 leaves {0} immediately.
The speed measure is defined so far only on ]0, oc[ and the formula in Theorem
C.6) gives for b > 0, the mean value of 7Ь л Tb for the process started at x e]0, b[.
We will show that the definition of G may be extended and that m({0}) may be
defined so as to give the mean value of Ть for the process started at x e [0, b[.
We first define a function s on ] — oc, oc[ by setting s(x) = s(x) for x > 0 and

§3. Linear Continuous Markov Processes 307
s(x) — -s(-x) for x < 0. For J = [0,b[, we define Gj as the function G,
defined for / = [—b, b] by means of J in lieu of s. For x, у > 0, we next define
a function Gj by
C.10) Proposition. One can choose m({0}) in order that for any x € J and any
positive Borel function f,
x\f " fiX^ds] = f
= / Gj(x,y)f(y)m(dy).
Jj
Proof Thinking of the case of reflected BM, we define a process X on ] — oo, oof
from which X is obtained by reflection at 0. This may be put on a firm basis by
using the excursion theory of Chap. XII and especially the ideas of Proposition
B.5) in that chapter. We will content ourselves here to observe that we can define
the semi-group of X by setting for x > 0 and А С К+
P,(x, A) = Ex [lA(X,)l,,<r0)] + -Ex [|д(Х,I(,>г„)]
and for А с /?-,
_ 1 r -,
i j " L A ' <'-7»>J
where A = —A. For x < 0, we set P,(x, A) = P,(-x, A). Using the fact that
To = t + To о в, on [To > t), it is an exercise on the Markov property to show
that this defines a transition semi-group. Moreover, excursion theory would insure
that X has all the properties we demanded of X at the start of the section. It is
easy to see that J is a scale function for X and the corresponding speed measure
m then coincides with m on ] — oo, Of and ]0, oof.
Let now / be the function equal to / on [0, oof and defined by f(x) —
— f(—x) on ] - oo, Of. By applying Corollary C.8) to X, we have
Ex j f(Xs)ds = Ex\l f(X,)ds = / Gj(x, y)f(y)rh(dy)
f '
J]o.b] ' 2
It remains to set m([0}) — m({0})/2 to get the result.
We observe that m({0}) < oo.
C.11) Definition. The point 0 is said to be slowly reflecting if m({0}) > 0 and
instantaneously reflecting ifm({0}) = 0.
If absorbing points were not excluded by regularity, it would be consistent to
set m([0}) = oo for 0 absorbing.
For the reflected BM, the point 0 is instantaneously reflecting; the Lebesgue
measure of the set {t : X, = 0} is zero which is typical of instantaneously reflecting

308 Chapter VII. Generators and Time Reversal
points. For slowly reflecting points, the same set has positive Lebesgue measure as
is seen by taking / — l@) in the above result. An example of a slowly reflecting
point will be given in Exercise B.29) Chap. X.
We next turn to the description of the extended infinitesimal generator A of
X; its domain (see Sect. 1) is denoted by Од. Here E is any subinterval of BL We
recall that the s-derivative of a function / at a point x is the limit, if it exists, of
the ratios (f(y) - f(x))/(s(y) - s(x)) as у tends to x. The notions of right and
left derivatives extend similarly.
о
C.12) Theorem. For a bounded function f of Шл and x e E,
dm ds
in the sense that
i) the s-derivative ^? exists except possibly on the set {x : m({x}) > 0},
ii) if X\ and X2 are two points for which this s-derivative exists
df. , df f*>
-r(x2) - -j-(-*i) = / Af{y)m{dy).
ds ds Л,
Proof. If / e Ю>д, by definition
M/ = f(X,) - f(X0) - f Af(Xs)ds
Jo
is a martingale. Moreover, |A//| < 2||/|| +г||Л/|| so that if T is a stopping time
such that EX[T] < oc, then m{aT is uniformly integrable under Px and therefore
ЕЛ/(*г)] - fix) = Ex \j Af{Xs)ds\.
о
For / = ]a, b[ с E and a < x < b, we may apply this to T — Ta л Ть, and, by
Corollary C.8), it follows that
(#) f(a)(s(b) - s{x)) + f(b)(s(x) - s(a)) - f(x)(s(b) - s(a))
= (s(b) - s(a)) [ G,(x, y)Af(y)m(dy).
By straightforward computations, this may be rewritten
fib) - f(x) fix) - f(a)
s(b)-s(x) s(x)-s(a)
where
= / H,(x,y)Afiy)m(dy)
s(y)-s(a)
na<y<x,
s(x) - s(a)
s(b)-s{y) .. ,
—¦ ^— if x < у < b,
s(b)-s(x)
0 otherwise.

§3. Linear Continuous Markov Processes 309
If we let b decrease to x, the integrand ^ll'^'j 1(X<V<A) tends to l(.vj; by an
application of Lebesgue's theorem, we see that the right s-derivative of / exists.
Similarly, the left ^-derivative exists. If we let simultaneously a and b tend to x,
we find, applying again Lebesgue's theorem, that
-j-(x) - -¦?-(*) = 2m({x))Af(x)
ds ds
which yields the part (i) of the statement.
To prove the second part, we pick h such that a < x + h < b. Applying (#)
to x and x + h and subtracting, we get
If the ^-derivative of / in x exists, letting h go to zero and applying once more
Lebesgue's theorem yields
f(b) - f(a) - (s(b) - ^
= -[ (s(y)-s(a))Af(y)m(dy)+ f (s(b) - s(y))Af(y)m{dy)
J a J v
/b p.x pb
s(y)Af{y)m(dy)+s(a) I Af(y)m(dy) + s(b) I Af(y)m(dy).
J a J v
Let x\ < X2 be two such points; by subtraction, we obtain
df df f!
-7-U2) - -j-(Jri) = / Af(y)m(dy)
ds ds Л,
which is the desired result.
Remarks. Iе) The reader will observe that, if.? is multiplied by a constant, since by
its very definition m is divided by the same constant, the generator is unchanged,
as it should be.
2°) For the linear BM, we get A — j j^ as we ought to. The reader can
further check, using the values for * and m found above, that for the BES3, we
have A = \ ^ + { ?; this jibes with the SDE satisfied by BES3 which was given
in Sect. 3 of Chap. VI.
3°) The fact that s(X)R is a local martingale agrees with the form of the
infinitesimal generator and Proposition A.6).
We now investigate what happens at the boundary point when E = [0, oc[.
The positive maximum principle shows that the functions of Ю>д must satisfy some
condition on their first derivative. More precisely, we have the

310 Chapter VII. Generators and Time Reversal
C.13) Proposition. If E = [0, oc[, for every bounded f e UA
Proof. Using Proposition C.10) instead of Corollary C.8), the proof follows the
same patterns as above and is left to the reader as an exercise. ?
For the reflected BM, we see that, by continuity, Af@) = \f"{0) and that
/'@) — 0 which is consistent with the positive maximum principle; indeed, a
function such that /'@) < 0, could have a maximum in 0 with /"@) > 0. It
is also interesting to observe that the infinitesimal generators of the reflected BM
and the absorbed BM coincide on C|(]0, oo[) (see Exercise C.16)).
As may be surmised, the map X -*¦ (.v, m) is one-to-one or in other words,
the pair (s, m) is characteristic of the process X. We prove this in a special case
which will be useful in Chap. XI.
C.14) Proposition. IfX and X are two Feller processes on [0, oc[ such that s = s
and m = in, then they are equivalent.
Proof By Propositions A.6) and A.7) on the one hand, and Theorem C.12) and
Proposition C.13) on the other hand, the spaces &А and i/A are equal and so are
the generators A and A. It follows from Exercise A.18) that the semi-groups of
X and X are equal whence the result follows. ?
C.15) Exercise. Prove that X is on its natural scale if and only if for any a < b
and xo — (a + b)/2,
PXl,[Ta<Tb}=\/2.
C.16) Exercise. Prove that the domain of the infinitesimal generator of the re-
reflected BM is exactly {/ e C02([0, oc[) : /'@) = 0}.
* C.17) Exercise. 1°) (Dynkin's operator). For x e E and h sufficiently small,
call /(A) the interval ]x - h, x + h[. For / e BA, prove that
Af(x) = Hm (Ex [f{Xaith))] - fix)) /mHh)(x).
The limit on the right may exist even for functions / which are not in Од. We
will then still call the limit Af, thus defining a further extension of the operator
A.
2°) For / =]a, b[cz E, define p,(x) = Px[Th < Ta]. Prove that Ap, = 0 on /
and Ant/ = -1 on /.
C.18) Exercise. If ф is a homeomorphism of an interval E onto an interval Ё and
if X is a process on E satisfying the hypothesis of this section, then X = ф(Х)
is a process on Ё satisfying the same hypothesis. Prove that s = s о ф~1 and that
m is the image of m by ф.

§3. Linear Continuous Markov Processes 311
C.19) Exercise. Suppose that X is on its natural scale and that E = [0, oo[; prove
that for any e > 0, /,0 , ym(dy) < oo.
# C.20) Exercise. Let X be a diffusion on R with infinitesimal generator
2 dx2 dx
where a and b are locally bounded Borel functions and о does not vanish. We as-
assume that X satisfies the hypothesis of this section (see Exercise B.10) Chap. IX).
1°) Prove that the scale function is given by
s(x)= / expl- / 2b(z)a l(z)dz\dy
Jc \ J с /
where с is an arbitrary point in K. In particular, if b — 0, the process is on natural
scale.
2°) Prove that the speed measure is the measure with density B/s'a2) with
respect to the Lebesgue measure where s' is the derivative of s.
[Hint: Use Exercise C.18).]
C.21) Exercise. 1°) If E =]/, r[ and there exist x. b with I < x < b < r such
that Px[Tb < oo] = 1 (for instance in the case of BESrf, d > 2), then s{l+) -
lima;/i-(a) = —oo. Conversely, if s(l+) = -oo, prove that Рх[Ть < oo] — 1 for
every / < x < b < r.
2°) If, moreover, s(r-) = lim^,. s(b) — oo, prove that X is recurrent.
3°) If instead of the condition in 2C), we have s(r—) < oo, prove that
Px |limA", = Л = Px [infX, > /j = 1
and find the law of у — inf, X, under Px. As a result, X is recurrent if and only
if s{l+) = -oo and s(r-) = oo.
4°) Under the conditions of 3°), i.e. s(l+) = —oo, .s(r-) < oo, prove that
there is a unique time p such that Xp = у.
[Hint: use the same method as in Exercise C.17) of Chap. VI, the hypotheses
of which, the reader may observe, are but a particular case of those in 3°).]
C.22) Exercise. Let X be a strong Markov process on К with continuous paths.
For a e R, set
D,,+ = T]a Ж[, Da- — Г]_ос.а[.
1°) Prove that either Pa(,Da+ = oo] = 1 or Pa[Da+ = 0] = 1 and similarly
with Da_. The point a is said to be regular (resp.: a left shunt, a right shunt,
a trap) if Pa[Da+ = 0, Da = 0] = 1 (resp.: Pa[Da+ = oo, Dfl_ = 0] = 1,
Pa[Da+ = 0, ?>a_ = oo] = 1, Pa[Da+ = oo, Da- = oo] = 1). Find examples of
the four kinds of points. Prove that a regular point is regular for itself in the sense
of Exercise B.25) in Chap. Ill, in other words Pa[Ta = 0] = 1. For a regular

312 Chapter VII. Generators and Time Reversal
process in the hypothesis of this section, prove that all the points in the interior
of / are regular.
2°) Prove that the set of regular points is an open set, hence a union of intervals.
Assume that ? = oo a.s. and show that the process can be restricted to any of
these intervals so as to obtain a process satisfying the hypothesis of this section.
C.23) Exercise. If E =]0, oof, ? — oo a.s. and X is on its natural scale, then 0
is a natural boundary.
C.24) Exercise. Let E = R and X be recurrent (Exercise C.21), 3°)). Choose s
such that s@) — 0. If ц is a probability measure on К such that
/ \s(x)\dn{x) < oo, / s(x)dn(x) - 0,
prove that there exists a stopping time T such that the law of Хт under Po is ц.
* C.25) Exercise (Asymptotic study of a particular diffusion). In the notation of
Exercise C.20) let a(x) = 2A + x2) and b(x) = 2x.
1° Prove that X = tan(y#,) where у is a BM1 and
H, =2 f A + X2)~] ds = inf\u : f A + tar\2(y,))ds > 2t\ .
Jo I Jo J
[Hint: Use Exercise C.20) and Proposition C.4).]
2°) Show that as t tends to oo,
lim H, = inf{M : \yu\ = л/2} a.s.,
and lim^so |X,| = oo a.s.
3°) Show that for t0 sufficiently large (for instance to — sup{/ :\X,\ — 1)), the
following formula holds for every / > /0,
ft pi
/0|+ / x;1B(i + x.?)I/2u?A+?-r0- / x;2ds
Jtn J la
where /J is a BM.
4°) Show that ?~'/2(log |X,| — t) converges in law as t tends to oo, to 2G
where G is a standard Gaussian r.v.
§4. Time Reversal and Applications
In this section, we consider a Markov process with general state space and con-
continuous paths on [0, ?[ and assume that X?_ exists a.s. on {? < oo} as is the case
for Feller processes (Theorem B.7) of Chap. III). Our goal is to show that, under
suitable analytic conditions, one can get another Markov process by running the
paths of X in the reverse direction starting from a special class of random times
which we now define.

§4. Time Reversal and Applications 313
D.1) Definition. A positive r.v. L on Q is a cooptional time if
i) {L < oo] с {L<t;};
ii) For every t > 0, L о в, = (L - t)+.
The reader will check that ? is a cooptional time and so is the last exit time
of a Borel set A defined by
LA(w) - sup{/ : Х,(ш) е A]
where sup@) = 0. We also have the
D.2) Proposition. If L is cooptional, then for any s > 0, the r.v. (L — s)+ is also
cooptional.
Proof. Condition i) of D.1) is obviously satisfied by (L — s)+ and moreover
(L-s)+o0, = (Loe,-s)+ = ((L-t)+-s)+ = (L-t-s)+ = ((L-s)+-t) +
which is condition ii). ?
In what follows, L is a fixed, a.s. finite and strictly positive cooptional time
and we define a new process X taking its values in Ea by setting for t > 0
~ \ A if L(a>) < t or L(a>) = oo,
X<{C0)-\ ХЦш).,(со) if 0<?<L(«),
and Xo = XL- if 0 < L < oo and Xo = Л otherwise.
We will set -V, — cf(Xs, s < t). On {L > t + и], we have, using Property ii)
of Definition D.1), L@U) — L-u > t, hence
^г(^м) = Хши)-1(ви) — Xl-ii-i+ii = Ki-
Kilt follows from the monotone class theorem that, if Г is in .7"t, then for every
м > О
We now introduce the set-up in which we will show that X is a Markov
process. We assume that:
i) there is a probability measure ц such that the potential v = jiU where U is
the potential kernel of X (Exercise B.29) Chap. Ill), is a Radon measure,
ii) there is a second semi-group on E, denoted by (P,), such that
a) if / e CK(E), then P,f is right-continuous in t;
b) the resolvents (Up) and (Up) are in duality with respect to v, namely
j Upf gdv = J f-Upgdv
for every p > 0 and every positive Borel functions / and g.

314 Chapter VI]. Generators and Time Reversal
Examples will be given later in this section. The last equality will also be
written
{UPf,g)v = (f,Ul,g)v.
If X is another Markov process with P, as transition semi-group, we say that X
and X are in duality with respect to v. Using the Stone-Weierstrass and monotone
class theorems, it is not difficult to see that this relationship entails that for any
positive Borel function ф on K+
(P<pf-g)v = (/. /W>,,.
where Рф/{х)=/™ф(ПР,/(х№.
Our goal is to prove that X is a Markov process with transition semi-group
(Pi). We will use the following lemmas.
D.3) Lemma. Given r > О, ф a positive Borel function on K+ and H a positive
^-measurable r.v., then for any positive Borel function f on E,
f фA)Ец [/(Х,+Л)Я] dt = f fh*dv
where Иф(х) — Ex [Нф(Ь — r)\\r<i]\. Moreover, for s > r,
f фA)Ец [/(*,+,)//] rff = f fPs-rh,pdv.
Proof. By considering (L — r)+ instead of L, we may make r = 0 in the equality
to be proven. The left-hand side is then equal to
/00 /»OC
EpL[f(XL-,)\lL>oH^(t)dt = / EJHf(Xu№(L-u)](L>u)]du
Jo
>0)) o0uf{Xu)]du
since, as a consequence of the definition of •>(o, we have H = H ови on (L > u).
Furthermore, by the Markov property of X, the last expression is equal to
duE,[h,p{Xu)f(Xu)\ =
which proves the first part of the lemma.
To prove the second part, observe that since .F^ с .i^, the r.v. H is also in.%
so that, by the first part, we may write
,+,)//] dt = J f(x)Exy>(L-s)H \ls<L)]v(dx).
But since l(r<L) ° &s~r = 1|.5<l} and H о 9r_s = H on {L > s), we have
ф(Ь —s)H l(.t<L) = (ф(Е — r)H \{r<L}) o&r-s and the second part follows from
the Markov property. " a

§4. Time Reversal and Applications 315
D.4) Lemma. Keeping the notation of Lemma D.3), if^i is another positive Borel
function on R+, then
Pil/h0 — h^/^ф = P<ph,[,.
Proof. From the above proof, it follows that
Ps-rh0 = E.[<j>(L-s)H \is<L)].
As a result
Р*Ьф = f № ~ r)Ps-rh0ds = ? \f if(s - rL>(L - s)H \ls<L)ds\
Г fL 1
= ?. M(r<L)H / '/Ф - г)ф(Ь -s)ds\ = Иф*ф,
L Jr A
which is the first equality. The second one follows by symmetry. ?
We may now turn to the main result of this section.
D.5) Theorem. Under P^, the process X is a Markov process with respect to (,^)
with transition function (P,).
Proof. We want to prove that E^ \f(Xl+s) \ J^l = P,f(Xs) P^-a.s. But using
the notation and result in Lemma D.3) we have
[ 4>(t)E, [/(Х,+,)Я] dt = (/, Р,-ГИФ)„
and on the other hand, Fubini's theorem yields
Let us compare the right members of these identities. Using the above lemmas
and the duality property, we get
/ ns - r)(f, Ps-^)vds = (f,
Jr
), = (f,
It follows that there is a Lebesgue-negligible set N(/, г, Н, ф) с R+ such that for
s > r, s i N(f, г, Я, ф),
Let N be the union of the sets N(f,r, Н,ф) where / runs through a dense
sequence in Ск(Е), ф runs through a sequence which is dense in CV(K+), r runs
through rational numbers > 0 and for each r, H runs through a countable algebra

316 Chapter VII. Generators and Time Reversal
of bounded functions generating cr(Xu, и < r); then, for s t? N, the last displayed
equality holds simultaneously for every / e CK(E), ф e СК(Ш+), r e Q+, s > r
and every H which is cr(Xu, и < r)-measurable and bounded. As a result, under
the same conditions
for almost every t. But, by the property ii) a) of P,, both sides are right continuous
in t and the equality holds for every t.
Next, because of the continuity of paths, the filtration (.^) is right and left
continuous (i.e. .i^"_ = .7~, = .yf+ for each t) up to sets of /^-measure zero; it
follows first that the last displayed equality is valid with H in .Vs and, finally,
since each s is a limit of points in Nc, that this equality holds without restriction.
As a result, for / e CK(E),
for every s and t, which is the desired result.
Remarks. Iе) This result does not show that X has good properties such as the
Feller or Strong Markov property. However, in many applications the semi-group
(A) is a Feller semi-group, which insures that X has good properties (see also
Exercise D.13)).
2°) If P, is already known to be the semi-group of a process X, then X under
Рц has the same law as X under Ai where Д = Xi-(Pfl) is the law of X^_
under Рц.
We will now give two applications of Theorem D.5) to the Bessel process of
dimension 3 which was introduced and studied in Sect. 3 Chap. VI. This process
and the BM killed at 0 (which we will denote by B°) are in duality with respect
to the measure v(dx) = 2x dx. Indeed, in the notation of Chap. VI, using the fact
that Q, is in duality with itself with respect to Lebesgue's measure, we have, for
f,g>0 andh(x) -x,
/•00 1-Х [ ^OC
/ f(x)P?g(x)xdx= f(x)-Q,(gh)(x)xdx= / Q,f(x)g(x)xdx
Jo Jo x Jo
which proves our claim. This is a particular instance of a more general result;
the process BES3 is the A-process of BM° as defined in Sect. 3 Chap. VI11 and
as such is in duality with BM° with respect to the measure h(x)dx (see Exercise
C.17) in Chap. VIII).
Furthermore, it was shown in the last section that v is precisely equal to
the potential measure t/@, •) of BES3@). Thus, we are exactly in the setting of
Theorem D.5) with ц = eo, and we may state the
D.6) Corollary (Williams). Let X be a BES3(O) and В a BM(fc) with b > 0, then
ifLb = sup{? : X, = b), the processes {XLh-,, 0 < t < Lh] and {B,, 0 < t < To)
have the same law.

§4. Time Reversal and Applications 317
Remarks. 1°) Another proof of this result relying on excursion theory will be
given in Chap. XII.
2°) This corollary implies that the law of L/, for BES3@) is the same as the
law of 7 for BM(fe), which was computed in Chap. II, Proposition C.7) and
Chap. Ill, Proposition C.7).
Our second application deals with the process BES3 killed when it first hits a
point b > 0. More precisely, if X is a BES3, we consider the process Xh defined
by
Xf = X, if / < Th and Xo e [0, b[, Xht = Л otherwise
where as usual Th = inffr > 0 : X, = b). It was shown in Exercise B.30) of
Chap. Ill that this is a Markov process on [0, b[ and clearly Th is the deathtime,
hence a cooptional time, for Xh .
D.7) Lemma. The processes Xb and b — Xh are in duality with respect to the
measure %(dx) — x(b — x)dx on [0, b].
Proof. We have already used the fact, that the potential U of X has the density
u(x, y) — inf(l/;c, 1/v) with respect to the measure 2y2dy. By a simple applica-
application of the strong Markov property, we see that the potential V of Xb is given by,
for x < b,
Vf(x) = Ex \J " f(X,)dt\ = Uf(x) - PThUf(x)
fh
= 2 (u(x.y)-u(b,y))y2dy-
Jo
in other words, V has the density v(x, y) = inf(I/jr, \/y) — \/b with respect to
the measure 2y2\@<v<b)dy. Clearly, the potential V of the process b — Xb has the
density v(b — x, b — y) with respect to the measure 2(b — yJ1 (o<v<b)dy. It is then
a tedious but elementary computation to check that for f,g>0,
j Vf-gd^ = J f-Vgdt;.
Now the mapping / ->• Vf (resp. / -» Vf) is bounded on the space of bounded
functions on [0. b] so that the result follows from Exercise D.17). ?
D.8) Proposition. If X is a BES3@) and b is strictly positive, the processes
(*Y,,-r. 0 < f < Th) and (b - X,,0 < t <Th) are equivalent.
Proof. The potential measure V@, dy) is equal, by what we have just seen, to
2A /y - \/b)y2dy = b%(dy). Thus the result follows at once from Theorem D.5)
and the above lemma. a
Bringing together Corollary D.6), Proposition D.8) and Theorem C.11) of
Chap. VI we obtain

318 Chapter VII. Generators and Time Reversal
BES3@)
Fig. 7.
D.9) Theorem (Williams' Brownian path decomposition). For b > 0, let be
given the four following independent elements:
i) a r.v. a uniformly distributed on [0, b];
ii) a standard BM B;
Hi) two BES3@) processes p and p',
and define
Ta = M{t :B,=a], gn = Ta + sup{/ : a - p(t) = 0}.
then, the process X defined for 0 <t <Ть by
X,=
a-p(t-Ta),
P'(t - gr,,),
is a BM@) killed when it first hits b.
0 < / < Ta
Та <Г <gTl,
gn<t < Tb
Proof. By Corollary D.6), a BM killed at time Tb is a time-reversed BES3@) to
which we apply the decomposition Theorem C.11) of Chap. VI. The time-reversed
parts are easily identified by means of Corollary D.6) and Proposition D.8). Here
again, the result is best described by Figure 7; it is merely Figure 5 of Chap. VI
put "upside down".

§4. Time Reversal and Applications 319
Remark. There are actually other proofs of the fact that BM taken between gTt
and Th (if gTh is the last zero before Tb) is a BES . If this result were known,
then the above decomposition theorem might be deduced from D.6) and Theorem
C.11) of Chap. VI without having to resort to Proposition D.8) above.
D.10) Exercise. If L and V are two cooptional times, then L V L' and L л L'
are cooptional times.
* D.11) Exercise. Let L be a cooptional time and &L be the family of sets Г e .W
such that for every и > 0,
гп([>и)=о;|(Г)п|ь«].
1°) Prove that •'?/, is a a -algebra (see also Exercise D.13) below) and that L
and XL are .'^-measurable.
2°) If A e '-0L, prove that the r.v. LA defined by
LA = L on A, LA=0 onAc,
is cooptional.
* D.12) Exercise. 1°) Let p, be the modulus of BM2 and suppose that p0 = r
with 0 < r < 1. Prove that there exists a BM1 у started at (—logr) such that
- logp, = yc, where C, = inf {u : /0" exp(—2ys)ds > ?}.
[Hint: Use the ideas of Sect. 2 Chap. V.]
2°) Let A" be a BES2@) and 7", = inf{? : X, = 1}. Prove that there exists a
BES3@), say Y, such that
(-logA:,,0 </ < 7-,) =
where A, = sup {и : /и°° exp(-2Ys)ds > t).
[Hint: Apply Corollary D.6) to the BM у of 1°), then let r converge to 0.]
3°) Extend the result of 1°) and 2°) to p = |BMJ| with d > 3. More precisely
prove that if X is a BESrf@)
where A, = sup [u : (d - 2)~2 /a°° Y-"ds > t}, and a = 2(d - \)/(d - 2).
* D.13) Exercise. With the notation of this section, let '?, be the a -algebra of sets
Г in .V such that for every и > 0
1°) Prove that (¦'?,) is a right-continuous filtration which is larger than (.Pf).
Check that Lemmas D.3) and D.4) are still valid with ('.?,) instead of {.Fi).
2°) Prove that if T is a (.'?,)-stopping time, then (L — T)+ is a cooptional
time.
3°) Prove that in Theorem D.5), one can replace (.Pp> by ('?,); then using 2°),
prove that X has the strong Markov property.

320 Chapter VII. Generators and Time Reversal
* D.14) Exercise. Let L be a cooptional time and set ф(х) = PX[L > 0].
1°) Prove that ф is an excessive function (see Definition C.1) of Chap. X).
2°) If / is excessive and finite, prove that one defines a new transition semi-
semigroup Pf by setting
p/{x,dy)= f-\x)P,(x,dy)f(y) if/(*)/0
= 0 otherwise.
(See also Proposition C.9) in Chap. VIII).
* 3°) Let Y,(a>) = X,(w) if t < L(w) and Y,(w) = Л if t > L(w), and prove that
for any probability measure ц, the process У is a Markov process with transition
semi-group p/.
* D.15) Exercise (Another proof of Pitman's theorem). Let В be the standard
linear BM, L its local time at 0 and as usual ri = inf{/ : L, > 1}. We call
G") the following property which is proved in Exercise B.29) of Chap. VI and in
Exercise D.17) of Chap. XII: the processes (|fi,|, t < т{) and (|ВГ|_,|. / < ri) are
equivalent. Call (P) the property proved in Pitman's theorem (Sect. 3 Chap. VI)
namely
BS, - B,,S,.t > 0) = (Z,,J,,r >0)
where Z is a BES3@) and J, — inft>, ZS. Call further (R) the time-reversal
property of Corollary D.6). The aim of this exercise is to show that together with
the Levy equivalence (S, - B,,t > 0) = (\B,\, L,,t > 0) proved in Sect. 2 of
Chap. VI and which we shall call (L), any two of the properties G"), (P), {R)
imply the third one.
1°) Let as usual 7", = inf{/ : B, = 1}; deduce from (L) that
(\BTl.u\.u <r,) = (-1+Sr,-H + A -BTl.u).u < Г,)
and conclude that (R) and (P) imply G").
2°) Using (L) (or Tanaka's formula) prove that
(Ям. и < h) = (Lu - \В„\,и < n)
and conclude that G") and {P) imply (/?).
[Hint: If (L) is known, (P) is equivalent to (P1), namely
(\BU\ + Llt,u >0)(= (Z,,.w>0).]
3°) Use G"), then (L), to prove that
(\BH\ + Lu,u <т,)(= (l-BT,-u.u <Tt).
Use the scaling invariance properties to deduce that for any a > 0,
(\BU\ + Lu,u < та) = (Zu,u < La)
and conclude that G") and (R) imply {P'),

§4. Time Reversal and Applications 321
* D.16) Exercise (On last passage times). Let L be a cooptional time.
Iе) In the notation of Exercise D.14) prove that the supermartingale Z, —
ф(Х,) (see Proposition C.2) Chap. X) is equal to PX[L > / | .i^] Px-a.s.
2=) Suppose that X is a Feller process on ]0, oof and that the scale function
.v is such that s@+) — — oo and s(oo) = 0 (see Exercise C.21)). For a > 0,
let L = La — sup{f : X, — a] and Лх be the family of local times of the local
martingale s(X). Prove that
is a local martingale (a particular instance of Meyer's decomposition theorem).
3°) Prove that for every positive predictable process H,
EX[HL] = —-EX\
2s (а) [Л)
This may be stated: jTuT)^^'0 's l^e dual predictable projection of lio<z.< ]•
4°) We now assume in addition that
i) there exists a continuous function /?• on ]0, oo[3 such that
P,(x,dy) = p,(x, y)m(dy);
ii) for every positive Borel function / on R+,
i /•
f(X,)ds= / f(y)A)(y)m(dy)
(see Exercise B.32) Chap. X and Sect. 1 Chap. XI). Prove that
9 s(u)
9/ '
5C) Show that
Px(La edt) = ——pt(x,a)dt.
2s (a)
An important complement to this exercise is Exercise A.16) in Chap. X where
it is shown how conditioning with respect to L is related to the distribution of the
bridges of X.
# D.17) Exercise. Г) Let V1' and V1' be two resolvents on (E. У,) such that the
kernels V = V° and V — V° are bounded on the space ЬУ, of bounded Borel
functions. Iff is a measure such that
f
= j f-Vgdt;
for every pair of positive Borel functions, prove that the two resolvents are in
duality with respect to ?.
[Hint: For p sufficiently small, V = XX p" V"+>.]

322 Chapter VII. Generators and Time Reversal
2°) If the two resolvents are the resolvents of two right- or left-continuous
processes with semi-groups P, and Pt, prove that for every /,
P,f-gdt;= I f-P,gd$.
D.18) Exercise. Let X be a Feller process on [0, oo[ such that 0 is not reached
from the other points and such that the restriction of X to ]0, oo[ satisfies the
hypothesis of Sect. 3. We call л and m the corresponding scale function and speed
measure and assume that .y@+) = —oo, 5@0) — 0 (see Exercise C.21)).
1°) Compute the potential kernel of X and prove that X is in duality with itself
with respect to m.
2°) Prove that for every b > 0, the process Y, = {X^_,, t < Lb} is under Po
a Markov process on ]0, b[ with initial measure eb and semi-group Q, given by
Q,f{x) = P,(fs)(x)/s(x).
As a result the law of Lb under Po is the same as the law of To for the process Y.
3°) Prove that (—\/s) is a scale function for the process with semi-group Q,
and that the corresponding speed measure is s2m.
Notes and Comments
Sect. 1. This section is ucvoted to the minimum of semi-group theory which is
necessary (for intuition more than for technical needs) in the sequel. For a detailed
account, we recommend the book of Pazy [1]; another exposition designed for
probabilists is that of Dellacherie-Meyer [1] vol. IV.
We are uncertain about the origin of Theorem A.13) but we can mention that
it is a special case of a much more general result of Kunita [1] and Roth [1];
the same is true of Exercise A.19). Exercise A20) comes from Chen [1] and
Ledoux [1].
In relation to Exercise A.21), the reader shall find more extensions of lto's
formuh in Kunita [5].
Most exercises of this section just record classical properties of semi-groups
and may be found in the textbooks on the subject.
Sect. 2. The bulk of this section is taken from Stroock-Varadhan [1] and Priouret
[1]. The systematic use of martingale problems in the construction of diffusions
is due to Stroock and Varadhan. Their ideas were carried over to other contexts
and still play a great role in present-day research although it is only of marginal
interest in our own exposition which favors the SDE aspect.
The exercises of this section have the same origin as the text. Exercise B.8)
is taken from a lecture course by Meyer.

Notes and Comments 323
Sect. 3. The material covered in this section appeared in a series of papers of
Feller. There are many extensive - much more so than ours - expositions of the
subject, e.g. in Dynkin [1], Ito-McKean [1], Freedman [1] and Mandl [1]. Our
exposition is borrowed from Breiman [1] with other proofs, however, where he
uses approximation by discrete time processes. The exercises are very classical.
Sect. 4. The main result of the section, namely Theorem D.5), is due to Nagasawa
[1]. In our exposition, and in some of the exercises, we borrowed from Meyer [2]
and Meyer et al. [1]. Corollary D.6) is stated in Williams [3]. The remarkable
Theorem D.9), which was the first of this kind, is from Williams [2] and [3].
Exercise D.12) is from Williams [3] and from Yor [16]. Exercise D.16) comes
from Pitman-Yor [1]; see Getoor [2] for the particular case of transient Bessel
processes.
In connection with Exercise D.15) let us mention that Pitman's theorem has
now been extended to other processes (see Tanaka ([2], [3]) for random walks,
Bertoin [5] for Levy processes, and Saisho-Tanemura ([1]) for certain diffusions;
see in particular Exercise A.29) Chap. XI).

Chapter VIII. Girsanov's Theorem
and First Applications
In this chapter we study the effect on the space of continuous semimartingales of
an absolutely continuous change of probability measure. The results we describe
have far-reaching consequences from the theoretical point of view as is hinted at
in Sect. 2; they also permit many explicit computations as is seen in Sect. 3.
§1. Girsanov's Theorem
The class of semimartingales is invariant under many operations such as compo-
composition with C2-functions or more generally differences of convex functions. We
have also mentioned the invariance under time changes. It is also invariant under
an absolutely continuous change of probability measures. This is the content of
Girsanov's theorem: If Q is a probability measure on (?2, .'7') which is absolutely
continuous with respect to P, then every semimartingale with respect to P is a
semimartingale with respect to Q.
The above theorem is far from intuitive; clearly, a process of finite variation
under P is also a process of finite variation under Q but local martingales may
lose the martingale property. They however remain semimartingales and one of
our goals is precisely to describe their decomposition into the sum of a local
martingale and a process with finite variation.
We will work in the following setting. Let (>^0), t > 0, be a right-continuous
filtration with terminal cr-field .%? and P and Q two probability measures on
.7?. We assume that for each / > 0, the restriction of Q to .7~t() is absolutely
continuous with respect to the restriction of P to .FJ°, which will be denoted by
Q <\ P. We stress the fact that we may have Q <] P without having Q <& P.
Furthermore, we call D, the Radon-Nikodym derivative of Q with respect to P on
.J7"°. These (classes of) random variables form a (->^0, /^-martingale and since
(.J^"°) is right-continuous, we may choose D, within its P-equivalence class so
that the resulting process D is a (.^"°)-adapted martingale almost every path of
which is cadlag (Theorem B.5) and Proposition B.6) Chap. II). In the sequel we
always consider such a version.
A.1) Proposition. The following two properties are equivalent:
i) the martingale D is uniformly integrable;
ii) Q«.Pon . JTO.

326 Chapter VIII. Girsanov's Theorem and First Applications
Proof. See Exercise B.13) Chap. II.
As was observed in Sect. 4 of Chap. I and at the beginning of Chap. IV, when
dealing with stochastic processes defined on (?>, .7^, P), one has most often to
consider the usual augmentation of (-^"°) with respect to P, or in other words to
consider the a-fields .^"obtained by adding to .7^°, 0 < / < oo, the P-negligible
sets of the completion .7^ of .7? with respect to P. If Q <] P but Q is not
absolutely continuous with respect to P on -7^, then Q cannot be extended to
.7^. Indeed, if P(A) - 0 and Q{A) > 0, then all the subsets of A belong to
each .7i and there is no reason why Q could be consistently defined on -У{А).
In contrast, if Q <? P on .7^ we have the following complement to Proposition
A.1).
A.Г) Proposition. The conditions of Proposition A.1) are equivalent to:
Hi) Q may be extended to a probability measure Q on ..7^ such that Q < P in
restriction to each of the completed a-fields (,7i), 0 < / < oo.
If these conditions hold, D, = E [dQ/dP | J^°] P-a.s.
Proof. It is easy to see that ii) implies iii). If iii) holds and if A e .7~? and
P(A) = 0, then A e .ЯГ, hence Q(A) = 0 since Q «; P on J*T and therefore
Q « P on .>??. n
In the sequel, whenever Q <JC P on -i^?, we will not distinguish between 2 and
Q in the notation and we will work with (.^-adapted processes without always
recalling the distinction. But if we have merely Q <\ P, it will be understood that
we work only with (.J^"°)-adapted processes, which in view of Exercise A.30)
Chap. IV is not too severe a restriction. The following two results deal with the
general case where Q <\ P only.
The martingale D is positive, but still more important is the
A.2) Proposition. The martingale D is strictly positive Q-a.s.
Proof. Let T = inf {/ : D, = 0 or D, = 0}; by Proposition C.4) in Chap. II, D
vanishes P-a.s. on [7\ oof , hence D, — 0 on {T < t} for every t. But since
Q - D, ¦ P on .3^"°, it follows that Q({T < t}) = 0 which entails the desired
conclusion. ?
We will further need the following remark.
A.3) Proposition. If Q <] P, then for every {.F^^-stopping time T,
Q = DT- P on .7? П [T < oo}.
IfQ « P. then Q = DT ¦ P on .7?.

§1. Girsanov's Theorem 327
Proof. The particular case Q <$C P follows from the optional stopping theorem.
For the general case, we use the fact that D is uniformly integrable on each [0,/].
Let A e .3^°; for every /, the event А Л (T < t) is in ¦^г°л, and therefore
Q{An(T<t)) = f E[D,\.Z?t]dP= [ DTAt
JAn{T<t) JAn(T<t)
[?t] [ TAtdP
An{T<t) JAn(T<t
f DTdP.
JАП(Г<о
Letting t tend to infinity, we get
Q(AD(T < oo)) = / DTdP
J АП(Т<00)
which completes the proof. ?
The martingale D plays a prominent role in the discussion to follow. But,
since the parti-pris of this book is to deal only with continuous semimartingales,
we have not developed the techniques needed to deal with the discontinuous case;
as a result, we are compelled to prove Girsanov's theorem only in the case of
continuous densities. Before we turn to the proof, we must point out that if X
and Y are two continuous semimartingales for both P and Q and if Q <] P, any
version of the process (X, Y) computed for P is a version of the process {X, Y)
computed for Q and this version may be chosen (.5^"°)-adapted (the reader is
referred to Remark A.19) and Exercise A30) in Chap. IV). In the sequel, it will
always be understood that {X, Y) is such a version even if we are dealing with Q.
The following theorem, which is the main result of this section, is Gir-
Girsanov's theorem in our restricted framework; it specifies the decomposition of
P-martingales as continuous semimartingales under Q.
A.4) Theorem (Girsanov's theorem). If Q <] P and if D is continuous, every
continuous {.Wt , P^-semimartingale is a continuous (-3^~°, Q)-semimartingale.
More precisely, if M is a continuous (-У^°, P)-local martingale, then
is a continuous (•3^°, Q)-local martingale. Moreover, if N is another continuous
P-local martingale
(M.N) = (M.N) = {M.N).
Proof. If X is a cadlag process and if XD is a (.3^°, P)-loc. mart., then X is
a (.37°, (?)-loc. mart. Indeed, by Proposition A.3), Dj is a density of Q with
respect to P on .3^°, and if (XD)T is a P-martingale, it is easily seen that XT
is a <2-martingale (see the first remark after Corollary C.6) Chap. II). Moreover a
sequence of (.>^)-stopping times increasing to +00 P-a.s. increases also to +00
Q-a.s., as the reader will easily check.

328 Chapter VIII. Girsanov's Theorem and First Applications
Let Т„ — inf [t : D, < \/n}\ it is easy to see that the process (D ¦ (M, D))T"
is P-a.s. finite and consequently, (A??)) " is the product of two semimarts. By the
integration by parts formula, it follows that
(MD)Tfn = M0D0 + f " MsdDs + f " DsdMs + (M, D)T:,At
Jo Jo
= MQD0 + f M,dDs + [ " DsdMs - (M, D)TnAt
Jo Jo
MsdDs + / DsdMs,
Jo
which proves that [MD) " is a P-loc. mart. By the first paragraph of the proof it
is also a Q-loc. mart.; but (Т„) increases <2-a.s. to +00 by Proposition A.2) and
a process which is locally a loc. mart, is a loc. mart. (Exercise A.30) Chap. IV)
which completes the proof of the second statement. The last statement follows
readily from the fact that the bracket of a process of finite variation with any
semimart. vanishes identically. ?
Furthermore it is important to note that stochastic integration commutes with
the transformation M —»• M. The hypothesis is the same as for Theorem A.4) and
here again the processes involved are (.^"°)-adapted.
A.5) Proposition. If H is a predictable process in L\QC(M), then it is also in
T^M = H ¦ M.
Proof. The first assertion follows at once from the equality (M, M) = (M, M).
Moreover, if H is locally bounded,
H ¦ M = H ¦ M - HD~[ -{M,D) = H ¦ M - D~l ¦ {H ¦ M, D) = bT~M.
We leave to the reader the task of checking that the expressions above are still
meaningful for H e ЦОС(М), hence that H ¦ M = H ¦ M in that case too.
With a few exceptions (see for instance Exercise A.22) in Chap. XI), Q is
usually actually equivalent to P on each .7^°, in which case the above results
take on an even more pleasant and useful form. In this situation, D is also strictly
positive P-a.s. and may be represented in an exponential form as is shown in
A.6) Proposition. If D is a strictly positive continuous local martingale, there
exists a unique continuous local martingale L such that
D, =exp
jL,

§1. Girsanov's Theorem 329
L is given by the formula
L, = logD0+
Jo °' dDs'
Proof. Once again, the uniqueness is a consequence of Proposition A.2) in
Chap. IV.
As to the existence of L, we can, since D is > 0 a.s., apply Ito's formula to
the process log D, with the result that
log Д = log Do + f D;]dDs-[- f D;2d{D,D)s
Jo l Jo
If P and Q are equivalent on each ->^, we then have Q = X(L), • P on .^"°
for every /, which we write simply as Q — X(L) ¦ P. Let us restate Girsanov's
theorem in this situation.
A.7) Theorem. If Q — X (L) ¦ P and M is a continuous P-local martingale, then
M = M - D~] ¦ {M, D) = M - (M, L)
is a continuous Q-local martingale. Moreover, P = <C(-L)Q.
Proof. To prove the first statement, we need only show the identity D -(M. D) —
{M, L) which follows from the equality L, = log Do + /„' D~ldDs.
For the second statement, observe that, because of the first, — L — — L + (L, L)
is a 2"'°cal martingale with (L, L) — (L, L). As a result
К {-L)t = exp j-L, + (L. L), - ~(L, L)\ = K (L) ;
consequently, P = X (-L) ¦ Q. a
We now particularize the situation still further so that P and Q can play totally
symmetric roles.
A.8) Definition. We call Girsanov's pair a pair (P. Q) of probability measures
such that Q ~ P on .K? and the P-martingale D is continuous.
If (P. Q) is a Girsanov pair, then the completion of (->^0) for Q is the same
as the completion for P and we may deal with the filtration (.Щ and forget about
(.>^). Moreover, by the above results, we see that (Q, P) is also a Girsanov pair
and that the class of continuous semimartingales is the same for P and Q.
A.9) Definition. If(P, Q) is a Girsanov pair, we denote the map M —* M from
the space of continuous P-local martingales into the space of continuous Q-local
martingales by Gp and we call it the Girsanov transformation from P to Q.

330 Chapter VIII. Girsanov's Theorem and First Applications
This map is actually one-to-one and onto as a consequence of the following
proposition; it moreover shows that the relation P ~ Q if (/>, Q) is a Girsanov
pair is an equivalence relation on the set of probability measures on .7^.
A.10) Proposition. If(P, Q) and (Q, R) are two Girsanov pairs, then (P, R) is
P = GRp.
) an
a Girsanov pair and Cg о G P = GRp. In particular,
GpQoGQP=GQpoGpQ =
<1Q
Proof. Since jp r = ~ r x -Jf T, the pair (P. R) is obviously a Girsanov
pair.
By definition, G^(M) is the local martingale part of M under R, but as M
and Gp(M) differ only by a finite variation process, G^(M) is also the local
martingale part of Ср(ЛУ) under R which proves the result.
Remark. This can also be proved by using the explicit expressions of Theorem
A.7).
In this context, Propositon A.5) may be restated as
A.11) Proposition. The Girsanov transformation Cp commutes with stochastic
integration.
Before we turn to other considerations, let us sketch an alternative proof of
Theorem A.7) in the case of a Girsanov pair, based on Exercises C.11) and C.14)
in Chap. IV. The point of this proof is that the process A which has to be subtracted
from M to get a B-local martingale appears naturally.
By Exercise C.14), Chap. IV, it is enough to prove that for any A, the process
exp Yk{M - A) - у (М, Af) | is a 2-local martingale, which amounts to showing
that
X2
exp ШМ - A) (M,M}\explL- -(L,L)
is a P-martingale. But this product is equal to
exp |XM + L - — (Л/, M)--{L.L)-kA
which, by Exercise C.11), Chap. IV, is equal to <C(XM + L) provided that A =
(M.L).
Remark. We observe that if Q is equivalent to P on each .3^"and D is continuous,
(P, Q) becomes a Girsanov pair if we restrict the time interval to [0, t] and
the above results apply. However, one should pay attention to the fact that the
filtrations to consider may not be complete with respect to P (or Q); when working
on [0, t], we complete with the negligible sets in .i^"in order to recover the above
set-up.

§1. Girsanov's Theorem 331
To illustrate the above results, we treat the case of BM which will be taken
up more thoroughly in the following section.
A.12) Theorem. If Q <\ P and if В is а (.уГ°, P)-BM, then В = В - (B,L) is
a (.5f\ 6)-BM.
Proof The increasing process of В is equal to that of B, namely /, so that P.
Levy's characterization theorem (Sect. 3, Chap. IV) applies.
Remarks. 1°) The same proof is valid for BM''.
2°) If (.>^) is the Brownian filtration, then for any Q equivalent to P on .3^,
the pair (P, Q) is a Girsanov pair as results from Sect. 3 in Chap. V. Even in that
case the filtration of В may be strictly smaller than the filtration of В as will be
seen in Exercise C.15) in Chap. IX.
3C) The same result is true for any Gaussian martingale as follows from Ex-
Exercise A.35) in Chap. IV.
In this and the following chapters, we will give many applications of Gir-
Girsanov's theorem, some of them under the heading "Cameron-Martin formula" (see
the notes and comments at the end of this chapter). In the usual setting D, or rather
L, is given and one constructs the corresponding Q by setting Q = D, ¦ P on .7^.
This demands that L being given, the exponential <C(L) be a "true" P-martingale
(which is equivalent to E [#(/.),] = 1) and not merely a local martingale (oth-
(otherwise, see Exercise A.38)). Sufficient criterions ensuring this property will be
given below.
When E [<v(L),j = 1 obtains, the formula Q — К (L)-P defines a set function
Q on the algebra |J( .^ which has to be extended to a probability measure on
.К*.. To this end, we need Q to be ст-additive on |J( .5f-
A.13) Proposition. Let X be the canonical process on Q = W = C(K+, W1) and
¦ W^ = a (Xs, л- < /)+/ if'E [<^(L),] = 1, there is a unique probability measure Q
on (&,.7?) such that Q = X{L) ¦ P.
This is a consequence of Theorem F.1) in the Appendix.
The above discussion stresses the desirability of a criterion ensuring that <6 (L)
is a martingale. Of course, if L is bounded or if ?[exp(L*)] < сю this property
obtains easily, but we will close this section with two criterions which can be
more widely applied. Once again, we recall that <C (L) is a supermartingale and
is a martingale if E [i^(L),j = 1 as L is always assumed to vanish at 0. The first
of these two criterions is known as Kazamaki's criterion.
A.14) Proposition. If L is a local martingale such that exp (|L) is a uniformly
integrable submartingale, then К (L) is a uniformly integrable martingale.
Proof. Pick a in ]0, 1[. Straightforward computations yield
l-u2
(aL), = (X (L)tf (z\a))

332 Chapter VIII. Girsanov's Theorem and First Applications
with Z,(a) — exp(aL,/(l +a)). By the optional stopping theorem for uniformly
integrable submartingales (Sect. 3 Chap. II), it follows from the hypothesis that
the family IZ^\ T stopping timej is uniformly integrable. If Г is a set in .7~
and T a stopping time, Holder's inequality yields
Е[\гУАа1)т] < E[X(L)tY E\\rZfX " <е\\г1^Л
since E [#(L)r] < 1. It follows that [<C(aL)T, T stopping time} is uniformly
integrable, hence К {aL) is a uniformly integrable martingale. As a result
1 = E [^(aL)oo] < E [#(L)oo]" E [Z^]1^" .
The hypothesis implies also that Lx exists a.s. and that exp (^Lx) is integrable.
Since Z^' < l(i.^<0) +exp(iLoo) 1а^>0), Lebesgue's dominated convergence
[-|1-и2
Zoo = 1. By letting a tend to 1 in the last
displayed inequality, we get E[^(L)X] > 1, hence ?[?T(L)oc] = 1 and the
proof is complete. D
Remarks. If L is a u.i. martingale and ?[ехр(^оо)] < сю, then exp(|L,) is
a u.i. submartingale and Kazamaki's criterion applies; this will be used in the
next proof. Let us also observe that if M is a local martingale, the condition
?[expM,] < oo for every / does not entail that exp(Af) is a submartingale;
indeed, if Z is the planar BM and 0 < a < 2 then exp (-a log \Z,\) — \Z,\~a has
an expectation decreasing in t and in fact is a supermartingale (see also Exercise
A.24) in Chap. XI).
From the above proposition, we may derive Novikov's criterion, often easier
to apply, but of narrower scope as is shown in Exercises A.30) and A.34).
A.15) Proposition. If L is a continuous local martingale such that
then <6 (L) is a u.i. martingale. Furthermore E [exp (^iJc)] < oo. and as a result,
L is in Hp for every p e [1, oo].
Proof. The hypothesis entails that (L, L)oc has moments of all orders, therefore,
by the BDG-inequalities, so has L*^; in particular L is a u.i. martingale. Moreover
exp Q^oo) = »a)^exp ((L,
so that, by Cauchy-Schwarz inequality,
E [exp Q*-oo)l < E [* (LU]172 E Гехр Q<L, JJ]

§1. Girsanov's Theorem 333
and since ?[^(L)oo] < 1, it follows that E [exp (^L^)] < сю. By the above
remarks, К (L) is a u.i. martingale.
To prove that E [exp (^L^)] < oo, let us observe that we now know that for
с < 1/2,
exp(cL,) = tf'-(L), exp
is a positive submartingale. Applying Doob's inequality with p — \/2c, yields
E sup exp ( -L, ) | < CpE \ 6 (L)^ exp ( -{
L '
Then, by Holder's inequality, we get
Г /1 M 7-/4 2Д2-С) Г
E sup exp I -L, < CPE \<C' '(/.)? c" 1 E exp
L ' \2 /J L
and since B — c)/4c < 1 for с > 2/5, the left-hand side is finite. The same
reasoning applies to —L, thus the proof is complete.
Remark. It is shown in Exercise A.31) that 1/2 cannot be replaced by A/2) — S,
with S > 0, in the above hypothesis.
The above results may be stated on the intervals [0, f]; by using an increasing
homeomorphism from [0, oo] to [0, t], we get
A.16) Corollary. If L is a local martingale such that either exp(^L) is a sub-
submartingale or E [exp (\{L, L),)] < oo for every t, then К (L) is a martingale.
Let us close this section with a few comments about the above two criterions.
Their main difference is that in Kazamaki's criterion, one has to assume that
exp(jL) is a submartingale, whereas it is part of the conclusion in Novikov's
criterion. Another difference is that Novikov's criterion is "two-sided"; it works
for L if and only if it works for —L, whereas Kazamaki's criterion may work for
L without working for —L. Finally we stress the fact that Kazamaki's criterion
is not a necessary condition for <C (L) to be a martingale as will be seen from
examples given in Exercise B.10) of Chap. IX.
# A.17) Exercise. Suppose that Q — D, ¦ P on .j^for a positive continuous martin-
martingale D. Prove the following improvement of Proposition A.2): the r.v. Y = inf, D,
is > 0 ?>-a.s. If moreover P and Q are mutually singular on .K^ then, under Q,
Y is uniformly distributed on [0, 1].
A.18) Exercise. Let />, / = 1, 2, 3 be three probability measures such that any
two of them form a Girsanov pair and call Dj = rC (l'\ the martingale such that
Pi = D) ¦ Pj. There is а Л-martingale M such that D\ = К (gp^{M)\. Prove
that L] = M + L].

334 Chapter VIII. Girsanov's Theorem and First Applications
A.19) Exercise. Call .//j(P) the space of cont. loc. mart, with respect to P. Let
(P, Q) be a Girsanov pair and Г a map from ../6(P) into .///(g).
1°) If (Г(М), N) = (M, N) for every M e . /t(P) and N e ..//4Q) prove
that Г = GQP.
2") If {Г(М), F(N)) = (Л/, N) for every M,N e. /?(P) there exists a map
7 from . /6{P) into itself such that {/(A/), J(N)) = (A/, N) and Г = GQP о J.
A.20) Exercise. If (P. 2) is a Girsanov pair with density D and if M is a P-
martingale, then MD~l is a Q-martingale. Express it as a Girsanov transform.
In relation with the above exercise, observe that the map M -+ MD'X does not
leave brackets invariant and does not commute with stochastic integration.
A.21) Exercise. Г) Let В be the standard linear BM and for a > 0 and b > 0
set ст„ b = inf {f : B, + bt = a]. Use Girsanov's theorem to prove that the density
of сто b is equal to аBл73)~'/2ехр (-(а -btJ/2t). This was already found by
other means in Exercise C.28) of Chap. Ill; compare the two proofs.
2°) Prove Novikov's criterion directly from the DDS theorem and the above
result.
A.22) Exercise. If for some s > 0, E [exp ((j + s) (M, A/)()] < oo for every /,
prove, using only Holder's inequality and elementary computations, that К (М) is
a martingale.
A.23) Exercise. Let В be the standard linear BM. For any stopping time T such
that E [exp (^ 7")] < oo, prove that
l \1
= 1.
A.24) Exercise. Г) Let В be the standard linear BM and prove that
r = inf{/ : B2= 1 -t]
is a stopping time such that P[0 < T < 1] = 1.
2°) Set Hs = -2BS ¦ 1(г>.о/A - sJ and prove that for every /,
Jo
Hfds < oo a.s.
3°) If M, =/0' HsdBs, compute M, - \{M.M), +A -/ л ТГ2ВТл,.
4°) Prove that ?[^(Л/),] < 1 and hence that X{M),, t e [0, I], is not a
martingale.
# A.25) Exercise. Let (.5f) be a filtration such that every (.3^)-martingale is con-
continuous (see Sect. 3 Chap. V). If Hn is a sequence of predictable processes
converging a.s. to a process H and such that \Hn\ < К where A" is a locally
bounded predictable process, prove that, for every / and for every continuous
(.>^)-semimartingale X,

§1. Girsanov's Theorem 335
P- lim f HndX = f H dX
"-*°°Уо Jo
[Hint: Use the probability measure Q - Р(-ПГ)/Р(Г) where Г is a suitable
set on which the processes Hn are uniformly bounded.]
A.26) Exercise. (Continuation of Exercise E.15) of Chap. IV). Prove that N
has the (.5^"K)-PRP. As a result every (.J^^-local martingale is continuous.
[Hint: Start with a bounded h and change the law in order that Y become a
BM.]
A.27) Exercise. Г) Let (P, Q) be a Girsanov pair relative to a filtration ()
Prove that if M is a f-cont. loc. mart, which has the (.3^-PRP, then GQP(M)
has also the (.>^)-PRP. It is shown in Exercise C.12) Chap. IX that this does not
extend to the purity property.
2°) Let В = (В1, В2) be a BM2 and set
X, =B} + [ B2ds.
Г, = Bj + f B2
Jo
Prove that B1 is not adapted to {.'St x).
[Hint: Use Q = К (-/0 B2dB\) ¦ P to prove that there is a BM1 which has
the X
A.28) Exercise. In the notation of this section, assume that Q <g; P on .^ and
that D is continuous. \fdQ/dP is in L2(P) prove that (in the notation of Exercise
D.13) of Chap. IV) any semimartingale of.b^(P) belongs to </\{Q).
A.29) Exercise. Prove that if (A/, A/)<» = со, then <6 (A/), converges a.s. to 0 as
t tends to +oo, hence cannot be uniformly integrable.
A.30) Exercise. Let В be the standard linear BM, T\ = inf [t : B, = I}, and set
t
т' = 7Г7лГ' iff<L T' = Tl lf'-L
Prove that M, — BTi is a continuous martingale for which Kazamaki's criterion
applies and Novikov's does not.
[Hint: Prove that K(-M) is not a martingale and observe that Novikov's
criterion applies to M if and only if it applies to —M.]
* A.31) Exercise. Retain the situation and notation of Exercises C.14) in Chap. II
and C.28) in Chap. Ill (see also Exercise A.21) above).
1°) Prove that
2°) Derive therefrom that, for any S > 0, there exists a continuous martingale
M such that ?[exp(j - S) {M, M)^] < +oo and К (М) is not a uniformly
integrable martingale.

336 Chapter VIII. Girsanov's Theorem and First Applications
* A.32) Exercise. Let M e BMO and Mo = 0; using Exercise A.40) in Chap. IV
prove that for any stopping time T,
E [X(M)xK(M)rl I-Щ > exp (-- ||W|||MO,J .
Prove that consequently X (M) is a uniformly integrable martingale.
* A.33) Exercise. For a continuous local martingale M vanishing at 0 and a real
number a, we set
G" = ех
g(ot) = sup{E[G"]; T stopping time}.
l°)Fora < p < 1, prove that g(fi) < g(a){i-^^-a), and that for 1 < a <
2°) If а ф 0 and T, = inf{s : (Af, M)s > /), then K(aM) is a uniformly
integrable martingale if and only if
lim E[K (аМ)ъ
/ —*¦ oc
=0.
3°) If а ф 1 and g(a) < oo, then K(aM) and K(M) are uniformly integrable
martingales.
4°) If M € BMO, then g(a) < oo for some аф\.
* A.34) Exercise. Let p, be the modulus of the planar BM started at a / 0. Prove
that L, = log(/0,/|a|) is a local martingale for which Kazamaki's criterion applies
(on the interval [0, t ]) and Novikov's criterion does not.
* A.35) Exercise. Assume that the filtration (.3^) is such that all (.^-martingales
are continuous. Let X and Y be two continuous semimartingales. Prove that their
martingale parts are equal on any set Г е .7^ on which X — Y is of finite variation.
(This is actually true in full generality but cannot be proved with the methods of
this book).
[Hint: Use Q = P{- П Г)/Р(Г).]
* A.36) Exercise. Let P be the Wiener measure on Q - C([0, 1], Ж), .V, =
o(w{s), s < t) and b be a bounded predictable process. We set
B = exp|/ b(s,co)dco(s)-- I b2(s,a>)ds\ ¦ P
[Jo 2 Уо )
and 0(?w), = a)(t) - /J fe(s, a))ds.
Prove that if (Af,,r < 1) is a (.ЯГ, f)-martingale then (Af, о в, t < 1) is a
(•Я", O)-martingale. For instance if/г is a function of class C21 such that |0 +
|f = 0 then /г F(co),,t) is a (.J*T, B)-martingale.
[Hint: If .Я" = <? (в(ш),,х < t), one can use the representation theorem to
prove that every LPi, 6J-martingale is a (.5Г, 6)-martingale.]

§1. Girsanov's Theorem 337
* A.37) Exercise. Let (f2,.Wt, P) be a filtered space such that every (.71, P)-
martingale is continuous. Let X be a (.^-adapted continuous process and л
the set of probability measures Q such that
i) Q\.% ~~ Pbr for every f
ii) X is a (.5^, <2)-local martingale.
Г) Show that if л- is non-empty, then X is a f-semimartingale with canonical
decomposition X = M + A such that dA <?d{M, M) a.s.
2a) Conversely if under P the condition of 1°) is satisfied we call h a good
version of dA/d{M1 M) (see Sect. 3 Chap. V). Assume that one can define a
probability measure Q on .Wx by setting
q\^=k(J hsdM5
prove then that Q is in n. ^
3°) If the conditions in 2°) are satisfied, describe the set n. Prove that Q is
the only element of n if and only if M has the PRP under P.
* A.38) Exercise. Let P and Q be two probability measures on a space with a
filtration (.7,) which is right-continuous and complete for both P and Q. Let
D, be the martingale density of Q with respect to 2~X(P + Q). Remark that
0 < D, < 2, P + Q-a.s. and set T = inf {/ : D, = 2).
1°) Prove that P[T — oo] = 1 and that the Lebesgue decomposition of Q
with respect to P on .>^~may be written
Q(B)= I Z,dP + Q(B<l(T </))
where Z, = D,/B- D,) on {/ < Г), Z, = 0 on [t > T]. Prove that Z is a positive
P-supermartingale and if G", Z') is another pair with the same properties, then
T' — T Q-a.s. and Z' = Z up to f-equivalence.
2°) Assume that D is continuous and prove Girsanov theorem for P and Q
and the filtration (.>W).
A.39) Exercise. Let Я be a predictable process with respect to the Brownian
filtration such that 0<c<#<C<oofor two constants с and C. For any /
in Ц0С(Ж+, ds), prove that
exp((c2/2)j f(sJds\ < e\ovp([ f(s)HsdB.X\
< exp ((C2/2) f f(sJds\.
[Hint: Apply Novikov's criterion.]

338 Chapter VIII. Girsanov's Theorem and First Applications
* A.40) Exercise (Another Novikov's type criterion). Let В be a (.>^)-BM and
H an adapted process such that
E[exp{aH?)]<c,
for every s < I and two constants a and с > 0.
Prove that for r < s < t and s — r sufficiently small,
E[(K {H ¦ B)s /K {H ¦ B)r) \ Щ = \,
and conclude that E [<5 (H ¦ B),] = 1. This applies in particular if Я is a Gaussian
process.
[Hint: Use a truncation of H and pass to the limit using Novikov's criterion
and Jensen's inequality.]
A.41) Exercise (A converse to Theorem A.12)). Let P be a probability mea-
measure on W such that, in the notation of this section, the law of X under Q is the
same as the law of X under P, for any Q equivalent to P. Prove that P is the
law of a Gaussian martingale.
[Hint: For any positive functional F,
EP[F({X. X))(dQ/dP)] = EP[F((X. X})] .]
§2. Application of Girsanov's Theorem
to the Study of Wiener Space
This section is a collection of results on Wiener space which may seem to be
loosely related to one another but are actually linked by the use of Girsanov's
transformation and the ubiquitous part played by the Cameron-Martin space (re-
(reproducing kernel Hilbert space of Exercise C.12) in Chap. I) and the so-called
action functional of BM.
We restrict the time interval to [0, T] for a positive real T and we will consider
the Wiener space W of Revalued continuous functions on [0, T] vanishing at 0.
We endow W with the topology of uniform convergence; the corresponding norm
is denoted by || H^. The Wiener measure will be denoted by W and the coordinate
mappings by /3,, 0 < / < T. As W is a vector space, we can perform translations
in W; for h e W, we call xh the map defined by
Let W/, be the image of W under rh. By definition, for any finite set of reals f, < T
and Borel sets Л, с M.d, we have
Wh If] {со : j3ti(co) eAi}\ = wlf\{a>: /3,,(co) + h(t,) e Л,} j

§2. Application of Girsanov's Theorem to the Study of Wiener Space 339
thus a probability measure Q is equal to Wf, if and only if, under Q, /3 = В + h
where В is a standard BM''.
We are going to investigate the conditions under which W/, is equivalent to
W. For this purpose, we need the following
B.1) Definition. The space H of functions h defined on [0, T] with values in Rd,
such that each component h; is absolutely continuous, /г,@) = 0, and
\h\s)\2ds = Y h\(sJds
~\ JO
1=1
is called the Cameron-Martin space.
For d = 1, this space was introduced in Exercise A.12) of Chap. I (see also
Exercise C.12) in the same chapter). The space Я is a Hilbert space for the scalar
product
* т
h'i(s)g-(s)ds,
= 51 [
;J0
and we will denote by \\h\\H the Hilbert norm (h,h)l/2. It is easy to prove, by
taking linear approximations, that H is dense in W for the topology of uniform
convergence.
If h is in H, the martingale
where the stochastic integrals are taken under W, satisfies Novikov's criterion
A.15) and therefore K(h' ¦ /3) is a martingale. This can also be derived directly
from the fact that W ¦ /3 is a Gaussian martingale, and has actually been widely
used in Chap. V.
B.2) Theorem. The probability measure Wf, is equivalent to W if and only ifh €
H and then Wh = X(h' ¦ fi)T ¦ W.
This may also be stated: W is quasi-invariant with respect to H and the
Radon-Nikodym derivative is equal to К (W ¦ fi)j.
Proof. If W/, is equivalent to W, then by the results in Sect. 3 Chap. V and the
last section, there is a continuous martingale M such that W/, — К (М)т • W;
by Girsanov's theorem, we have /3' = B' + (/3', M) where В is a UVBrownian
motion. But by definition of Wh, we also have /3 = В + h where В is a W/,-BM.
Under Wh, the function h is therefore a deterministic semimartingale; by Exercise
A.38) (see also Exercise B.19)) Chap. IV, its variation is bounded. Thus, under
Wh, we have two decompositions of the semimartingale /3 in its own filtration; by
the uniqueness of such decompositions, it follows that /?,(/) = {/3\ M), a.s.
Now by the results in Sect. 3 of Chap. V, we know that there is a predictable
process ф such that /or \<j)s\2ds < oo W-a.s. and M, = ?,¦ /0' <$>[д$[. Consequently,

338 Chapter VIII. Girsanov's Theorem and First Applications
* A.40) Exercise (Another Novikov's type criterion). Let В be a (,5f)-BM and
H an adapted process such that
E[exp(aH?)]<c,
for every s < t and two constants a and с > 0.
Prove that for r < s < t and s — r sufficiently small,
E[(# (H ¦ B)s /& {H ¦ B)r) [&] = h
and conclude that E [% (H ¦ B),] — 1. This applies in particular if Я is a Gaussian
process.
[Hint: Use a truncation of H and pass to the limit using Novikov's criterion
and Jensen's inequality.]
A.41) Exercise (A converse to Theorem A.12)). Let P be a probability mea-
measure on W such that, in the notation of this section, the law of X under Q is the
same as the law of X under P, for any Q equivalent to P. Prove that P is the
law of a Gaussian martingale.
[Hint: For any positive functional F,
EP[F({X, X)){dQ/dP)] = EP[F{{X, X))] .]
§2. Application of Girsanov's Theorem
to the Study of Wiener Space
This section is a collection of results on Wiener space which may seem to be
loosely related to one another but are actually linked by the use of Girsanov's
transformation and the ubiquitous part played by the Cameron-Martin space (re-
(reproducing kernel Hilbert space of Exercise C.12) in Chap. I) and the so-called
action functional of BM.
We restrict the time interval to [0, T] for a positive real T and we will consider
the Wiener space W of Kd-valued continuous functions on [0, T] vanishing at 0.
We endow W with the topology of uniform convergence; the corresponding norm
is denoted by || ||oo. The Wiener measure will be denoted by W and the coordinate
mappings by p,, 0 < t < T. As W is a vector space, we can perform translations
in W; for h s W, we call т> the map defined by
P, (rh(co)) = p,(co) + h(t).
Let Wh be the image of W under rA. By definition, for any finite set of reals tt <T
and Borel sets А, С Rd, we have
со : fl» + *(*,) e A
,}J ;

§2. Application of Girsanov's Theorem to the Study of Wiener Space 339
thus a probability measure Q is equal to Wh if and only if, under Q, P = В + h
where В is a standard BM''.
We are going to investigate the conditions under which Wh is equivalent to
IV. For this purpose, we need the following
B.1) Definition. The space H of functions h defined on [О, T] with values in Rd,
such that each component Л, is absolutely continuous, A,-@) = 0, and
I
T d
\h'(s)\2ds = V / h-isfds < +00
is called the Cameron-Martin space.
For d = 1, this space was introduced in Exercise A.12) of Chap. I (see also
Exercise C.12) in the same chapter). The space Я is a Hilbert space for the scalar
product
(h,g) = T h'i(s)g',(s)ds,
J
and we will denote by \\h\\u the Hilbert norm (h, h}1^2. It is easy to prove, by
taking linear approximations, that H is dense in W for the topology of uniform
convergence.
If h is in H, the martingale
where the stochastic integrals are taken under W, satisfies Novikov's criterion
A.15) and therefore % (h' ¦ fi) is a martingale. This can also be derived directly
from the fact that h' ¦ fi is a Gaussian martingale, and has actually been widely
used in Chap. V.
B.2) Theorem. The probability measure Wh is equivalent to W if and only ifh e
H and then Wh = <f(h' ¦ fi)T ¦ W.
This may also be stated: W is quasi-invariant with respect to H and the
Radon-Nikodym derivative is equal to %(h' ¦ P)T.
Proof. If Wh is equivalent to W, then by the results in Sect. 3 Chap. V and the
last section, there is a continuous martingale M such that Wh = %{M)T ¦ W;
by Girsanov's theorem, we have /5' = B' + {fi', M) where В is a WVBrownian
motion. But by definition of Wh, we also have /3 — В + h where В is a Wh-BM.
Under Wh, the function h is therefore a deterministic semimartingale; by Exercise
A.38) (see also Exercise B.19)) Chap. IV, its variation is bounded. Thus, under
Wh, we have two decompositions of the semimartingale fi in its own filtration; by
the uniqueness of such decompositions, it follows that A,@ = (fi', M), a.s.
Now by the results in Sect. 3 of Chap. V, we know that there is a predictable
process ф such that /0 \<t>s\2ds < oo W-a.s. and M, = J2i /0' Ф№Р'3- Consequently,

340 Chapter VIII. Girsanov's Theorem and First Applications
we get hi(t) = f'04>'sds. This proves on the one hand that h e H, on the other
hand that ф' can be taken equal to the deterministic function h\ whence the last
equality in the statement follows.
Conversely, if h e H, the measure Q = % (ti ¦ P)T ¦ W is equivalent to W
and under Q we have
F' = Bj + f h'i(s)ds = Bi +hi
Jo
where В is a BM, which shows that Q = W/,. a
Remark. Whatever h may be, the process p is, under Wh, a Gaussian process
and therefore, by a general result, Wh and W are either equivalent or mutually
singular. Thus if h ? H then W and Wh are mutually singular.
Let us now recall (see Sect. 1 Chap. XIII) that the a -algebra .Щ generated by
/S is the Borel a -algebra of W and that the support of a measure is the smallest
closed subset which carries this measure.
B.3) Corollary. The support of the measure W is the whole space W.
Proof. The support contains at least one point; by the previous results, it contains
all the translates of that point by elements of H. One concludes from the density
of H in W. Q
We next supplement the representation Theorem C.4) of Chap. V, which we
just used in the proof of Theorem B.2). Let (J^) be the W-complete filtration
generated by /S; if X is in L2 (.Щ, W), then
X = E[X]
Jo
for а ф e L2y>(PT). On the other hand, X is a.s. equal to a function, say F, of the
path w e W such that E [F(aJ] < oo. Our aim is to compute ф as a function
of F.
If ir e W, ft» + ir is also in W and F(w + ф) makes perfect sense. We will
also sometimes write F(/3) instead of F(u>). We will henceforth assume that F
enjoys the following properties:
i) there exists a constant К such that
for every ф е W;
ii) there exists a kernel F' from Q to [0, T] such that for every ф е Н
fT
lim e~l (F(P + еф) - F@))= I F'(p,dt№(t), for a.e. p.
«->0 Jq

§2. Application of Girsanov's Theorem to the Study of Wiener Space 341
It is worth recording that condition i) implies the integrability condition
E[F(fiJ] < oo. Moreover, if F is a differentiable function with bounded deriva-
derivative on the Banach space W, then, by the mean value theorem, condition i) is in
force and since a continuous linear form on W is a bounded measure on [О, Т],
ii) is also satisfied.
Under conditions i) and ii), we have Clark's formula:
B.4) Theorem. The process ф is the predictable projection of F'(/8, ]t, T]).
Proof. Let и be a bounded (.^-predictable process and set
f ~ Г
, — I usds, fi=\ usd(ls;
Jo Jo
for each со, the map t -*¦ i/rt (an) belongs to W.
For e > 0, fj'(e\lf),, t < T, is, by Novikov's criterion, a uniformly integrable
martingale and if Q — %(ет]/)т ¦ W, then, under Q, the process /5 - sij/ is a BM.
As a result E[F(fi)] = /w F(/5 - s^)dQ, where, we recall, E is the expectation
with respect to W and consequently
E[F@)] = E
This may be rewritten
E [(Ftf - sf) - F(P)) {X{.ef)T ~ 1)] + E[F(p - sf) - F{p)]
Let us divide by e and let e tend to zero. By condition i), the modulus of the first
part is majorized by KT\\u\\E [\%(e\Js)T - l|] and this converges to zero as e goes
to zero; indeed, '6{e^r)T converges to 1 pointwise and &(е\]/)т < ехр(е|т/гг|).
Moreover we will show at the end of the proof that, for a sufficiently small,
E [exp^Vj)] < oo. Thus, the necessary domination condition obtains and one
may pass to the limit in the expectation. In the second term, conditions i) and ii)
allow us to use the dominated convergence theorem. Finally
)T- \)= / %
Jo
and %(e-*jr)u converges to и in L2(dW <8) ds). Indeed, by Doob's inequality,
E Lp ((# (ef)s - lJ и2)] < Ци\\2Е [(ЙГ(е^)г - IJ]
and this last term converges to zero as s tends to zero because the integrand is dom-
dominated by (exp (a^r) + l) for e < a. By the L2-isometry property of stochastic
integrals, it follows that e~' (%(е~ф)т - l) converges in L2(W) to f0T usdfis.

342 Chapter VIII. Girsanov's Theorem and First Applications
The upshot of this convergence result is that
usdps] =E\fTF'(p,
hence, using integration by parts,
-T -[ г fT rT -| г ГТ -I
usF'(p,]s,T])ds \.
0 J U0 ./0 J UO J
The result now follows from the definition of a predictable projection.
It remains to prove the integrability property we have used twice above. But,
by the DDS Theorem, there is a BM у such that i/r = y^ ^ and since (ф, ф)т < с
for some constant c, we get
E [exp (a (V4J)l < E [exp (a (yc*J)l < °o,
for a sufficiently small. D
Remark. If one wants to avoid the use of predictable projections, one can state
the above result by saying that ф is equal to the projection in the Hilbert space
L2(dW <8) dt) of the process F'(/S, ]•, T]) on the subspace of equivalence classes
of predictable processes (see Exercise E.18) Chap. IV).
The remainder of this section will be devoted to asymptotic results on Brownian
motion.
B.5) Definition. The functional It on W defined by
1 Гт 1
1т(Ф) = г/ 4>\sJds = -m2H if*e//,
I Jo 2-
= +oo otherwise,
is called the action functional of Brownian motion.
We will often drop T from the notation. We begin with a few remarks about
h.
B.6) Lemma. A real-valued function ф on [0, T] is absolutely continuous with
derivative </>' in L2([0, T]) if and only if
М(ф) = sup]T @(r,-+!) - (/>(f,)J /(f, + i - /,¦),
where the supremum is taken on all finite subdivisions of[0, T], is finite and in that
case
П
b'(sJds = М(ф).
f
Jo

§2. Application of Girsanov's Theorem to the Study of Wiener Space 343
Proof. If ф' е L2, a simple application of Cauchy-Schwarz inequality proves that
fT
М(ф) < I tfisfds.
Jo
Suppose conversely that М(ф) < oo; for disjoint intervals (a,, ft) of [0, T],
Cauchy-Schwarz inequality again gives us
< М(фI'2
which proves that ф is absolutely continuous. If (a") is the и-th dyadic par-
partition of [0, T], we know, by an application of Martingale theory (Exercise
B.13) Chap. II) that the derivative ф' of ф is the limit a.e. of the functions
irn(s) = B"/Т)(ф(а^1) -</>«)) l[a«,a»+1[(j). By Fatou's lemma, it follows
that
I tf{sJds < lim / {f"{s)) ds < М(ф).
J0 n Jo
B.7) Proposition, i) The functional ф —> 1т(Ф) is lower semi-continuous on W.
ii) For any X > 0, the set Kk = {</> : 1т(Ф) < А.} й compact in W.
Froo/ The result can be proved componentwise so it is enough to prove it for
d — 1. Moreover, i) is an easy consequence of ii). To prove ii), we pick a sequence
{</>„} converging to ф in W and such that 1т(фп) < ^ for every и. It is easy to see
that М(ф) < supn М(ф„) < 2X, which, by the lemma, proves our claim. Q
Our first asymptotic result is known as the theorem of large deviations for
Brownian motion. We will need the following definition.
B.8) Definition. The Cramer transform of a set А с W is the number Лт(А)
(< oo) defined by
In what follows, T will be fixed and so will be dropped from the notation.
We will be interested in the probability W[sfi e A], as e goes to zero. Clearly,
if A is closed and if the function 0 does not belong to A, it will go to zero and
we shall determine the speed of this convergence. We will need the following
lemmas known as the Ventcell-Freidlin estimates where В(ф, S) denotes the open
ball centered in ф and with radius S in W.
B.9) Lemma. //<? e H.for any S > 0,
lime2logW[efi e В(ф,8)] > -/(</>).

344 Chapter VIII. Girsanov's Theorem and First Applications
Proof. With y, = p, - e *<j>(t), we have
W[e0 е В(ф, 8)] =
and, by Theorem B.2), this is equal to
e S@,
f exp\-- [ ф'ШР,-^ [ 4>'isJds\
Jmo.se-1) [ s Jo 2e Jo J
= ехр(-в~21(ф)) f expf-i/" tf(s)d
JBiO.Se-1) V s JO
dW
Now on the one hand W[B(Q, Ss~1)] > 3/4 for e sufficiently small; on the other
hand, TchebichefFs inequality implies that
w
which implies
W
As a result
\J ф'(*Ш>
1 i
2^2/@) <
J ~ 8/@)
-е-1 /
1
> 2 exP I ~?
for e sufficiently small. The lemma follows immediately. D
B.10) Lemma. For any S > 0,
Tim e2 log W [p (efi, Kx) > 8] <-X
?->0
where p is the distance in W.
Proof. Set a = T/n and let 0 = t0 < ^ <...<<„= Г be the subdivision of
[0, T] such that r*+i - ^ = a. Let Lf be the function which componentwise is
affine between t^ and ^+i and equal to ?/? at these times. We have
W\p(ep,Le)> 8] < У^ИЧ max |e0,-Lf|>S
max Ie0, — Le.\ > 8\
0<t<a J
< nW
= nW
max
0<r<a
fit Pa
a
max 10,1 > Be)"'S

§2. Application of Girsanov's Theorem to the Study of Wiener Space 345
because I max \pt\ > УеуЧ] Э I max Ifl, - LBa\ > Ss~l \, and by the expo-
[0<t<a J [0<t<a a^"l~ J
nential inequality (Proposition A.8) Chap. II), this is still less than
2ndexp(-n82/8e2d T).
If n is sufficiently large, we thus get
Пт e2 log W \p(eB, Le) > 8] < -X.
Let us fix such an и; then because
{p(efi, Kx) >S}C {I(LS) >X}U {p(eB, V) > 8},
it is enough, to complete the proof, to prove that
Пт s2 log IV [l(L?) > X] < -X.
But
where the щ are independent standard Gaussian variables. For every В > О,
E exp (-^ r\] \ \= Cp < oo, and therefore, by Markov inequality,
W[I(Le) > X] = P
exp
2 Y
whence lim e2 log W[I(LS) > X] < ~(l - B)X and, as В is arbitrary, the proof of
the lemma is complete. D
We can now state
B.11) Theorem (Large deviations). For a Borel set А с W,
-Л(А) < lim e2 log W[sB e Д] < Пт e2 log W[sB e A] < -Л(А).
Proof. If A is open, for any ф е Н П A, and 8 sufficiently small,
W[sBeA]> W[p(ep,4>) < 8],
and so, by Lemma B.9),
lim e2 log W[ep e A] > - inf[/W>), tf> e A} = -
Let now A be closed; we may suppose Л(А) > О as the result is otherwise
obvious. Assume first that Л(А) < oo. For Л(А) > у > 0, the sets A and

346 Chapter VIII. Girsanov's Theorem and First Applications
= ЛГл(А)-х are disjoint. Since К is compact, there is a number S > 0 such that
K) >& for every ф е A; by Lemma B.10), we consequently get
ilm e2 log W[ep e А] < -Л(А) + у
and, since у is arbitrary, the proof is complete. If Л(А) = +оо, the same reasoning
applies with К = KM for arbitrary large M. a
We will now apply the preceding theorem to the proof of a beautiful result
known as Strassen's functional law of the iterated logarithm. In what follows, we
set g{n) = Bnlog2n)-l/2, n > 2, and Xn{t) = g{n)pnl, 0 < t < T. For every
ca, we have thus defined a sequence of points in W, the asymptotic behavior of
which is settled by the following theorem. The unit ball of the Hilbert space H
which is equal to the set K\/2 will be denoted by U.
B.12) Theorem. For W-almost every со, the sequence {Х„(-, со)} is relatively com-
compact in W and the set of its limit points is the set U.
Proof We first prove the relative compactness. For S > 0, let Ks be the closed set
of points w in W such that p(w, U) < S. Using the semi-continuity of IT, it may
be seen that Л ((Ks) ) > 1/2, and thus, for fixed S, we may choose у such that
1 < у < 2Л ({Ks)c). Pick X > 1 and set n(m) = [Xm]; by the scaling property
ofBM,
W [Xnim) ?KS]
thus by Theorem B.11), for m sufficiently large,
W [Xn(m) i Ks] < exp (-y log2 n(m)) < ((/и - 1) logA)^ .
It follows that W-a.s., Xn(m) belongs to Ks for m sufficiently large. As this is true
for every S, it follows that the sequence {Х„(т)} is a.s. relatively compact and that
its limit points are in U. Clearly, there is a set В of full W-measure such that for
ш e B, all the sequences Xn(m)(a) where X ranges through a sequence S = {Xk}
decreasing to 1, are relatively compact and have their limit points in U. We will
prove that the same is true for the whole sequence {Xn{co)}. This will involve no
probability theory and we will drop the со which is fixed throughout the proof.
Let M = sup/,^ ||й||оо and set b(t) = «Ji; observe that
sup\h(t)-h(s)\ <b(\t-s\)
thanks to the Cauchy-Schwarz inequality.
Fix X e S; for any integer n, there is an m such that n(m) < n < n(m + 1) and
we will write for short JV = n(m + 1). We want to show that for a fixed S > 0,
we have p(Xn, U) < 8 for и sufficiently large which we will do by comparing
Х„ and XN. Indeed
p(Xn, U) < p(XN, U) + p(Xn, XN).

§2. Application of Girsanov's Theorem to the Study of Wiener Space 347
Pick e such that 0 < e < 6/5. The number X being fixed, for n sufficiently large,
we have p(XN, U) < e. On the other hand
p(Xn,XN) = sup
t<T
g{n)
8(N)
g(n)
g(n)
g(N)
- 1
- 1
t<T
XN(~t)-XN(t)
Again, for fixed X, we may find n sufficiently large to have
~l and
For such an и, we therefore have
p(Xn,U)<4s-
- \)(M
By taking X sufficiently close to 1, this is less than S and we are done. Thus,
we have proved the relative compactness of the sequence {Xn} and that the limit
points are in U. It remains to prove that all the points of U are limit points of the
sequence {Xn}.
We first observe that there is a countable subset contained in {h e H; \\h \\H < 1}
which is dense in U for the distance p. Therefore, it is enough to prove that if
h e H and ||А||Я < 1,
\ lim p(Xn,h) = 0\ - 1.
W
To this end, we introduce, for every integer k, an operator Lk on W by
ifO< t < T/k,
if T/k<t < T.
For к > 2 and да > 1, we may write
p(Xk»,h) < sup 1X^@1+ sup
t<T/k t<T/k
+ \\Lk(Xkm-h)\\O0.
Since {Xn} is a.s. relatively compact, by Ascoli's theorem, we may for a.s. every
ft» choose a &(ft>) such that for к > к(со) the first four terms of this inequality are
less than any preassigned S > 0 for every m. It remains to prove that for a fixed k,
W
Г lim
|_т->оо
11^№--ЛI1о0 = 0 =1.

348 Chapter VIII. Girsanov's Theorem and First Applications
But, on the one hand, the processes LkiX^m) are independent as m varies; on the
other hand, by the invariance properties of BM,
< e
W[\\Lk{Xkn,-h)\\O0<e\=W\ sup
[_0<(<ГA-
where hit) = hit + T/k), 0 < / < T{\ - I/it). Since
/•7Ч1-1Д) fT
/ h\sJds < I h'isfds < 1,
Jo Jo
the large deviation theorem applied for open sets and on [О, ГA — \/k)] instead
of [0, T] shows that for m sufficiently large
W[\\Lk (X*. -A)ll«, 1 < e] > rap(-ylog2*m) = (mlogfc)^
for some у < 1. An application of Borel-Cantelli's Lemma ends the proof.
Remark. We have not used the classical law of the iterated logarithm of Chap. II
which in fact may be derived from the above result (see Exercise B.16)).
B.13) Exercise. 1°) In the setting of Theorem B.2), prove that for t <T,
dWh
dW
= exp h>iO/j; - W h'lMfids -\t \h'is)\2ds
It is noteworthy that this derivative may be written without stochastic integrals.
2°) Conversely, prove that W is the only measure on W which is quasi-
invariant with respect to H and which admits the above derivatives.
[Hint: Consider functions h with compact support in ]0, t[.]
B.14) Exercise. P) Prove that for the BMd, d > 2, absolutely continuous func-
functions are polar in the sense of Exercise A.20) of Chap. I (see also Exercise B.26)
in Chap. V).
2°) Prove that if В is a Brownian Bridge of dimension d > 2 between any
two points of W1, then for any x e Rd,
P[3t e]0, 1[: В, =*] = 0.
[Hint: Use the transformations of Exercise C.10) Chap. I.]
3°) If in 2°) we make d = 2 and Bo = Bx = 0, the index /(*, B) of x Ф 0 with
respect to the curve B,, 0 < t < 1, is well defined. Prove that Р[/(дг, В) = п] > 0
for every n 6 Z.
[Hint: Extend the result of Corollary B.3) to the Brownian Bridge and use it
together with the fact that x has the same index with respect to two curves which
are homotopic in R2\{;c}.]

§3. Functionals and Transformations of Diffusion Processes 349
B.15) Exercise. 1°) Recall from Exercise C.26) Chap. Ill that there is a.s. a
unique time a such that S\ — Ba. Prove that
S\ = у/2/л + I ф^В5
Jo
where ф5 is the predictable projection of 1(<T>J) (or its L2-projection on the pre-
predictable a -field).
[Hint: With the notation of Theorem B.4), prove that F'(/8, •) = ea.]
2°) Prove that there is a right-continuous version of P [a > t \ .5f] which is
indistinguishable from ф, and conclude that for Ф(х) — f°°g\(y)dy,
p,=2<p((St -S,)
[Hint: P[a>t\#\ = P[Ta<l- t]a=Si_Bi .]
3°) (Alternative method). Compute E [/(Si) | -Я"] where / is a positive Borel
function and deduce directly the formula of 1°).
B.16) Exercise. 1°) In the setting of Theorem B.12), let Ф be a real-valued
continuous function on W, and prove that
W \Шф (Xn(-)) = sup <P(h)\ = 1.
L " lei/ J
2°) Derive therefrom the classical law of the iterated logarithm.
B.17) Exercise. Let (?„) be a sequence of independent identically distributed real
random variables with mean 0 and variance 1 and set Sn = Y^," ?*• Define a process
S, by
S, = A - t + [t])S[t] + (t- [r])S[/]+i.
Prove that the sequence Xn(t) = g(n)Snt, 0 < t < T, has the same property as
that of Theorem B.12).
[Hint: Use the result in Exercise E.10) of Chap. VI.]
§3. Functional and Transformations of Diffusion Processes
In the study of diffusions and stochastic differential equations, Girsanov's theorem
is used in particular to change the drift coefficient. One reduces SDE's to simpler
ones by playing on the drift or, from another view-point, constructs new Markov
processes by the addition of a drift. This will be used in Sect. 1 Chap. IX.
In this section, we will make a first use of this idea towards another goal,
namely the computation of the laws of functionals of BM or other processes. We
will give a general principle and then proceed to examples.
The situation we study is that of Sect. 2 Chap. VII. A field a (resp. b) of
d x d symmetric matrices (resp. vectors in Rd) being given, we assume that for

350 Chapter VIII. Girsanov's Theorem and First Applications
each x € Rd there is a probability measure Px on Q = С(Ш+,Ша) such that
(Q,.^°, X,, Px) is a diffusion process in the sense of Definition B.1) of Chap.
VII with a — aa'. By Theorem B.7) Chap. VII, for each Px, there is a Brownian
motion В such that
(, = x + / a(Xs)dBs + f b(Xs)ds.
Jo Jo
We moreover assume that Px is, for each x, the unique solution to the martingale
problem n(x, aa', b).
Suppose now given a pair (/, F) of functions such that D, = exp(/(X,)—
/(Xo)-/J F(Xs)ds) is a (.^°, P,)-continuous martingale for every x. By Propo-
Proposition A.13) we can define a new probability p/ on ,J^ by p/ — DtPx on .^.
If Z is an ,^"-measurable function on J2, we will denote by Epx[Z \ X, = •] a
Borel function ф on RJ such that EX[Z | X,] =
C.1) Proposition. The term (i2,.iF^°, X,, Pi) is a Markov process. For each x,
and t > 0, the probability measures P, (x, dy) and P,{x, dy) are equivalent and,
for each x, the Radon-Nikodym derivative is given by
~~~ = exp (/()>) - f(x))EPx Гехр (- j' F(Xs)ds^ \ X, = y\.
Proof. The measurability of the map x —> p/ is obvious. Let g be a positive
Borel function and Y a J^"°-measurable r.v. Because D,+s = D, ¦ Ds о в,, we have,
with obvious notation,
E{[Yg(Xl+s)] = Ex[Yg(X,+s)Dt+s]
= Ex [YD,EXl [g(X,)Ds]] = E{ [yEXi [g(Xs)]\
which proves the first claim. The second follows from the identities
p/g(x) = Ex [D,g(X,)] = Ex Г#(Х,)ехр (f(Xt) - f(X0) - j F(Xs)dsX\
= Ex \g(X,) exp (/(X,) - f{x)) Ex Г exp (- j F{Xs)ds\ \ X, 11.
In the above Radon-Nikodym derivative, three terms intervene, the two semi-
semigroups and the conditional expectation of the functional exp(—/0' F(Xs)ds). This
can be put to use in several ways, in particular to compute the conditional expec-
expectation when the two semi-groups are known. This is where Girsanov's theorem
comes into play. Since D, = %{M), for some local martingale M (for each
Px\ then Girsanov's theorem permits to compute the infinitesimal generator of
P/, hence at least theoretically, the semi-group p/ itself. Conversely, the above
formula gives p/ when the conditional expectation is known.

§3. Functional and Transformations of Diffusion Processes 351
We now give a general method to find such pairs (/, F) and will afterwards
take advantage of it to compute the laws of some Brownian ftinctionals.
The extended generator L of X is equal on C2-functions to
L = -
where a = aal. We recall that if / e C2 then
Mf = /(*,) - f(X0) - f Lf(Xs)ds
Jo
is a continuous local martingale and we now show how to associate with / a
fiinction F satisfying the above hypothesis.
C.2) Definition. The operateur carre du champ Г is defined on С2 х C2 by
Г(/, g) = L(fg) - fLg - gLf.
C.3) Proposition. Iff,g€ C2, then, under each Px,
,= / nf,g)(Xs)ds.
Jo
Proof. Let us write A, ~ B, if A - В is a local martingale. Using the integration
by parts formula, straightforward computations yield
(VJ ~ f L(f2)(Xs)ds + (f Lf(Xs)dsj ~2f(X,)f Lf(Xs)ds
~ f L(f2)(Xs)ds - 2 f (f Lf){Xs)ds = f Г(/, f)(Xt)ds.
Jo Jo Jo
The proof is completed by polarization. D
As a consequence of this proposition, if / € C2, then
&(Mf), = explf(X,)-f(X0)-J'Lf(Xs)ds-l-jr(f,f)(,Xs)ds\
= (h(Xt)/h(Xo))ex.p(f (Lh(Xs)/h(Xs))ds\ ,
if h = exp(/). If this local martingale turns out to be a true martingale, then
we may define the probability measures p/ as described at the beginning of the
section, with F = Lf + \F(f, /). In this setting, we get
C.4) Proposition. IfL is the extended generator of X, the extended generator of
the Р?-process is equal on C2 to L + F(f, ¦).

352 Chapter VIII. Girsanov's Theorem and First Applications
Proof. If ф € C2, then ф(Х,) - ф(Х0) - /0' Ьф{Хх^$ is a /^-local martingale
and Girsanov's theorem implies that
ф(Х,)-ф(Х0)- f Ьф(Х^з-(Мг,Мф),
Jo
is a p/-local martingale. The proof is completed by means of Proposition C.3).
D
We proceed by applying the above discussion to particular cases. Let us sup-
suppose that / is a solution to Lf = 0, which, thinking of the special case of BM,
may be expressed by saying that / is "harmonic". Then Г(/, /) = L(f2) and
F = jL(/2). The extended generator of the /"/-process is equal on ф е С2 to
We see that the effect of the transformation is to change the drift of the process.
If the P-process is a BMrf and / is harmonic in the usual sense, then F =
| V/|2 and the generator is given by ~Аф + (У/, Уф). We will carry through some
computations for particular cases of harmonic functions. Let for instance S be a
vector in M.d; then f(x) — (8, x) is a harmonic function, and plainly К (М?) is a
martingale. We get a Markov process with generator
= -Лф+ E, Уф),
which is the Brownian motion with constant drift 8, namely B, +t8. Let us call Pf
instead of p/ the corresponding probability measures. By the above discussion
P* = exp Us, X, - Xo) - 1^-11 • Px on .3T°
and the semi-group Pts is given by
Pf{x, dy) = exp I (8, у - x) - ^ j • P,(x, dy),
where P, is the Brownian semi-group. Of course in this simple case, the semi-
semigroup Pf may be computed directly from Pt. Before proceeding to other examples,
we shall study the probability measure Pq .
We suppose that d > 2; since Phx is absolutely continuous with respect to Px on
¦^° and since the hitting time of a closed set is a (.5^"°)-stopping time, it follows
from the polarity of points for BM2 that the hitting times of points under Px are
also a.s. infinite. Thus, we may write a.s. X, — р,в, for all t > 0, where p, — \Xt\
and the process в takes its values in the unit sphere. We set .^й, = a (ps, s < t)
C.5) Lemma. For each t > 0, the r.v. в, is, under Po, independent of .
uniformly distributed on the unit sphere Sd~l.

§3. Functionals and Transformations of Diffusion Processes 353
Proof. Suppose d = 2 and let Z be .^oo-measurable and > 0 and G be a positive
Borel function on Sd~[. Because of the invariance of Pq, i.e. the Wiener measure,
by rotations, for every ф e [0, 2л],
and integrating with respect to ф,
J
^ J G
since the Lebesgue measure on S1 is invariant by multiplication by a given point
of S1, we get
Eo [ZG(e,)] = E0[Z] (J^
\.
For d > 2 there is a slight difficulty which comes from the fact that Sd~[ is only
a homogeneous space of the rotation group. The details are left to the reader (see
Exercise A.17) Chap. III). D
We henceforth call \xd the uniform distribution on Sd~l and if ф is a positive
Borel function on Rd, we set
Мф(р) = f 4>{pu)nd{du).
C.6) Corollary. On the o-algebra .#,„
\S\2t
with ф(х) = exp{(E, x)}.
Proof. Since ,*0, С ^°, we obviously have
Po5 = Eo Гехр ({8, X,) - l-~\ | .^,
on .
and it is easily checked, taking the lemma into account, that the conditional ex-
expectation is equal to exp ( — ^-4 Мф(р,). ?
We may now state
C.7) Theorem. Under P?, the process p, is a Markov process with respect to
{.??,). More precisely, there is a semi-group Q, such that for any positive Borel
function f onR+,
Es0 [f(p,+s) | Л?п\ = Qsf(pt).

354 Chapter VIII. Girsanov's Theorem and First Applications
Proof. Pick A in J&t; we may write, using the notation of Corollary C.6),
? / (pt+s) dP$ = exp (~~(t+ s)\ j f (a+.v) Мф (p,+s) dP0,
and by the Markov property under Po, this is equal to
- — (t+s)J ] EXlU{Ps)M<t>(Ps)}dPu
= exp (-~(t + s)j I Eo [Ex, [f(Ps)M<t>(ps)] | Щ dP0.
By the same reasoning as in Corollary C.6), this is further equal to
expl- —V +
with ir(x) — Ex [/(Pi)Af(/)(pj)]. Thus we finally have
/ f(pl+s)dP? = / exp (-—) (Mt(p,)/W(P,))dPi.
Ja Ja \ l )
This shows the first part of the statement and we now compute the semi-group
Q,
Plainly, because of *he geometrical invariance properties of BM, the function
ij/ depends only on |x| and consequently Мф = ф. Thus, we may write
Eo [f(Pi+s) I •*#»] = exp I —-—- J ф(р,)/Мф(р,)
= Ер, /(р.,)Мф(рх) exp I — j IМф{р,)
where Pa is the law of the modulus of BM started at x with |x| = a. We will see
in Chap. XI that this process is a Markov process whose transition semi-group has
a density pds (a, p). Thus
where Qs(a, dp) = jjj^ exp (-^) Pds (a, p)dp. Since /?f is the density of a
semi-group, it is readily checked that Q, is a semi-group, which ends the proof.
D
Remark. The process p, is no longer a Markov process under the probability
measure P% for x Ф 0.

§3. Functionals and Transformations of Diffusion Processes 355
We now turn to another example, still about Brownian motion, which follows
the same pattern (see also Exercise A.34)). Suppose that d — 2 and in complex
notation take f(z) = a log |г| with a > 0. The function / is harmonic outside the
polar set {0}. Moreover, for every t,
sup%(Mf)s <sup|Zs|a;
since the last r.v. is integrable, it follows that the local martingale <5(М?) is
actually a martingale so that our general scheme applies for Pa if а Ф 0. With the
notation used above, which is that of Sect. 2 in Chap. V,
_.
From Ito's formula, it easily follows that under Pa
1 /•' _
Pt=Po + P, + - Ps
*¦ Jo
where ft, = /0' p~l (XsdXs + YsdYs) is a linear BM. But we also know that
log p, =log/o0+ / p;ldps,
Jo
hence (P,\ogp), = f^p~lds. Thus, Girsanov's theorem implies that under p/,
the process /8, = /5, — a J^p~xds is a BM, and consequently p, = p0 + /8, +
^Y^ /0' p~xds. The equations satisfied by p under Pa and p/ are of the same
type. We will see in Sect. 1 Chap. XI, how to compute explicitly the density p\
of the semi-group of the solution to
Pi = Po + fit + -^r- / P^ds.
*¦ Jo
All this can be used to compute the law of в,, the "winding number" of Z, around
the origin. As f(z) depends only on \z\, the discussion leading to Proposition C.1)
may as well be applied to p as to Z with the same function /. As a result, we
may now compute the conditional Laplace transform of C, = /„' p~2ds. We recall
that /„ is the modified Bessel fiinction of index v.
C.8) Proposition. For every a and а ф 0,
Еа[ехр((а(в,-в0))\р,=р] = Еа\екр(~сЛ\р,=р\
- '-(?)/'¦(?)•
Proof. The first equality follows from Theorem B.12) in Chapter V and the second
from Proposition C.1) and the explicit formulas of Chap. XI, Sect. 1.

356 Chapter VIII. Girsanov's Theorem and First Applications
Remark. From this result, one may derive the asymptotic properties of в, proved
in Theorem D.1) Chap. X (see Exercise D.9) in that chapter).
Our next example falls equally in the general set-up of Proposition C.1).
Suppose given a function F for which we can find a C2-function / such that
then our general scheme may be applied to the computation of the conditional
expectation of exp ( —/0' F(Xs)ds j given X,.
Let us apply this to the linear BM with drift b, namely the process with
generator
Again, it is easily seen that Г(/, /) = f'2, so that if we use the semi-
semigroup (P, ) associated wi
exp (- /0' F(Xs)ds) with
group (P,) associated with /, we can compute the conditional expectation of
F(x) = l-f"(x) + b(x)f'(x) + \f'(xf.
By playing on b and /, one can thus get many explicit formulas. The best-known
example, given in Exercise C.15), is obtained for b(x) — kx and leads to the
Cameron-Martin formula which is proved independently in Chap. XI.
We proceed with some other examples in which we consider not only / but
its product vf by a constant v and call Pvx instead of Pvxf the corresponding
probability measures; we furthermore replace /' by ф in the notation. Moreover,
we take b(x) = аф(х) for a constant a and assume that ф satisfies the differential
equation
Eq. C.1) ф2 = -^-ф> + у
for some constant y. The function F is then given by
(v2 +2av) у v2
F(x) - V , ; - — ф'(х),
2 4«
the general formula of Proposition C.1) reads
Xi=yh WMexp
and the infinitesimal generator of the /"'-process is given by
Lvg = -g" + (a + vL>g',
as follows from Proposition C.4).
v ГФШи i •

§3. Functionals and Transformations of Diffusion Processes 357
By solving Eq. C.1) for ф, we find for which drifts and functions the above
discussion applies. This is done by setting ф = —h'/2ah and solving for h. Three
cases occur.
Case 1. у = 0. In that case, ф(х) = ^ j~g where A and В are constants. If, in
particular, we take a = 1/2, A = 1, В = 0, the generator V is then given by
2° ' \2 ) x
The /"-process is thus a Bessel process which will be studied in Chap. XI.
Case 2. y < 0. Then ф(х) = V\vl^tilZZ where m = ~2<*Ш In the
special case у = -1, A = 1, В = 0, we get as generator
Lvg(x) = -g"(x) + (a + v)cotBax)g'(x)
which is the generator of the so-called Legendre process.
Case 3. у > 0. Then ф(х) - y/Y^ZtH^Z where m = 2a Jy. For у = 1,
Л = 1, В — 0, the generator we get is
Lvg(x) = -g"(x) + (a + v)cofhBax)g'(x).
The corresponding processes are the so-called hyperbolic Bessel processes.
We proceed to other important transformations of diffusions or, more generally,
Markov processes.
Let h be "harmonic", that is, as already said, h is in the domain of the extended
infinitesimal generator and Lh — 0. Suppose further that h is strictly positive. If
we set / = log/г, our general scheme applies with F = 0 provided that h is
P, (x, -Hntegrable for every x and t. In that case one observes that h is invariant
by the semi-group P,, namely P,h — h for every t.
The semi-group obtained from P, by using this particular function / namely
log/г, will be denoted by Pth and the corresponding probability measures by P*.
Plainly, Р,иф = /?~'Р,(й(/>). The process X under the probability measures P,A is
called the h-process of the P,-process and is very important in some questions
which lie beyond the scope of this book. We will here content ourselves with the
following remark for which we suppose that the P,-process is a diffusion with
generator L.
C.9) Proposition. The extended infinitesimal generator of the h-process is equal
on the C2-function ф to

358 Chapter VIII. Girsanov's Theorem and First Applications
Proof. lf<peCl, then
Р,нф(х) - ф(х) - f Psh (h-'L(h4>)) ds
Jo
= /Г'(х) Р,Щ){х) - к{х)ф{х) - f Ps(L^))ds] = 0,
L Jo J
and one concludes by the methods of Sect. 2, Chap. VII. D
In the case of Brownian motion, the above formula becomes
1}ф = -40 + A~'(VA, V0>
which is again the generator of Brownian motion to which is added another kind
of drift. Actually, we see that the A-process is pulled in the direction where A VA
is large. This is illustrated by the example of BES3; we have already observed
and used the fact that it is the A-process of the BM killed at 0 for A(x) = x
(see Exercise A.15) in Chap. Ill, Sect. 3 in Chap. VI and Exercise C.17) in this
section).
Finally, we observe that in Proposition C.1) we used only the multiplicative
property of Dt. Therefore, given a positive Borel function g we may replace D,
by N, = exp ( — f^g(Xs)ds) which has the same multiplicative property. Thus,
we define a new semi-group P, and probability measures Px by
Again, X is a Markov process for the probability measures Plxg) and
Pt{s)(x, dy) = Ex |"exp (- j' g(Xs)ds\ \Xt=y~\ P,(x, dy).
This transformation may be interpreted as curtailment of the life-time of X or
killing of X and g appears as a killing rate. Evidence about this statement is also
given by the form of the extended infinitesimal generator of the new process which
is denoted by L(g).
C.10) Proposition (Feynman-Kac formula). //</> e C2,
L(*V = Ьф~ 8ф.
Proof. Let ф е C<g; then, Мф is a /^-martingale and the integration by parts
formula gives
И,ф{Х,) = ф{Х0)- f' ф(Х,)Ых8(Х5№ + f N^{X,)ds+ f NsdMf.
Jo Jo Jo
The last term is clearly a Px -martingale and integrating with respect to Px yields
Exg) [ф(Х,)] = ?<g) [ф(Х0I
Using Proposition B.2) in Chap. VII, the proof is easily completed. D

§3. Functionals and Transformations of Diffusion Processes 359
Let us finally observe that these transformations are related to one another.
Indeed, if g = Lf + |Г(/, /), we have, with the notation of Proposition C.1),
Ptl8\x, dy) = cxp ( - f(x))P,f(x, dy) exp (f{y)).
Thus, the semi-group P,is) appears as the /г-transform of the semi-group P/ with
h = exp(/).
C.11) Exercise. In the situation of Theorem C.7) but with x Ф 0 prove that the
bidimensional process (p,, /0' p~2ds) is a Markov process with respect to (.*?,)
under fj.
t C.12) Exercise (Time-inversion). If X is a process indexed by t > 0, we define
Xhy
X,=tXl/t, t > 0.
1°) With the notation of this section, prove that if P% is the law of X, then
f/ is the law of X. In other words, for the BM with constant drift, time-inversion
interchanges the drift and the starting point.
2°) Suppose that d = 2, x — 0 and 8^0, namely X is the complex BM with
drift <5 started at 0. Prove that X, = p, ехр(г'}/д1) where p, = \X,\, у is a linear
BM independent of .^oo and A, = /r°° p~2ds.
3°) If for r > 0 we set T = inf [t : p, = r), then XT is independent of .^j.
As a result, Xr and T are independent. Observe also that this holds equally for
5 = 0 (see also Exercise A.17) Chap. HI).
4°) For r — 1, prove that .XV follows the so-called von Mises distribution of
density C$ exp({<5,9)) with respect to the uniform distribution on Sl, where Cg is
a normalizing constant and (8, в) is the scalar product of 8 and в as vectors of R2.
C.13) Exercise. Replace Eq. C.1) by the equation ф2(х) = Рф'(х) + у where ft
is a constant independent of a, and carry the computations as far as possible.
* C.14) Exercise. Г) Let X be the standard rf-dimensional BM and p = \X\. Prove
that Proposition C.1) extends to
and prove that the infinitesimal generator of the transformed process is given on
Ф € C2, by
We call Px the corresponding probability measures.
2°) Prove that under Px, the process X satisfies the SDE
X,=x + B,+X I Xsds
Jo

360 Chapter VIII. Girsanov's Theorem and First Applications
where В is a rf-dimensional BM. Deduce therefrom that it can be written
ekt (x + B (A - e'2kt) /2A.)), where В is a standard BM<*, and find its semi-group.
[This kind of question will be solved in a general setting in the next chapter, but
this particular case may be solved by using the method of Sect. 3 Chap. IV].
The process X may be called the rf-dimensional OU process.
3°) For d = 1 and Л < 0 the process X is an OU process as defined in Exercise
A.13) Chap. III. Check that it can be made stationary by a suitable choice of the
initial measure.
4°) Prove that
Ex exp ( - — \ p]ds\ | p, = p\
Xt f\x\2 + p\, \ Пх\рк\ I. (\x\p\
= — exp I A - Xt coth Xt) L I / /„
sinhAr *\ 2t 7 \smhXtj/ \ t )
where v — (d/2) — 1. The reader will observe that for d = 2, this gives the law
of the stochastic area S, studied in Exercise B.19) Chap. V. For x — 0 or p — 0,
and k2 = \x\2 + p2, the right-hand side becomes
/ kt Y+1 /k2 \
. . . exp —A - Xt coth Xt)\.
\smhkt/ \2t J
5°) For d — 2, prove that
Eo [exp(iA.S,) \B,=z] = -r^— exp (-l-^-
sinhA? V 2r
C.15) Exercise. Prove that the extended generator of the semi-group of BES3
(Exercise A.15) Chap. Ill and Sect. 3 Chap. VI) is equal on C2(]0, oo[) to
ф — ]-ф"(х) + -ф'(х).
2 x
[Hint: Use the form of the generator of BM killed at 0 found in Exercise A.22)
Chap. VII.]
C.16) Exercise. If (P,) and (Pt) are in duality with respect to a measure ? (see
Sect. 4 Chap. VII), then the semi-group of the /г-process of X, and (P,) are
in duality with respect to the measure h%. This was implicitly used in the time
reversal results on BES3 proved in Sect. 4 of Chap. VII.
C.17) Exercise (Inverting Brownian motion in space). In i^\{0}, d > 3, put
0(X) = X/\X}2.
1°) If В is a BM^fl) with а ф 0, prove that ф(В) is a Markov process with
transition function
Q,(x, dy) = {2nt)-dl2\yr2d exp (-\x - y\2/2t\x\2\y\2) dy.

§3. Functionals and Transformations of Diffusion Processes 361
2°) Call (т,) the time-change associated with A, = /0' \Bs\~4ds (see Lemma
C.12) Chap. VI) and prove that Y, = 0(Br,), t < A», is a solution to the SDE
[ (YS/\YS\2)ds,
o
where j8 is a BM stopped at time A^ = inf{t : Y, = 0}.
3°) Prove that the infinitesimal generator of Y is equal on С2A^\{0}) to
and that Y is the /г-process of BMrf associated with h(x) = |jc|2~rf.
4°) Prove that under Pa, the law of A^ is given by the density
(r(v)B\a\2)v t^)~l exp(-Bt\a\2)-])
with v = (rf/2) - 1.
5°) Conditionally on Аж = и, the process (K,, t < и) is a Brownian Bridge
between a/|a|2 and 0 over the interval [0, и].
This exercise may be loosely interpreted by saying that the BMrf looks like a
Brownian Bridge on the Riemann sphere.
C.18) Exercise. Let У be a rf-dimensional r.v. of law к and set B^ — B, + tY
where В is a standard BMrf independent of Y. Call Px (resp. P°) the law of Bf
(resp. B).
1°) Prove that Pk <] P° with density hx(B,, t) where
Ал(х
. r) = f exp ({x, j) - t\y\2/2) Щу).
2°) Prove that in the filtration \-^gk) the semimartingale decomposition of
Bx is given by
f
Jo
Jo
Write down the explicit value of this decomposition in the following two cases:
i) d= 1 andA. = E_m+5m)/2;
ii) d = 3 and к is the rotation-invariant probability measure on the sphere of
radius r > 0, in which case if R, is the radial part of Bf-,
R, = y, +r I coth(rRs)ds
Jo
for a BM y.
C.19) Exercise. Let ц > 0 and set Bj1 — B,+fj.t where В is a BM1 @). Call LM
the local time of B" at 0. In particular B° = В and L° = L. Set X" = |В'' | + LM
and X° = X.

362 Chapter VIII. Girsanov's Theorem and First Applications
1°) Using the symmetry of the law of B, prove that for any bounded func-
functional F
E [F {X* ,s<t)] = E[F{Xs,s < r) cosh (ji\Bt |) exp (-/x2t/2)] .
2°) Using Levy's identity and Corollary C.6) of Chapter VI, prove that these
expressions are yet equal to
E [F (Xs, s < /) (sinh (fiX,) /цХ,) exp [~n2t/2)] .
3°) Put Sf = sup (Д", s <t) and prove that the processes X* and 2S" - B"
have the same law, namely, the law of the diffusion with infinitesimal generator
(l/2)d2/dx2 + Ц, coth(ixx)d/dx.
4°) Prove that for /x Ф 0 the processes IB"! and S" - B^ do not have the
same law. [Hint: Their behaviors for large times are different.]
Prove more precisely that for every bounded functional F,
E [F (|B,"|, s<t)] = E[F {S? - Bf, s < t) Д]
where A, = exp (-Sr") (l + exp B/x (S? - В?)) /2).
5°) Prove however that
(|B,"|, s < т, | x, < ex)) (=' {Sf -B?,s< T,),
where r, - inf [s > 0 : L% > t} and T, = inf{s >0:B?= t}. Give closed form
formulae for the densities of т, and T,.
6°) Use the preceding absolute continuity relationship together with question
4°) in Exercise D.9) Chapter VI to obtain a closed form for the Laplace transform
of the law of
inf {5 : S," - B," = a}.
Notes and Comments
Sect. 1. What we call Girsanov's theorem - in agreement with most authors - has
a long history beginning with Cameron-Martin ([1] and [2]), Maruyama ([1] and
[2]), Girsanov [1], Van Schuppen-Wong [1]. Roughly speaking, the evolution has
been from Gaussian processes to Markov processes, then to martingales. Cameron
and Martin were interested in the transformation of the Brownian trajectory for
which the old and new laws were equivalent; they first considered deterministic
translations and afterwards random translations so as to deal with BM with non
constant drift. The theory was extended to diffusions or more generally Markov
processes by Maruyama and by Girsanov. Proposition A.12) is typical of this
stage. Finally with the advent of Martingale problems developed by Stroock and
Varadhan, it became necessary to enlarge the scope of the results to martingales
which was done by Van Schuppen and Wong; Theorem A.4) is typically in the line

Notes and Comments 363
of the latter. This already intricate picture must be completed by the relationship
of Girsanov's theorem with the celebrated Feynman-Kac formula and the Doob's
h -processes which are described in Sect. 3.
For the general results about changes of law for semimartingales let us mention
Jacod-Memin ([1], [2]) and Lenglart [3] and direct the reader to the book of
Dellacherie and Meyer [1] vol. II. Other important papers in this respect are
Yoeurp ([3], [4]) which stress the close connection between Girsanov's theorem
and the theory of enlargements of filtrations. In fact the parenthood between the
decomposition formulas of martingales in this theory and in the Girsanov set-up
is obvious, but more precisely, Yoeurp [4] has shown that, by using Follmer's
measure, the enlargement decomposition formula could be interpreted as a special
case of a wide-sense Girsanov's formula; further important developments continue
to appear in this area.
The theory of enlargements of filtrations is one of the major omissions of this
book. By turning a positive random variable into a stopping time (of the enlarged
filtration), it provides alternative proofs for many results such as time reversal
and path-decomposition theorems. The interested reader may look at Jeulin [2]
and to Grossissements de filtrations: exemples et applications, Lecture Notes in
Mathematics, vol. 1118, Springer A985). A thorough exposition is also found
in Dellacherie, Maisonneuve and Meyer [1]. Complementing Yoeurp's theoretical
work mentioned above, enlargement of filtrations techniques have been used to-
together with Girsanov's theorem; see for example Azema-Yor [4], Follmer-Imkeller
[1] and Mortimer-Williams [1].
The theory of enlargements also allows to integrate anticipating processes
with respect to a semimartingale; a different definition of the integral of such
processes was proposed by Skorokhod [3] and continues to be the subject of
many investigations (Buckdahn, Nualart, Pardoux, ...).
The Girsanov pair terminology is not standard and is introduced here for the
first time. The proof of Kazamaki's criterion (Kazamaki [2]) presented here is due
to Yan [1] (see also Lepingle-Memin [1]).
Exercise A.24) is taken from Liptser-Shiryaev [1] as well as Exercise A.40)
which appears also in the book of Friedman [1].
Sect. 2. The fundamental Theorem B.2) is due to Cameron-Martin. We refer
to Koval'chik [1] for an account of the subject and an extensive bibliography.
Theorem B.4) is from Clark [1], our presentation being borrowed from Rogers-
Williams [1]. The result has been generalized to a large class of diffusions by
Haussman [1], Ocone [2], and Bismut [2] for whom it is the starting point of his
version of Malliavin's calculus.
Part of our exposition of the large deviations result for Brownian motion is
borrowed from Friedman [1]. Theorem B.11) is due to Schilder [1]. The theory of
large deviations for Markov processes has been fully developed by Donsker and
Varadhan in a series of papers [1]. We refer the reader to Azencott [1], Stroock [4]
and Deuschel and Stroock [1]. The proof of Strassen's law of the iterated logarithm

364 Chapter VIII. Girsanov's Theorem and First Applications
(Strassen [1]) is borrowed from Stroock [4]. In connection with the result let us
mention Chover [1] and Mueller [1].
Exercise B.14) is from Yor [3]. The method displayed in Exercise B.16) is
standard in the theory of large deviations. The use of Skorokhod stopping times
in Exercise B.17) is the original idea of Strassen.
Sect. 3. The general principle embodied in Proposition C.1) and which is the ex-
expression of Girsanov's theorem in the context of diffusions is found in more or less
explicit ways in numerous papers such as Kunita [1], Yor [10], Priouret-Yor [1],
Nagasawa [2], Elworthy [1], Elworthy-Truman [1], Ezawa et al [2], Fukushima-
Takeda [1], Oshima-Takeda [1], Ndumu [1], Truman [1], Gruet [3], among others.
The operateur carre du champ was introduced by Kunita [1] and by Roth [1].
Clearly, it can be denned only on sub-algebras of the domain of the (extended)
infinitesimal generator and this raises the question of its existence when one studies
a general situation (see Mokobodzki [1] who, in particular, corrects errors made
earlier on this topic).
Proposition C.4) may be found in Kunita [1] and Theorem C.7) in Pitman-Yor
[1]. For Exercise C.12), let us mention Kent [1], Wendel [1] and [2], Pitman-Yor
[1] and Watanabe ([1], [4]). Exercise C.14) comes from Pitman-Yor [1] and Yor
[ 10]; the formula of 4°) was the starting point for the decomposition of Bessel
Bridges (see Pitman-Yor [2] and [3]). In connection with Exercise C.14) let us
also mention Gaveau [1] whose computation led him to an expression of the
heat semigroup for the Heisenberg group. It is interesting to note that P. Levy's
formula for the stochastic area plays a central role in the probabilistic proof given
by Bismut ([4] and [5]) of the Atiyah-Singer theorems.
Exercise C.17) is taken from Yor [16], following previous work by L. Schwartz
[1]. It is also related to Getoor [2], and the results have been further developed
by Carne [3].

Chapter IX. Stochastic Differential Equations
In previous chapters stochastic differential equations have been mentioned several
times in an informal manner. For instance, if M is a continuous local martingale,
its exponential # (M) satisfies the equality
%(M), = 1+ f '6{M)sdMs\
Jo
this can be stated: %(M) is a solution to the stochastic differential equation
XsdMs,
f
Jo
which may be written in differential form
dX,=X,dM,, XQ=\.
We have even seen (Exercise C.10) Chap. IV) that >5(M) is the only solution to
this equation. Likewise we saw in Sect. 2 Chap. VII, that some Markov processes
are solutions of what may be termed stochastic differential equations.
This chapter will be devoted to the formal definition and study of this notion.
§1. Formal Definitions and Uniqueness
Stochastic differential equations can be defined in several contexts of varying
generality. For the purposes of this book, the following setting will be convenient.
As usual, the space C(R+, Rrf) is denoted by W. If w(s), s > 0, denote the
coordinate mappings, we set ,jffit = a(w(s),s < t). A function / on R+ x W
taking values in W is predictable if it is predictable as a process defined on W with
respect to the filtration (J9,). If X is a continuous process denned on a filtered
space (??,.5f, P), the map s -*¦ Xs(co) belongs to W and if / is predictable,
we will write f(s, X) or f(s, X (ft))) for the value taken by / at time s on the
path t -*¦ X,(<d). We insist that we write X,((o) here and not Xs(a>), because
f(s, X,(co)) may depend on the entire path X,(o>) up to time s. The case where
f(s, w) = a(s, w(s)) for a function a defined on R+ x M.d is a particular, if
important, case and we then have f(s, X_) = a(s, Xs). In any case we have

366 Chapter IX. Stochastic Differential Equations
A.1) Proposition. IfX is (J^)-adapted, the process f(t, X.(co)) is (.^-predict-
(.^-predictable.
Proof. Straightforward.
A.2) Definition. Given two predictable functions f and g with values in d x r
matrices and d-vectors, a solution of the stochastic differential equation e(f, g) is a
pair (X, B) of adapted processes defined on a filtered probability space (Q, &j, P)
and such that
i) В is a standard (.^)-Brownian motion in W;
Ц) far i = 1, 2,..., d,
W + f g,(s,
' Jo
X.)ds.
Furthermore, we use the notation ex(f, g) if we impose the condition Xq = x a.s.
on the solutions.
We will rather write ii) in vector form
X,=X0+ f /E, X)dBs + f g(s, X.)ds
Jo Jo
and will abbreviate "stochastic differential equation" to SDE. Of course, it is
understood that all the integrals written are meaningful i.e., almost-surely,
f YfMs,X.)ds < ex), f \g(s,X.)\ds < ex).
Jo j~f Jo
Consequently, a solution X is clearly a continuous semimartingale.
As we saw in Sect. 2 of Chap. VII, diffusions are solutions of SDE's of the
simple kind where
f(s, X.) = <r(s, X,), g(s, X.) = b(s, Xs)
for a and b defined on R+ x Rrf, or even in the homogeneous case
f(s,X.)=a(Xs), g(s,X.)
Such SDE's will be denoted e(a, b) rather than e(f, g).
We also saw that to express a diffusion as a solution to an SDE we had to
construct the necessary BM. That is why the pair (X, B) rather than X alone is
considered to be the solution of the SDE.
We have already seen many examples of solutions of SDE's, which justifies
the study of uniqueness, even though we haven't proved yet any general existence
result. There are at least two natural definitions of uniqueness and this section will
be devoted mainly to the study of their relationship.

§ 1. Formal Definitions and Uniqueness 367
A.3) Definitions. 1°; There is pathwise uniqueness/or e(f, g) if whenever (X, B)
and (X1', B') are two solutions defined on the same filtered space with В = В' and
Xo = X'o a.s., then X and X' are indistinguishable.
2°) There is uniqueness in law for e(f, g) if whenever (X, B) and (X', B') are
two solutions with possibly different Brownian motions В and B' (in particular if
(X, B) and (A", B') are defined on two different probability spaces (?2,, V, J? P)
and (Q1, &', .$?', P')) and Xo = X'o, then the laws of X and X' are equal. In
other words, X and X' are two versions of the same process.
Uniqueness in law is actually equivalent to a seemingly weaker condition.
A.4) Proposition. There is uniquenes in law if, for every x 6 M.d, whenever (X, B)
and (X', B') are two solutions such that Xq = x and X'o — x a.s., then the laws of
X and X' are equal.
Proof Let P be the law of (X, B) on the canonical space C(R+, W+r). Since
this is a Polish space, there is a regular conditional distribution P(co, •) for P with
respect to J$<j. For almost every со the last r coordinate mappings /3' still form a
BMr under P{co, •) and the integral
f f{s,t)dps+ f
Jo Jo
g(s,i)ds,
where ? stands for the vector of the first d coordinate mappings, makes sense. It
is clear (see Exercise E.16) Chap. IV) that, for almost every со, the pair (?, fl)
is under P(co, •) a solution to e(f, g) with ?0 = ?(«) P(v, -)-a.s. If (X\ B') is
another solution we may likewise define P'(co, •) and the hypothesis implies that
P(co, •) = P'(co, •) for со in a set of probability 1 for P and P'. If Xo = X'o we
get P — P' and the proof is complete. о
Remark. The reader will find in Exercise A.16) an example where uniqueness in
law does not hold.
The relationship between the two kinds of uniqueness is not obvious. We will
show that the first implies the second, but we start with another important
A.5) Definition. A solution (X, B) ofe(f,g) on (J2,.F\ J*T, P) is said to be a
strong solution ifX is adapted to the filtration (-^B) i.e. the filtration of В com-
completed with respect to P.
By contrast, a solution which is not strong will be termed a weak solution.
For many problems, it is important to know whether the solutions to an SDE
are strong. Strong solutions are "non-anticipative" functionals of the Brownian
motion, that is, they are known up to time / as soon as В is known up to time t.
This is important in as much as В is often the given data of the problem under
consideration.
We now prepare for the main result of this section. Let W| = C(R+, Rd) and
W2 = C(R+, Rr). On Wi x W2 we define, with obvious notation, the ст-algebras

368 Chapter IX. Stochastic Differential Equations
we observe that .j9' = .Щ v ,^?/ for each t.
Let (X, B) be a solution to e(f, g) and B the image of P under the map
ф : со -> (X.(w), B.(w)) from J2 into W| x W2. The projection of Q on W2 is the
Wiener measure and, as all the spaces involved are Polish spaces, we can consider
a regular conditional distribution Q(w2, •) with respect to this projection, that is,
Q(w2, •) is a probability measure on Wi x W2 such that Q (w2l W| x [w2]) = 1
Q-a.s. and for every measurable set А с Wj x W2, Q(w2, A) = Eq [\a | ,j^2]
Q-a.s.
A.6) Lemma. If A € .B}, the map w2 -> Q(w2, A) is .J&,2-measurable up to a
negligible set.
Proofs If. &\,. 0i2,. •# are three ст-algebras such that, -&\ v. #2 is independent
of.. ^ under a probability measure m, then for A e . H\,
m (A 1, 62) = m (a \. ii2v. <i\ m-a.s.
Applying this to .^', .^5*,2, .^2 and Q we get
Q(-, A) = EQ [1д I ,^2] = EQ [\ a \ .Д2] Q-a.s.,
which proves our claim.
A.7) Theorem. Ifpathwise uniqueness holds for e(f, g), then
i) uniqueness in law holds for e(f, g);
ii) every solution to ex(f, g) is strong.
Proof. By Proposition A.4), it is enough to prove i) to show that if (X, B) and
(X', B') are two solutions defined respectively on (??, P) and (?2'', P') such that
Xq = x and X'o — x a.s. for some x e Rd, then the laws of X and X' are equal.
Let Wi and W, be two copies of C(R+, Rd). With obvious notation derived
from that in the previous lemma, we define a probability measure n on the product
W, x W, x W2 by
л" (dw], dw\, dw2) — Q (w2, dw\) Q' (w2,dw[) W(dw2)
where W is the Wiener measure on W2. If ,Щ = о (w\ (s), w\ (s), w2(s), s < t)
then under n, the process w2(t) is an (,^)-BMr. Indeed we need only prove that
for any pair E, t), with s < t, w2(t) — w2(s) is independent of У5. Let A e .Щ,
A! e .Я}', В е .Д2; by Lemma A.6), for ? e W,

§1. Formal Definitions and Uniqueness 369
x [exp(i (f,W2(/) - w2(s))) \AlA,\B]
= / exp (/ (?, ш2@ - tu2(j))) Q (W2, Л) б' (ш2, A')
= exp (-|?|2(? - s)/2) f Q (w2, A) Q' (ш2, Л') W(dw2)
Jb
B(Ах А'х В)
which is the desired result.
We now claim that (w\, w2) and (u/p w2) are two solutions to ex(f, g) on the
same filtered space (Wi x W', x W2, ,'Щ, sz\ Indeed, if for instance
X, = x + [ f(s, X )dBs + f g(s, X)ds
Jo Jo
under P, then
wi(t)=x+l f(s,W[)dw2(s)+ I g(s,w\)ds
Jo Jo
under n because the joint law of (f(s, X), g(s, X), B) under P is that of
(f(s, w\), g(s, u>i), w2) under л (see Exercise E.16) in Chap. IV). Since more-
moreover u/,@) = x 7r-a.s., the property of path uniqueness implies that w\ and w\
are jt-indistinguishable, hence w\(n) — w\(n), that is: X(P) — X'(P') which
proves i).
Furthermore, to say that w\ and w[ are 7r-indistinguishable is to say that л is
carried by the set {(wi, w[, u>2) : u>i = w\). Therefore, for W-almost every w2,
under the probability measure Q (w2, dw\) ® Q' (ш2, dw[) the variables w\ and
w\ are simultaneously equal and independent; this is possible only if there is a
measurable map F from W2 into Wi such that for W-almost every w2,
Q (w2, •) = Q' (ш2, •) = eF(W2).
But then the image of P by the map ф defined above Lemma A.6) is carried by
the set of pairs (F(iu2), ш2), hence X = F{B) a.s. By Lemma A.6), X is adapted
to the completion of the nitration of В. Р
Remarks. 1) In the preceding proof, it is actually shown that the law of the pair
(X, B) does not depend on the solution. Thus stated, the above result has a con-
converse which is found in Exercise A.20).
On the other hand, property i) alone does not entail pathwisc uniqueness which
is thus strictly stronger than uniqueness in law (see Exercise A.19)). Likewise the
existence of a strong solution is not enough to imply uniqueness, even uniqueness
in law (see Exercise A.16)).
2) We also saw in the preceding proof that for each x, if there is a solution
to ex(f, g), then there is a function F(x, •) such that F(x, B) is such a solution.
It can be proved that this function may be chosen to be measurable in x as well,
in which case for any random variable Xo, F(Xo, B) is a solution to e(f, g) with
Xo as initial value.

370 Chapter IX. Stochastic Differential Equations
We now turn to some important consequences of uniqueness for the equations
e((i, b) of the homogeneous type. In Proposition B.6) of Chap. VII we saw that if
(X, B) is a solution of ex(a, b), then X(P) is a solution to the martingale problem
n(x, a, b) with a — aa1. By Proposition A.4) it is now clear that the uniqueness
of the solution to the martingale problem л(х,а,Ь) for every x e Rd, implies
uniqueness in law for e(a, b).
In what follows, we will therefore take as our basic data the locally bounded
fields a and b and see what can be deduced from the uniqueness of the solution
to 7r(jt, a, b) for every x e Rd. We will therefore be working on W = C(R+, Rd)
and with the filtration (.j&t) = (cr(Xs,s < ?)) where the Xt's are the coordinate
mappings. If r is a (.:^,)-stopping time, the cr-algebra .J9T is countably generated
(see Exercise D.21), Chap. I) and for any probability measure P on W, there is a
regular conditional distribution Q(w, •) with respect to .J9T.
A.8) Proposition. IfP is a solution to ж(х, a, b) and r is a bounded stopping time,
there is a P-null set N such that for w ? N, the probability measure et(Q(w, •))
is a solution to n (XT(w), a, b).
Proof. For a fixed ш, let A = {u/ : Xq(w') = Хт(ш)}. From the definition of a
regular conditional distribution, it follows that
ez(Q(w, -))(A) = Q (w, {w' : Xz(w') - XT(w))) - 1.
Thus, by Definition B.3) in Chap. VII, we have to find a negligible set N such
that for any / e C?\ any t > s,
= М{ овт Q(w, -)-a.s.
for w ? N. Equivalently, for ш ? N, we must have
(*) / M,f o6T(w')Q(w,dw')= I M{ oeT(u>')Q(w,dw')
Ja Ja
for any A e 9~l(.J!?s) and / > s.
Recall that, by hypothesis, each M^ is a martingale. Let / be fixed and pick
В т.??т; by definition of Q, we have
E \lB(w) f M[ o9T{w')Q{w,dw')\ = ?
Since 1в-1д is JS'r+s-measurable as well as М{ and since, by the optional stopping
theorem, м{+1 is a .J?r+l -martingale, this is further equal to
E [lB • lA (м[+г - M/)] = E UB(w)J Ml oer(w')Q(w,dw')\.
As a result, there is a P-null set N(A, f,s,t) such that (*) holds for w
N(A,f,s,t).

§1. Formal Definitions and Uniqueness 371
Now the equality (+) holds for every / in C?? if it holds for / in a countable
dense subset !& of C??; because of the continuity of X it holds for every 5 and
t if it holds for s and t in Q. Let if, be a countable system of generators for
e;\J$sY, the set
^=U U \jN{A,f,s,t)
s.teQ fed AeZs
is P-negligible and is the set we were looking for.
A.9) Theorem. If for every x e Rd, there is one and only one solution Px to the
martingale problem n(x,a,b) and, if for every A e .J&(Rd) and t > 0 the map
x -*¦ PX[X, e A] is measurable, then (X,, Px, x e Rd) is a Markov process with
transition Junction P,(x, A) = Px[Xt e A].
Proof. For every event Г е .i^oo» every bounded .^-stopping time г and every
x e Rd we have, with obvious notation,
the uniqueness in the statement together with the preceding result entails that
Making x =t and integrating, we get the semi-group property. ?
Remark. With continuity assumptions on a and b, it may be shown that the semi-
semigroup just constructed is actually a Feller semi-group. This will be done in a
special case in the following section.
Having thus described some of the consequences of uniqueness in law, we
want to exhibit a class of SDE's for which the property holds. This will provide
an opportunity of describing two important methods of reducing the study of SDE's
to that of simpler ones, namely, the method of transformation of drift based on
Girsanov's theorem and already alluded to in Sect. 3 of the preceding chapter
and the method of time-change. We begin with the former which we treat both in
the setting of martingale problems and of SDE's. For the first case, we keep on
working with the notation of Proposition A.8).
A.10) Theorem. Let a be afield of matrices, b and с fields of vectors such that a,
b and (с, ас) are bounded. There is a one-to-one and onto correspondence between
the solutions to the martingale problems n{x,a,b) and n(x,a,b + ac). If P and
Q are the corresponding solutions, then
dQ
JP "=eXP
\j [c{Xs),dXs)-X-j (c,ac){Xs)ds\
where X, = X, - /J b{Xs)ds.
The displayed formula is the Cameron-Martin formula.

372 Chapter IX. Stochastic Differential Equations
Proof. Let P be a solution to n{x, a, b). By Proposition B.4) in Chap. VII, we
know that under P the process X is a vector local martingale with increas-
increasing process f'Qa(Xs)ds. If we set Y, = fg(c(Xs),dXs), we have (Y, Y), =
/„' (c,ac) (Xs)ds and, since (с, ас) is bounded, Novikov's criterion of Sect. 1
Chap. VIII asserts that %(Y) is a martingale. Thus one can define a probability
measure Q by Q = %'(Y),.P on .^5",, which is the formula in the statement. We
now prove that Q is a solution to л(х,а, b + ac) by means of Proposition B.4)
in Chap. VII.
For в e Rd, the process Mf = @, X, — x) is a P-local martingale with in-
increasing process A, = /0' @,a(.XjH)rf.s. Thus by Theorem A.4) in Chap. VIII,
Me — [Me, Y) is a б-local martingale with the same increasing process A,. It is
furthermore easily computed that
-К
As a result, (#, X, - x — /„' b(Xs)ds - /0' ac(Xs)rf5| is a (?-local martingale with
increasing process A, which proves our claim.
The fact that the correspondence is one-to-one and onto follows from Propo-
Proposition A.10) in Chap. VIII applied on each subinterval [0, t]. D
The above result has an SDE version which we now state.
A.11) Theorem. Let f (resp. g, h) be predictable functions on W with values in
the symmetric non-negative d x d matrices (resp. d-vectors) and assume that h is
bounded. Then, there exist solutions to ex(f, g) if and only if there exist solutions
t° ?x(/. g + fh). There is uniqueness in law for e(f, g) if and only if there is
uniqueness in law for e(f, g + fh).
Proof. If (X, B) is a solution to e(/, g) on a space (?2, .5^, P), we define a proba-
probability measure Q by setting Q = &(M),. P on Jf where M, = /„' h(s, X,)dBs. The
process B, = B,— /0' h(s, X)ds is a BM and (X, B) is a solution of e(/, g + fh)
under Q. The details are left to the reader as an exercise. D
The reader will observe that the density Щ | w is simpler than in the Cameron-
Martin formula. This is due to the fact that we have changed the accompanying
BM. One can also notice that the assumption on h may be replaced by
E exp[-/ \\h(s, X,)fds\ < oo for every t > 0.
A.12) Corollary. Assume that, for every s and x, the matrix o(s, x) is invertible
and that the map (s, x) -*¦ cr(s, xyl is bounded; ife(cr, 0) has a solution, then for
any bounded measurable b, the equation e(a, b) has a solution. If uniqueness in
law holds for e(cr, 0), it holds for e{a, b).

§1. Formal Definitions and Uniqueness 373
Proof. We apply the previous result with /E, X.) = <r(s, Xs), g(s, X.) = b{s, Xs)
and h(s, X.) = -orE, Xs)~lb(s, Xs).
Remark. Even if the solutions of e(f, 0) are strong, the solutions obtained for
e(/, g) by the above method of transformation of drift are not always strong as
will be shown in Sect. 3.
We now turn to the method of time-change.
A.13) Proposition. Let у be a real-valued Junction on Rd such that 0 < к < у <
К < oo; there is a one-to-one and onto correspondence between the solutions
to the martingale problem n{x,a,b) and the solutions to the martingale problem
n(x, у a, yb).
Proof. With the notation of Proposition A.8) define
A,= f y(Xs)ds,
Jo
and let (t,) be the associated time-change (Sect. 1 Chap. V). We define a measur-
measurable transformation ф on W by setting X (<p(w)), = Xtl(w). Let P be a solution
to the martingale problem n(x, a, b); for any pair (s, t), s < t, and A e .jfts, we
have, for / e Cg,
f (f{X,)-f(Xs)- I y(Xu)Lf{Xu)du)d(t>(P)
Ja\ Js /
= f (f(XTl) - /(XTl) - f y(XtJLf(Xtu)du) dP
J<I>-4a) \ Л /
= f (f(XZl) - /(XTt) - f ' Lf{Xu)du)dP
J<t>-'(A) \ Л, /
thanks to the time-change formula of Sect. 1 Chap. V. Now since ф~1(А) е :j8%!s
the last integral vanishes, which proves that ф(Р) is a solution to ж{х, у a, yb).
Using y~] instead of y, we would define a map т/f such that ¦ф(ф{Р)) = P which
completes the proof. D
Together with the result on transformation of drift, the foregoing result yields
the following important example of existence and uniqueness.
A.14) Corollary. If a is a bounded function on the line such that \cr\ > e > 0
and b a bounded function on K+ x K, there is existence and uniquenss in law for
the SDE e(a, b). Moreover, if Px is the law of the solution such that Xq = x, for
any A € J9(R), the map x -> PX[X, e A] is measurable.
Proof. By Corollary A.12) it is enough to consider the equation e(a, 0) and since
the BM started at x is obviously the only solution to ex(l, 0), the result follows
from the previous Proposition. The measurability of P\Xt € A] follows from the
fact that the Px's are the images of Wx under the same map.

374 Chapter IX. Stochastic Differential Equations
Remark. By Theorem A.9) the solutions of e(cr, b) form a homogeneous Markov
process when b does not depend on л. Otherwise, the Markov process we would get
would be non homogeneous. Finally it is worth recording that the above argument
does not carry over to d > 1, where the corresponding result, namely that for
uniformly elliptic matrices, is much more difficult to prove.
A.15) Exercise. Let (Y, B) be a solution to e = ey(f,g) and suppose that /
never vanishes. Set t
A,= f B + Ys/(l+\Ys\))ds
and call T, the inverse of A,. Prove that X, = BTi is a pure local martingale if
and only if У is a strong solution to e.
A.16) Exercise. 1°) Let a(x) = 1 л |x|" with 0 < a < 1/2 and В be the standard
linear BM. Prove that the process f^a'2(Bs)ds is well-defined for any t > 0; let
r, be the time-change associated with it.
2°) Prove that the processes X, = BTl and Xt = 0 are two solutions for eo(a, 0)
for which consequently, uniqueness in law does not hold. Observe that the second
of these solutions is strong.
A.17) Exercise. A family Xх of Revalued processes with Xх = x a.s. is said
to have the Brownian scaling property if for any с > 0, the processes c~lX*2[
and Xct x have the same law. If uniqueness in law holds for e(cr, b) and if Xх is
a solution to ex(cr,b), prove that if ct(cjc) = cr(x) and cb(cx) — b(x) for every
с > 0 and x € Rd, then Xх has the Brownian scaling property. In particular if
ф is a function on the unit sphere and b(x) = \]х\\~1ф(х/\\х\\), the solutions to
ex(ald, b) have the Brownian scaling property.
A.18) Exercise. In the situation of Corollary A.14) let (X, B) be a solution to
ex(a, b) and set Y, = X, - x - /„' b(s, Xs)ds.
1°) Let W* be the space of continuous functions ioont+, such that ш@) — x.
For w e\Vx, set
\frn(t,w) = cr'l(x) for 0 < Г < 2~",
cr-[(ws)ds for kl~" <t < {k+\J-\
(i,n(s,X,)-cr-](Xs))dY
4
2°) Prove that there is an adapted function Ф from W* to W° which depends
only on the law of X and is such that В = Ф(Х).
3°) Derive from 2°) that if there exists a strong solution, then there is pathwise
uniqueness.
[Hint: Prove that if (X, B) and (A", B) are two solutions, then (X, B) =
(X\ B).]

§2. Existence and Uniqueness in the Case of Lipschitz Coefficients 375
# A.19) Exercise. 1°) If/9 is a BM@) and B, = fQsgp{ps)dps, prove that (ft B)
and (-/3, B) are two solutions to eo(sgn, 0).
More generally, prove that, if (е„, и > 0) is predictable with respect to the
natural filtration of/3, and takes only values +1 and —1, then (e^ft, B,;t>6) is
a solution to eo(sgn, 0), where g, = sup{s < t : ft = 0}.
2°) Prove that eo(sgn, 0) cannot have a strong solution.
[Hint: If X is a solution, write Tanaka's formula for |X|.]
A.20) Exercise. 1°) Retain the notation of Lemma A.6) and prove that (X, B) is
a strong solution to ex(f, g) if and only if there is an adapted map F from W2
into Wi such that
Q{w2, ¦) = eF(u,2) ?>-a.s.
2°) Let (X, B) and (X', B) be two solutions to ex(f, g) with respect to the
same BM. Prove that if
i) (X, B) (=> (X', B\
ii) one of the two solutions is strong,
then X = X'.
§2. Existence and Uniqueness
in the Case of Lipschitz Coefficients
In this section we assume that the functions / and g of Definition A.2) satisfy
the following Lipschitz condition: there exists a constant К such that for every t
and w,
\f(t, w) - f(t, w')\ + \g{t, w) - g(t, w')\ < К sup \w(s) - w'(s)\
where | | stands for a norm in the suitable space. Under this condition, given a
Brownian motion В in Шг. We will prove that for every x e Rd, there is a unique
process X such that (X, B) is a solution to ex(f, g); moreover this solution is
strong. As the pair (Bt, t) may be viewed as a r + 1-dimensional semimartingale
we need only prove the more general
B.1) Theorem. Let (?2, .5f, P) be a filtered space such that (.W,) is right-continu-
right-continuous and complete and Z a continuous r-dimensional semimartingale. If f satisfies
the above Lipschitz condition and if for every y, /(•, 3^) is locally bounded where
y{t) = y, then for every x e Rd, there is a unique (up to indistinguishability)
process X such that
X, = x + [' f(s, X)dZs.
Jo
Moreover, X is (,^z)-adapted.

376 Chapter IX. Stochastic Differential Equations
Proof. We deal only with the case d = 1, the added difficulties of the general
case being merely notational. If M + A is the canonical decomposition of Z we
first suppose that the measures d(M, M), and \dA\, on the line are dominated by
the Lebesgue measure dt.
Let x be a fixed real number. For any process U with the necessary measura-
bility conditions, we set
{SU),=x+ f f(s,U.)dZs.
Jo
If V is another such process, we set
Because any two real numbers A and к satisfy (A + kJ < 2(A2 + k2), we have
<t>,(SU,SV) < 2? sup(Y (/(r, U.)-f(r, V.))dMr\
+ supf I \f{r,U.)-f(r,y)\ \dA\r
s<t \Jo ,
and by the Doob and Cauchy-Schwarz inequalities, it follows that
0,(SU,SV) < 8?| ( f (/(r, V.)-f{r, К))<шЛ
l/(r,?/.)-/(r,V.)|2 МА|Л1
< 8? / (/(r, C/) - /(r, V)J rf(M, M)r
+2tE\( \f(r,U.)-f(r,V.)\2 \dA\r]
Uo J
¦> Г /" 7 1
< 2^-^D + t)E\ I sup\Us - Vs\zdr\
UO s<r J
= 2Л:2D + П / 0r(U,V)dr.
Jo
Let us now define inductively a sequence (Xn) of processes by setting X° = x
and Xn = S(X"-1); let us further pick a time Г and set С = 2K2D + Г). Using
the properties of / it is easy to check that D = ФТ(Х°, X1) is finite. It then
follows from the above computation that for every t < T and every n,
Ф,{Хп-\Хп) <DC"Tn/nl.
Consequently

§2. Existence and Uniqueness in the Case of Lipschitz Coefficients З77
00
E
n=\
sup I*?-
<oo.
2
Thus, the series $^li SUP«</ \%" ~~ %" 'I converges a.s. and as a result, X" con-
converges a.s., uniformly on every bounded interval, to a continuous process X. By
Theorem B.12) Chap. IV, X = SX, in other words X is a solution to the given
equation.
To prove the uniqueness, jve consider two solutions X and Y and put Tk =
inf{t : \X,\ or |V,| > k]. Let S be defined as S but with Z7* in lieu of Z. Then it
is easily seen that XTl = S(XTt) and likewise for Y, so that for / < T
~ /"'
Ф, (X7*, Yn) = Ф, (SX7*, 5ГГ*) <С Ф5 {XTt, YTk) ds.
Jo
Since by the properties of /, the function <P,(XTk, YTk) is locally bounded, Gron-
wall's lemma implies that ФХХТк, YTk) is identically zero, whence X = Y on
[0, Tk л T] follows. Letting к and T go to infinity completes the proof in the
particular case.
The general case can be reduced to the particular case just studied by a suitable
time-change. The process A't = t + (M, M), + /0' \dA\s is continuous and strictly
increasing. If we use the time-change Ct associated with A't, then M, = Me, and
A, = Ac, satisfy the hypothesis of the particular case dealt with above. Since
C, < t, one has
|/(С„ U.) - f(C,, V)\ < К sup \US - V,
s<t
and this condition is sufficient for the validity of th? reasoning in the first part of
the proof, so that the equation X, = x + /0' f(Cs, X,)dZs has a unique solution.
By the results of Sect. 1 Chap. V, the process X, = A"A; is the unique solution to
the given equation. ?
As in the case of ordinary differential equations, the above result does not
provide any practical means of obtaining closed forms in concrete cases. It is
however possible to do so for the class of equations defined below. The reader may
also see Exercise B.8) for a link between SDE's and ODE's (ordinary differential
equations).
B.2) Definition. A stochastic equation is called linear if it can be written
YsdXs
f
Jo
f
Jo
where H and X are two given continuous semimartingales.
It can also be written as
dY, = dHt + Y,dX,, Yo = Яо.

378 Chapter IX. Stochastic Differential Equations
An important example is the Langevin equation dV, — dB, — pV,dt, where В is
a linear BM and p a real constant, which was already studied in Exercise C.14)
of Chap. VIII. Another example is the equation Yt = 1 + /0' YsdXs for which we
know (Sect. 3 Chap. IV) that the unique solution is Y = %(X). Together with the
formula for ordinary linear differential equations, this leads to the closed form for
solutions of linear equations, the existence and uniqueness of which are ensured
by Theorem B.1).
B.3) Proposition. The solution to the linear equation of Definition B.2) is
Y, = %{X), I Ho + / %(XOl {dHs - dlH, X)s) ) ;
\ Jo J
in particular, if(H, X) — 0, then
Y, = X{X), (h0 + I %{X);ldHs\ .
Proof. Let us compute /0 YsdXs for Y given in the statement. Because of the
equality %(X)sdXs — d%(X)s and the integration by parts formula, we get
YsdXs
о
= Ho f %{X),dXs + I %{X)sdX, ( f ^(X)~l (dHu - d(H, X)u))
Jo Jo \Jo /
= -Ho + HOC6{X), + C6{X), I &(X)~l (dHs - dlH, X)s)
Jo
f
Jo
f %(X);l(dHs-d(H,x)s)
, X)t -1 f %\x)
\Jo
= Yt-Ht
= у,-н„
which is the desired result. G
Remark. This proof could also be written without prior knowledge of the form of
the solution (see Exercise B.6) 2°).
One may also prove that if Я is a progressive process and not necessarily a
semimart., the process
Y = H -
j H d (%(X)~l)
is a solution of the linear Equation B.2). The reader will check that this jibes with
the formula of Proposition B.3) if Я is a semimart..

§2. Existence and Uniqueness in the Case of Lipschitz Coefficients 379
The solution to the Langevin equation starting at v is thus given by
For /3 > 0 this is the OU process with parameter fi of Exercise A.13) of Chap.
Ill (see Exercise B.16)). The integral f0 Vsds is sometimes used by physicists
as another mathematical model of physical Brownian motion. Because of this
interpretation, the process V is also called the OU velocity process of parameter
f}. In physical interpretations, ft is a strictly positive number. For fi — 0, we
get V, — v + Br. In all cases, it also follows from the above discussion that the
infinitesimal generator of the OU process of parameter fi is the differential operator
We now go back to the general situation of the equation e(f, g) with / and g
satisfying the conditions stated at the beginning of the section. We now know that
for a given Brownian motion В on a space (Q, .^, P) and every jc e К"', there
is a unique solution to ex(f, g) which we denote by Xх. We will prove that Xх
may be chosen within its indistinguishability class so as to get continuity in jc.
B.4) Theorem. Iff andg are bounded, there exists a process Xxt, x e Rd, t e R+,
with continuous paths with respect to both variables t and x such that, for every x,
<x=x+ f
Jo
f(s, Xx)dBs
f
Jo
g(s, Xx)ds P-a.s.
Proof. Pick p > 2 and t > 0. Retain the notation of the proof of Theorem B.1)
writing Sx, rather than S to stress the starting point jc.
Because \a + b + c\p < 3P~[ (\a\p + \b\p + \c\p), we have
|x - y\" + sup / (f{r, U.) - f(r, V)) dBr
s<t \Jo
+ sup If (g(r,U)-g(r.V.))dr"
s<r \J0
Thanks to the BDG and Holder inequalities
e\suP f (f(r,U.)- f{r,V))dBr
< CPE
(f\f(r,U.)-f{r,V.)JdrJ
< Cpt(p~2)/2E Г Г \f(r, U.) - f(r,
Uo
< KpCpt(p~2)l2E
0 s<r

380 Chapter IX. Stochastic Differential Equations
Likewise,
sup f (g(r, U) - g(r, V))dr < Kptp~x ( sup\Us - Vs\pdr.
s<t \Jo Jo s<r
Applying this to U = Xх and V — Xy where Xх (resp. Xy) is a solution to
eAf, g) (resp. ey(f, g)) and setting h(t) = E [sups<, \XX - Xys\p], it follows that
h(t) < bf\x - y\p + cf I h(s)ds
Jo
for two constants bp and cf. Gronwall's lemma then implies that there is a constant
af depending only on p and / such that
1
By Kolmogorov's criterion (Theorem B.1) of Chap. 1), we can get a bicontinuous
modification of X and it is easily seen that for each jc, the process Xх is still a
solution to ex(f, g).
Remark. The hypothesis that / and g are bounded was used to insure that the
function h of the proof is finite. In special cases, h may be finite with unbounded
/ and g and the result will still obtain.
From now on, we turn to the case of e(a, b) where a and b are bounded
Lipschitz functions of jc alone. If we set
we know from Theorem A.9) that each Xх is a Markov process with transition
function P,. We actually have the
B.5) Theorem. The transition function Pt is a Feller transition function.
Proof. Pick / in Co. From the equality P, f(x) ~ E [f(Xx)], it is plain that the
function P,f is continuous. It is moreover in Co; indeed,
\Pf(x)\< sup \f(y)\ + \\f\\P[\Xx -x\>r]
and
JjT'wH'+|jf
< 2k2r~2(t + t2)
where A: is a uniform bound for a and b. By letting jc, then r, go to infinity, we
get ПгП|Л|^оо Pi fix) = 0.
On the other hand, for each jc, t —> P,f(x) is also clearly continuous which
completes the proof.
2Г2Е

§2. Existence and Uniqueness in the Case of Lipschitz Coefficients 381
With respect to the program outlined at the end of Sect. 1 in Chap. VII we
see that the methods of stochastic integration have allowed us to construct Feller
processes with generators equal on C\ to
1
-
2 ij i
whenever a and b art bounded and Lipschitz continuous and а = а а'.
# B.6) Exercise (More on Proposition B.3)). P) Denoting by * the backward in-
integral (Exercise B.18) of Chap. IV), show that in Proposition B.3) we can write
which looks even more strikingly like the formulas for ordinary linear differential
equations.
2°) Write down the proof of Proposition B.3) in the following way: set Y =
<f(X)Z and find the conditions which Z must satisfy in order that У be a solution.
B.6) Bis Exercise (Vector linear equations). If X is a d x d matrix of cont.
semimarts. and H a r x d matrix of locally bounded predictable processes, we
define the right stochastic integral (H-dX), — fQ HsdXs as the rxd matrix whose
general term is given by
к
Likewise, there is a left integral dX • H provided that H and X have matching
dimensions.
1°) If Y is a r x d matrix of cont. semimarts., we define (Y, X) as the r x d
matrix whose entries are the processes
Prove that
d(YX) = dYX + YdX + d{Y, X),
and that, provided the dimensions match,
(H-dY,X) = H-d{Y,X), (Y,dX-H) =d{Y,X)-H.
2°) Given X, let us call <f (X) the d x d matrix-valued process which is the
unique solution of the SDE
U, = Id+f UsdX,,
Jo
and set K'{X) = %'(X')'. Prove that %'{X) is the solution to a linear equation
involvine left inteerals. Prove that <?'(—X + (X. X)) is the inverse of the matrix

382 Chapter IX. Stochastic Differential Equations
(which thus is invertible, a fact that can also be proved by showing that its
determinant is the solution of a linear equation in dimension 1).
[Hint: Compute d(%(xyS'{-X + (X, X))).]
3°) If H is a r x d matrix of cont. semimarts., prove that the solution to the
equation
Y, = H,+ f YsdXs,
Jo
is equal to
(dHs - d(H, X)s) ?{Х8уЛ tf (X),.
J
J
State and prove the analogous result for the equation
Y, = H,+ f dXsYs.
Jo
B.7) Exercise. Let F be a real-valued continuous function on Ш and / the solution
to the ODE
f'(s) = F(s)f{s); /@)=l.
Г) Let G be another continuous function and X a continuous semimartingale.
Prove that
Z, = f(t)\z
is the unique solution to the SDE
Zt = z+ I F(u)Zudu+ [ G(u)dXu.
Jo Jo
Moreover
fit) 7 ^ [' fit) _. . ,v
Z+ Giu)dXu.
7 ^ [' fit) _. .
s+ —— Giu)
Js fiu)
Js
2°) If X is the BM, prove that Z is a Gaussian Markov process.
3°) Write down the multidimensional version of questions 1°) and 2°). In
particular solve the d-dimensional Langevin equation
dXt =adB, +px,dt
where a and J} are d x d matrices.
[Hint: use Exercise B.6) bis.]
B.8) Exercise (Doss-Siissman method). Let a be a C2-function on the real line,
with bounded derivatives a' and a" and b be Lipschitz continuous. Call hix,s)
the solution to the ODE
— ix,s)=<r(hix,s)), h(x,0)=x.

§2. Existence and Uniqueness in the Case of Lipschitz Coefficients 383
Let X be a continuous semimartingale such that Xo = 0 and call D, the solution
of the ODE
dD, f /•*'(<*> 1
— =b{h (Dt, X,(a>))) exp - / a'(h{Dt, s))ds\; Do = y.
Prove that h(D,, Xt) is the unique solution to the equation
Yt = y+ I <r(Y,)o dXs + f b{Ys)ds
Jo Jo
where о stands for the Stratonovich integral. Moreover D, — h(Y,, —X,).
B.9) Exercise. For semimartingales H and X, call f?x{H) the solution to the
linear equation in Definition B.2).
1 °) Prove that if К is another semimartingale,
%X(H + K-X) = 8X(H + K)~ K.
2°) Let X and Y be two semimartingales and set Z = X + Y + (X, Y).
For any two semimartingales H and K, there is a semimartingale L such that
%x(H) ¦ %y{K) = %z(L). For H = К = 1, one finds the particular case treated
in Exercise C.11) Chap. IV.
3°) If X, Y, H are three semimartingales, the equation Z = H + (Z,Y) ¦ X
has a unique solution given by
* B.10) Exercise (Explosions). The functions a and b on Rd are said to be locally
Lipschitz if, for any n e N, there exists a constant Cn such that
1Идс) - a(y)\\ + \\b(x) - b(y)\\ < C||x - y\\
for jc and у in the ball S@, и).
1°) If a and b are locally Lipschitz, prove that for any jceR1* and any BM В
on a space {Q,.W, P) there exists a unique (.^B)-adapted process X such that,
if e = inf{? : |X,| = +oo}, then X is continuous on [0, e[ and
f, = x + / a{Xs)dBs + [ b(Xs)ds
Jo J
[ b(Xs)
Jo
on {t < e]. The time e is called the explosion time of X.
[Hint: Use globally Lipschitz functions an and fon which agree with a and b
on S@, и).]
2°) If there is a constant К such that
for every jc e M.d, prove that E [\Xt |2] < oo and conclude that P[e = oo] = 1.
[Hint: If Tn = inf[t : \X,\ > и}, develop |Х,л:г,, | by means of Ito's formula.]

384 Chapter IX. Stochastic Differential Equations
3°) Let W be the space of functions w from Ш.+ to Rd U {A} such that if
?(ш) = infjf : w(t) = A}, then w(s) = A for any s > t, and w is continuous on
[0, C(w)[. The space W contains the space C(R+, Rd) on which f is identically
+oo^ Call У the coordinate process and, with obvious notation, let Qx be the law
on W of the process X of 1°) and Px be the law of BM(jc). For d = 1 and a = 1
prove that
As a result, <f (/^(К,)^) is a true martingale if and only if Qx{t, < oo) = 0.
4°) Prove that for any a e Ш, % (a f0 BsdBx) is a true martingale although
Kazamaki's criterion does not apply for t > a~l.
B.11) Exercise (Zvonkin's method). Г) Suppose that a is a locally Lipschitz
(ExerciseB.10)) function on R, bounded away from zero and b is Borel and
locally bounded; prove that the non-constant solutions h to the differential equation
\a2h" + bh! = 0 are either strictly increasing or strictly decreasing and that
g — (ah') о A is locally Lipschitz.
2°) In the situation of 1 °) prove that pathwise uniqueness holds for the equation
e{a, b).
[Hint: Set up a correspondence between the solutions to e(o\ b) and those to
e(g, 0) and use the results in the preceding exercise.]
This method amounts to putting the solution on its natural scale (see Exercise
C.20) in Chap. VII).
3°) Prove that the solution to e(a, b) where a is bounded and > s > 0 and b
is Lebesgue integrable, is recurrent.
B.12) Exercise (Brownian Bridges). The reader is invited to look first at ques-
questions 1°) and 2°) in Exercise C.18) of Chap. IV.
F) Prove that the solution to the SDE
Xх. = p, + / J ds, t € [0, 1[, x e R,
Jo 1 - *
is given by
2°) Prove that lim,ti Xf = x a.s. and that if we set Xх = x, then Xх, t e [0, 1],
is a Brownian Bridge.
[Hint: If g is a positive continuous decreasing function on [0, 1] such
that g(\) = 0, then for any positive / such that /J f(u)g(u)du < +oo,
lim(t,g(O/d Я«М" = 0; аРР!У this to 8@ = 1 - t and /(/) =
I0i - AIO-0-]
3°) If Px is the law of Xх on C([0, 1], R), the above questions together with
Г) in Exercise C.18) of Chap. IV give another proof of Exercise C.16) in Chap. I,

§2. Existence and Uniqueness in the Case of Lipschitz Coefficients 385
namely that Px is a regular disintegration of the Wiener measure on C([0, 1], R)
with respect to a(B{) where В is the canonical process.
4°) We retain henceforth the situation and notation of Exercise C.18) in
Chap. IV. Prove that Xх and P have the same filtration.
5°) Prove that the (&,)-predictable processes are indistinguishable from the
processes K(B\(w), s, w) where A" is a map on R x (R+ x Q) which is .jS'(R) x
^(^-measurable, dfiifi) being the ст-algebra of predictable sets with respect to
the filtration of ft. Prove that any square-integrable (.(?,)-martingale can be written
/(/?,)+ f Ksdj}s
Jo
where / and К satisfy some integrability conditions.
B.13) Exercise. 1°) Let В be the linear BM, b a bounded Borel function and set
Y, = B,- f b(Bs)ds.
Jo
Prove that .^Y = .WtB for every /.
[Hint: Use Girsanov's Theorem and Exercise B.11) 2° above.]
** 2°) If b is not bounded, this is no longer true. For example if Y, = B, —
/0' B~lds where the integral was defined in Exercise A.19) of Chap. VI, then
E[Bi \a(Ys,s < l)] = 0.
[Hint:For/e L2{[0, 1]), prove that E[B\^(f-Y)\] = Oand use the argument
of Lemma C.1) Chap. V. One may also look at 4°) in Exercise A.29) Chap. VI.]
* B.14) Exercise. (Stochastic differential equation with reflection). Let a(s, x)
and b(s,x) be two functions on R+ x M+ and В a BM. For jc0 > 0, we call
solution to the SDE with reflection eXf)(a, b) a pair (X, К) of processes such that
i) the process X is continuous, positive, .Я"в-adapted and
X, = x0 + [ a(s, Xs)dBs + f b(s, Xs)ds + Kt.
Jo J
s + f b(s, Xs)
Jo
ii) the process К is continuous, increasing, vanishing at 0, .3^B-adapted and
poo
XsdKs = 0.
г
Jo
If a and b are bounded and satisfy the global Lipschitz condition
\o(s, x) - o{s, y)\ + \b(s, x) - b(s, y)\ < C\x - y\
for every s, x, у and for a constant C, prove that there is existence and uniqueness
for the solutions to exo {a, b) with reflection.
[Hint: Use a successive approximations method with Lemma B.1) of Chap.
VI as the means to go from one step to the following one.]

386 Chapter IX. Stochastic Differential Equations
* B.15) Exercise (Criterion for explosions). We retain the situation of Exercise
B.11) and set
s(x) =
exp(- Jy 2b(z)<J-2{z)dz\dy,
m(x) = 2 f ехр(Г2b{z)a'2(z)dz)
k(x) = / m{y)s(dy).
Jo
-1(y)dy,
The reader is invited to look at Exercise C.20) in Chap. VII for the interpretations
of s and m.
1°) If f/ is the unique solution of the differential equation ~cr2U" + bU' = U
such that U{0) — 1 and U'@) = 0, check that
f
Jo
and prove that
U(x) = 1 + / ds(y) f U(z)dm(z)
Jo J
1 +k < U <exp(k).
2°) If e is the explosion time, prove that exp(—/ Ae)U(XtM), where X is the
solution to ex{o,b), is a positive supermartingale for every x. Conclude that if
k(—oo) = k(+oo) — oo, then Px[e = oo] = 1 for every x.
3°) If as usual 7b - inf {t : X, = 0}, prove that exp(-/ л T0)U(XtATl)) is a
bounded /^-martingale for every x and conclude that if either k(—oo) < oo or
fc(oo) < oo, then Px[e < oo] > 0 for every x.
4°) Prove that if Px[e < oo] — 1 for every x in R, then one of the following
three cases occurs
i) k{—oo) < oo and Ц+oo) < oo,
ii) k{—oo) < oo and s(+oo) = oo,
iii) &(+oo) < oo and s(—oo) = oo.
5°) Assume to be in case i) above and set
G(x,y) = (s{x) - s(-oo))(s(+oo) - s(y))/(s(+oo) - s(-oo)) ifx<y
= G(y,x) ify<x,
and set U\ {x) = / G(x, y)m{y)dy (again see Sect. 3 Chap. VII for the rationale).
Prove that U\(XtM) +l Ae is a local martingale and conclude that Ex[e] < oo,
hence Px[e < oo] = 1.
6°) Prove that in the cases ii) and iii), P[e < oo] = 1.
B.16) Exercise. 1°) Prove that for J} > 0, the solution V, to the Langevin equation
is the OU process with parameter 0 and size 1/2$.

§2. Existence and Uniqueness in the Case of Lipschitz Coefficients 387
2°) Prove that if Xo is independent of B, the solution V such that Vb = Xo is
stationary if Xo ~ . V'{0, 1/2/8). In that case
where В and Г are two standard linear BM's such that Bu = иГ\/и.
** B.17) Exercise. Exceptionally in this exercise we do not ask for condition i) of
Definition A.2) to be satisfied.
1°) Prove that the stochastic equation
X, = B, + f — ds
Jo s
has a solution for / e [0, 1].
[Hint: Use a time reversal in the solution to the equation of Exercise B.13).]
2°) Prove that for 0 < e < t < 1
' f' dBu
e Je и
3°) Prove that X is not J*"B-adapted.
[Hint: If the solution were strong, the two terms on the right-hand side of the
equality in 2°) would be independent. By letting e tend to zero this would entail
that ?[exp(j'AX,)] is identically zero.]
4°) By the same device, prove that if ф is a locally bounded Borel function
on ]0, oo[, the equation
X, = Bt + f 4,(s)X,ds
Jo
does not have an .Ув -adapted solution as soon as
/ exp B I 4>(s)ds\du = 00.
B.18) Exercise (Laws of exponentials of BM). Let 0 and у be two independent
idard linear BM's and for /
1°) Prove that the process
0Г') (xo + / exp{-0У
' \ ./0
is for every real jco, a diffusion with infinitesimal generator equal to
(A + jc2)/2) d2/dx2 + (х(ц + A/2)) + v)d/dx.
2°) Prove that X has the same law as sinh(F,), / > 0, where У is a diffusion
with generator
(\/2)d2/dy2 + (/x tanhC>) + vcoshty)-1) d/dy.
standard linear BM's and for ц, v in К set Д.(д) = 0, + /xt and y/v) = y, + vt.

388 Chapter IX. Stochastic Differential Equations
3°) Derive from 2°) that for fixed t,
Jo
In particular we get (Bougerol's identity): for fixed t,
i:
exp(Ps)dys = sinh(ft).
Jo
[Hint: For a Levy process X, the processes Xs, s < t, and X, — X,-s, s < t,
are equivalent.]
4°) Prove that on .3^, when v — 0 and Yo = 0, the law fM of Y has the density
cosh(X,)Mexp(-((/x - д2)/2) / cosh(Xs)'2ds\ exp(-(^i2/2)t)
with respect to the Wiener measure. Prove as a consequence that for fixed /,
exp(Ps + s)dys = sinh(/8, + et),
where e is a Bernoulli r.v. independent of p.
[Hint: See case i) in 2°) Exercise C.18) Chapter VIII.]
B.19) Exercise (An extension of Levy's equivalence). 1°) Let X e M. Prove
that pathwise uniqueness holds for the one-dimensional equation e(\,bx), where
bx(x) = -A.sgn(jc).
[Hint: Use Exercise B.11)].
2°) We denote by Xх a solution to e(\,bk) starting from 0, and by BK a
Brownian motion with constant drift k, starting from 0, and Sf = supJ<r B^.
Prove that:
(Sf- B^t >0) = {\X^\,t >0)
[Hint: Use Skorokhod's lemma B.1) in Chap. VI].
§3. The Case of Holder Coefficients in Dimension One
In this section, we prove a partial converse to Theorem A.7) by describing a
situation where uniqueness in law entails pathwise uniqueness, thus extending the
scope of the latter property. We rely heavily on local times and, consequently,
have to keep to dimension 1. We study the equation e(a,b) where a is locally
bounded on R+ x R and consider pairs of solutions to this equation defined on the
same space and with respect to the same BM. We recall that any solution X is a
continuous semimartingale and we denote by LX{X) the right-continuous version
of its local times.
What may seem surprising in the results we are about to prove is that we get
uniqueness in cases where there is no uniqueness for ODE's (ordinary differential

§3. The Case of Holder Coefficients in Dimension One 389
equations). This is due to the regularizing effect of the quadratic variation of BM
as will appear in the proofs. One can also observe that in the case of ODE's, the
supremum of two solutions is a solution; this is usually not so for SDE's because
of the appearance of a local time. In fact we have the following
C.1) Proposition. If Xх and X2 are two solutions ofe(a, b) such that Xx0 = X%
a.s., then Xх v X2 is a solution if and only ifL°(Xl - X2) vanishes identically.
Proof. By Tanaka's formula,
X) v X2 = X) + {X2 - Xj)+ = X) + f hx>>x])d (X2 - Xl)s
Jo
+ 1l?(x2-x').
and replacing X1, i = 1,2 by X'o + fo'a(s, X[)dBs + fQ b(s, X's)ds, it is easy to
check that
X) v X2 = (Xl0 vXl)+ f a (s, X\ v X,2) dBs+ f b (s, Xxs v X]) ds
Jo Jo
which establishes our claim. D
The following result is the key to this section.
C.2) Proposition. If uniqueness in law holds for e(o, b) and ifL°(Xl — X2) = 0
for any pair (X1, X2) of solutions such that Xq = Xq a.s., then pathwise uniqueness
holds for e(<7, b).
Proof. By the preceding proposition, if Xх and X2 are two solutions, X1 v X2 is
also a solution; but X1 and X1 v X2 cannot have the same law unless they are
equal, which completes the proof. ?
The next lemma is crucial to check the above condition on the local time. In
the sequel p will always stand for a Borel function from ]0, oo[ into itself such
that /0+ da/p{a) = +oo.
C.3) Lemma. If X is a continuous semimartingale such that, for some e > 0 and
every t
A, = f ll0*x,<,)p(Xs)-ld{X, X)s < oo a.s.,
Jo
then L°(X) = 0.
Proof. Fix t > 0; by the occupation times formula (Corollary A.6) Chap. VI)
A, =
Jo
If L°t(X) did not vanish a.s., as Ц(X) converges to L®(X) when a decreases to
zero, we would get A, = oo with positive probability, which is a contradiction.

390 Chapter IX. Stochastic Differential Equations
C.4) Corollary. Let bh i = 1, 2 be two Borel functions; if
\a(s,x)-o(s,y)\2 <p(\x-y\)
for every s,x, у and if X', i = 1,2, are solutions to e(o, bj) with respect to the
same BM, then L°(Xl - X2) = 0.
Proof. We have
X) - A-,2 = A-' - A-2 + f (o(s, X]s) - o(s, X2)) dBs
Jo
+ f (bi(s,X})-b2(s,X*))ds,
Jo
and therefore
f p(X] - X2rll(Xl>X2}d(Xl - A, A - A)v
Jo
= I p{X\ - A) (a(s, A"') - a(s, A)J l&^
П
We may now state
C.5) Theorem. Pathwise uniqueness holds for e(cr, b) in each of the following
cases:
i) \o(x) — o(y)\2 < p(\x - y\), \a\ > e > 0 and b and о are bounded;
ii) \cr(s, x) — a(s, y)\2 < p{\x — y\) and b is Lipschitz continuous i.e., for each
compact H and each t there is a constant K,, such that for every x, у in H
and s < t
\b(s,x)-b(s,y)\ <K,\x-y\;
Hi) \o(x) — o(y)\2 < \f(x)-f(y)\ where f is increasing and bounded, a > e > 0
and b is bounded.
Remark. We insist that in cases i) and iii) a does not depend on s, whereas in ii)
non-homogeneity is allowed.
Proof, i) By Corollary A.14), since |a| > e, uniqueness in law holds for e(o, b).
The result thus follows from Proposition C.2) through Corollary C.4).
ii) Let Xх and A be two solutions with respect to the same BM and such that
Xx0 = Xq a.s. By the preceding corollary,
|A",1 -X2\= [ sgn(X] - X])d[X\ - X2).
Jo
The hypothesis made on a and b entails that we can find a sequence (Т„) of
stopping times converging to oo, such that, if we set Y' = (A"'O", i = 1,2, for
fixed n, then о (s, Y^) is bounded and

§3. The Case of Holder Coefficients in Dimension One 391
Ms,Y})-b(s,Y?)\<C,\Ysl-Y?\
for s < t and some constant C,. As a result,
\< ~ У'\ - f sgn (К,1 - К2) (b (s, Y}) - b (s, Y})) ds
Jo
is a martingale vanishing at 0 and we have
Using Gronwall's lemma, the proof is now easily completed.
iii) By Corollary A.14) again, the condition a > s implies uniqueness in law
for e(a, b); we will prove the statement by applying Corollary C.4) with p(x) = x
and Proposition C.2). To this end, we pick a S > 0 and consider
llw_X2>S)d(Xl - X\ Xх - X2)
\j (X\ - X2srllw_X2>S)d
< E [jf (f{X\) - f{X2s)) {X\ - Xs2
We now choose a sequence {/„} of uniformly bounded increasing С '-functions
such that limn fn{x) — f(x) for any x which is not a discontinuity point for
/. The set D of discontinuity points for / is countable; by the occupation times
formula, the set of times s such that X) or X2S belongs to D has a.s. zero Lebesgue
measure, and consequently
lim(/n(X') - /„(X?)) = f{X]) - /(X,2)
for almost all s < t. It follows that K(f), = limn К(/„),.
For и € [0, 1], set Z" = X2 + м(Х" - X2); we have
K(fn), =
* f!ifo f-iz")ds]du
because Z,"(w) = Z(a>) + f^au(s,a>)dBs(co) + /„' b"(s,co)ds where o" > e.
Moreover, \a"\ + \b"\ < M for a constant M; by a simple application of Tanaka's
formula
sup?[L?(ZM)] = C <oo,
а,и
and it follows that

392 Chapter IX. Stochastic Differential Equations
K(fn), <e-2Csup||/n||.
n
Hence K(f), is bounded by a constant independent of 8; letting 8 go to zero, we
see that the hypothesis of Lemma C.3) is satisfied for p(x) = x which completes
the proof.
Remarks. 1°) In iii) the hypothesis a > e cannot be replaced by \o\ > e; indeed,
we know that there is no pathwise uniqueness for e(cr, 0) when o{x) = sgn(x).
2°) The significance of iii) is that pathwise uniqueness holds when a is of
bounded quadratic variation. This is the best that one can obtain as is seen in
Exercise A.16).
3°) The hypothesis of Theorem C.5) may be slightly weakened as is shown
in Exercises C.13) and C.14).
At the beginning of the section, we alluded to the difference between SDE's
and ODE's. We now see that if, for instance, a(x) — <J\x\, then the ODE
dX, — a{X,)dt has several solutions whereas the SDE e(a,b) has only one.
The point is that for SDE's majorations are performed by using the increasing
processes of the martingale parts of the solutions and thus it is (o(x) — cr(y)J
and not \a(x) — o{y)\ which comes into play. We now turn to other questions.
A consequence of the above results together with Corollary A.14) is that for
bounded b, the equation e{\,b) has always a solution and that this solution is
strong (another proof is given in Exercise B.11)). We now give an example,
known as Tsirel'son example, which shows that this does not carry over to the
case where b is replaced by a function depending on the entire past of X.
We define a bounded function г on K+ x W in the following way. For a strictly
increasing sequence (tk,k € —N), of numbers such that 0 < tk < 1 for к < 0,
to = 1, lim^-oo tk = 0, we set
\w(tk) - w(tk-\)~\
= —
L h-h-\ J
= 0 if t = 0 or / > 1,
where [x] is the fractional part of the real number x. This is clearly a predictable
function on W. If (X, B) is a solution to e(l, r), then on ]tk, tk+\]
X, - X,k = B, Bh + ГХ;*~Х"-'1 (t - tk);
I tk - h-\ J
if we set for tk < t < tk+\,
r,, = {X, - A-,,) l(t - tk), e, = (B, - B,k) /(t - tk),
we have

§3. The Case of Holder Coefficients in Dimension One 393
C.6) Proposition. The equation e(l, z) has no strong solution. More precisely
i) for every t in [0, 1], the r.v. [r]t] is independent of.^xB and uniformly dis-
distributed on [0, 1];
ii) for anyO < s <t, .^x = a ([r?,]) v ,Wt B.
Proof. The first statement is an obvious consequence of properties i) and ii), and
ii) follows easily from the definitions. We turn to proving i).
Let p e Z - {0} and set dk = E [exp {2/л-р^}]; by (*), we have
dk = E[cxp{2inp(elk + [rll
— dk-iE [exp (linpetk)] = dk-\ ехр{-2л-2/?2 (tk - tk-{I),
because e,k is independent of Д_,, where (,Yt) is the filtration with respect to
which (X, B) is defined. It follows that
Ш < |flk-i|exp{-27rV} < ... < \dk_n\exp{-lnn2p2},
and consequently, dk = 0 for every k. This proves that [r],k] is uniformly distributed
on [0, 1].
Define further .j$? = a {Bu - Bv,tn < и < v < tk); then
E [
= E [exp [linp (etk + eh_, + ... + е„_„+1 + пп-.)} \Л~"\
= exp {linp (e,,+... + elt_n+l)} dk-n
since r)tk_n is independent of./%~". The above conditional expectation is thus zero;
it follows easily that E [exp {2inpri,k\ \ ^kB] = 0 and since ,W,k is independent
of [B, - Blk,t >tk},
Finally for tk < t < tk+\, we have
E [exp{2inpr,,} | ,7[B\ = exp {2inpe,} E [exp {2inpr,lt} | .^B] - 0
and this being true for every p Ф 0, proves our claim.
Remark. As a result, there does exist a Brownian motion В on a space (?2,.%, P)
and two processes Xх, X2 such that (X!, B), i = 1, 2, is a solution of e(l, r). The
reader will find in Exercise C.17) some information on the relationship between
Xх and X1. Moreover, other examples of non existence of strong solutions may
be deduced from this one as is shown in Exercise C.18).
We will now use the techniques of this section to prove comparison theorems
for solutions of SDE's. Using the same notation as above, we assume that either
(a(s, x) — a(s, y)J < p(\x — y\) or that a satisfies hypothesis iii) of Theorem
C.5).

394 Chapter IX. Stochastic Differential Equations
C.7) Theorem. Let b', i = 1, 2, be two bounded Borelfunctions such that bl > b2
everywhere and one of them at least satisfies a Lipschitz condition. IfX', / = 1,2,
are solutions to e{a, b') defined on the same space with respect to the same BM
and if X^ > Xl a.s., then
P[XJ >X2forallt >0] = 1.
Proof. It was shown in Corollary C.4) and in the proof of Theorem C.5) that, in
each case, L°(Xl - X2) = 0, and therefore
l(xi>xl)(b2(s,X2)-bl(s,Xl))ds]
< E |jf 1(X?>4) (b^s, X2) - b\s, Xl))
Thus, if bl is Lipschitz with constants Kt,
ф@ < K,E \j \{X]>xl) \X2 - X'l^l = K, f <l>(s)ds,
and we conclude by using Gronwall's lemma and the usual continuity arguments.
If b2 is Lipschitz, using the same identity again, we have
e\J l(X>>xi)\b2(s, X2) - b2(s, X])\ds
+ f \(Xl>xl) (b2(s, Xl) - bl(s, Xl)) ds
Jo
E U 1{Хг>х1) \b2(s, X2) - b2(s, X\)\ds
since b2 < b\ and we complete the proof as in the first case. D
With more stringent conditions, we can even get strict inequalities.
C.8) Theorem. Retain the hypothesis of Theorem C.7). If the functions b' do not
depend on s and are continuous and a is Lipschitz continuous and if one of the
following conditions is in force:
i) bx > b2 everywhere,
ii) \o | > e > 0, either bl or b2 is Lipschitz and there exists a neighborhood V(x)
of a point x such that
I lF'(a)=62(a))^a = 0,
JV(x)
then, if(X', B) is a solution ofex(a, b'),
P [Xj > Xf for all / > 0] = 1.

§3. The Case of Holder Coefficients in Dimension One 395
Proof. In case i) one can suppose that either b1 or b2 is Lipschitz continuous,
because it is possible to find a Lipschitz function b3 such that bx > b3 > b2.
We now suppose that bl is Lipschitz, the other case being treated in similar
fashion. We may write
where
H,= [ (bl(X2)-b2(X2))ds
Jo
and
Jo
The hypothesis made on a and b' entails that H and M are continuous semi-
martingales; by Proposition B.3) we consequently have
f
Jo
and it is enough to prove that for every t > 0, H, > 0 a.s. This property is
obviously true in case i); under ii), by the occupation times formula,
e~2
/• + 00
/ (bx(a) - b2(a)) Lat{X2)da.
J-oo
If L*(X2) > 0 for all t > 0, the result will follow from the right-continuity in a
of L°. Thus we will be finished once we have proved the following
C.9) Lemma. If X is a solution ofex(o,b) and, moreover \a\ > e > 0, then
almost surely, Lxt{X) > Ofor every t > 0.
Proof. We may assume that X is defined on the canonical space. By Girsanov's
theorem, there is a probability measure Q for which X is a solution to e(a, 0).
The stochastic integrals being the same under P and Q, the formula
\X,-x\= f sgn(A-.5 - x)dXs + L*
Jo
shows that the local time is the same under P and Q. But under Q we have
X, =x+ f o(s,Xs)dBs,
Jo
hence X = fiAt where /8 is a BM(jc) and A, a strictly increasing process of time-
changes. The result follows immediately. D

396 Chapter IX. Stochastic Differential Equations
# C.10) Exercise (Stochastic area). Г) Give another proof of 3°) in Exercise
B.19) Chap. V in the following way: using the results of this chapter, prove that
(•&fp) С (.^^) where f} is defined within the proof of Theorem B.11) Chap. V
and compute (S, S) and E, f}).
2°) Prove that
E[eas>] =
where P is a linear BM. The exact value of this function of Л and t is computed
in Sect. 1 Chap. XI.
* C.11) Exercise. (Continuation of Exercise D.15) of Chap. V). Let Tt" be the
time-change associated with {Mn, M").
1°) Let Zr" = B"tl and prove that
(Z,"J=cn
Jo
where с„ and dn are two constants and /3" is the DDS Brownian motion of M".
2°) If n is odd, prove that Mn is pure.
[Hint: Use Theorem C.5) to show that T" is ,J^°-measurable.]
If n is even it is not known whether Mn is pure.
C.12) Exercise. 1°) Retain the notation of Exercise A.27) Chap. VIII and suppose
that (.3^") = (.^B) where В is a BM under P. Let т be the Tsirel'son drift and
define
Set
\BA))ds,
./o
and call Tt the inverse of A. Using Exercise A.15) prove that Gp(B)r is not pure,
hence that a Girsanov transform of a pure martingale may be not pure.
2°) In the situation of Proposition C.6), let U, be the inverse of the process
/„' A + t(s, X,)) ds and set M, = Ви,. Prove that M is not pure although .j^f =
a (•3^M,^OB) for every e > 0. This question is independent of the first. The
following question shows that the situation is different if we replace purity by
extremality.
3°) Let M be a («^")-loc. mart, with the following representation property: for
every e > 0 and every X e L2(,3^c) there is a suitable predictable process Фе
such that
X = E[X\m]+\ 0s(s)dMs.
Je
Prove that M has the

§3. The Case of Holder Coefficients in Dimension One 397
C.13) Exercise. Prove that Theorem C.5) is still true if in iii) we drop the hy-
hypothesis that / is bounded or if we replace the hypothesis a > e by : for every
r > 0, there is a number er > 0 such that a > er on \—r, r].
C.14) Exercise. Prove that parts, i) and ii) of Theorem C.5) are still true if the
hypothesis on a reads: there are locally integrable functions g and с and a number
S > 0 such that for every x, for every у € [x - S, x + <5],
(cr(s, x) - cr(s, y)f < (c(s) + g(x)cr\s, x)) p(\x - y\).
C.15) Exercise. Let у be a predictable function on W and т be the Tsirel'son
drift; define
Y(s, w) = x{s, w) + у I s, w — I t(m, w)du I .
\ Jo )
Let (X, B) be a solution to eo{l,T) on the space (П,,^, Р).
1°) If p = X - /0 z(u, X)du, prove that (,5fB) с (J^).
2°) Find a probability measure Q on (П,.^), equivalent to P, for which /3
is a BM and (X, P) a solution to eo(l> *). Derive therefrom that (X, B) is not a
strong solution to eo(l, T).
# C.16) Exercise. 1°) If В is a standard linear BM, prove that the process Z, — B?
satisfies the SDE
Jo
and derive therefrom another proof of the equality J^" = ,3^~ of Corollary B.2)
in Chap. VI.
2°) More generally if В is a BMrf@), show that \B\ and the linear BM
have the same filtration (see Sect. 3 Chap. VI and Sect. 1 Chap. XI).
3°) If Л is a symmetric d x ^-matrix and В a BMd@), prove that the local
martingale fo{ABs, dBs) has the same filtration as a BMr where r is the number of
distinct, non zero eigenvalues of A. In particular prove that a planar BM (В1, В2)
has the same filtration as (IB1 + B2\, \B} - B2\).
[Hint: Use Exercises A.36) and C.20) in Chap. IV.]
C.17) Exercise. 1°) Suppose given a Gaussian vector local martingale В in M.d on
a space {П, .J^~, P) such that (B1, BJ), = ptjt with p,,, = 1. For each i, we suppose
that there is an (.^")-adapted process X1 such that (X', B') is a solution of e(l, т)
with Xq = Bq = 0. With obvious notation derived from those in Proposition C.6)
prove that the law of the random vector [r],] = (W,], i = 1,..., d) is independent
of t and is invariant by the translations л:,- —*¦ xj + и, (mod. 1) if ?] /?,м,- — 0 for

398 Chapter IX. Stochastic Differential Equations
any (pt) e Zd such that ? PtjPiPj = 0. Prove further that this random variable
is independent of J^"B.
2°) Suppose from now on that all the components B' of В are equal to the
same linear BM fi, and let a be a vector random variable independent of .^,
whose law is carried by ([0, 1 [)d and is invariant under translations дг,- -* x,¦ + и
(mod. 1) for every «el (not Kd!). Set [??,_,] = a and prove that one can define
recursively a unique process r\ such that
It = e, +[ritk] fort e]tk,tk+l].
For any t, the vector random variable [tj,] is independent of .^*.
3°) Prove that the family of ст-algebras ^ = .5^* v<r([(|,]) is a nitration and
that /J is a (.f^,)-Brownian motion.
4°) If, for / e ]th tl+l], we define
the process X is (^f)-adapted and (X1, f}) is for each i a solution to e(l, r).
5°) Prove that for any Z e L2(.^[x, P), there is a (.^"*)-predictable process
ф such that E f0 <j^:ds < oo and
= Z0+
Jo
where Zo is .^x-measurable.
* C.18) Exercise. This exercise is designed to give an example of an SDE without
drift and with the same kind of properties as the Tsirel'son equation. With the
same sequence (tk) as for the Tsirel'son drift we define a predictable function /
onWby
f(t,w) = sgn(w(tk)-w(tk-i)) iftk<t< tkJrX.
1°) Let В be an (,^)-BM on (Q,.TU P) and (X',B),/ = 1,2, ...,n be
solutions to eo(/, 0); if we define r\ by
r]\: = sgn(x;-x;t) \uk<t<tk+u
then, for any t e ]0, 1], the law of r\, is invariant by the symmetry {x\,..., xn) ->
(-*!,..., — xn), tj, is independent of .9^B and almost-surely
ff(Xv,J < 0 =^B vcr(^) for any s<t.
2°) Conversely, if a is a r.v. taking its values in {—1, 1}", which is •im-
•immeasurable, independent of .^B, and such that its law is invariant by the above
symmetry, then there exists a filtration (,^) on ?2 and a (-^-adapted process X
such that:
i) В is a
ii) for each /, (X'\ B) is a solution to eo(f, 0) and a' = sgn(X;o - X'Ui).

Notes and Comments 399
C.19) Exercise. (Continuation of Exercise B.24) of Chap. VI). With obvious
definition prove that there is path uniqueness for the equation (*).
[Hint: One can use Exercise A.21) Chapter VI.]
C.20) Exercise. Retain the situation and notation of Theorem C.8).
1°) For /0 as in Lemma C.3), prove that, for every t > 0,
/'
Jo
p(X\ - X2) ]ds = oo a.s.
[Hint: Use the expression of X1 — X2 as a function of M and H given in the
proof of Theorem C.8).]
2°) If in addition b\ —b2 > a > 0 and if now p is such that /0+ p(u)~]du < oo,
then
f p(X] - X2)~]ds < oo a.s.
Jo
** C.21) Exercise. Retain the situation and notation of Theorem C.7) and suppose
that a and b\ satisfy some Lipschitz conditions. Prove that when a tends to zero,
for every t > 0 and s e ]0, l/2[,
L1(X] - X2) = 0(a1/2"?) a.s.
[Hint: Use Corollary A.9) in Chap. VI and the exponential formulas of the
proof of Theorem C.8).]
Notes and Comments
Sect. 1. The notion of stochastic differential equation originates with ltd (see ltd
[2]). To write this section, we made use of Ikeda-Watanabe [2], Stroock-Varadhan
[1] and Priouret [1]. For a more general exposition, see Jacod [2]. The important
Theorem A.7) is due to Yamada and Watanabe [1]. The result stated in Remark 2)
after Theorem A.7) was proved in Kallenberg [2].
Exercise A.15) is taken from Stroock-Yor [1] and Exercise A.16) from Gir-
sanov [2]. The result in Exercise A.18) is due to Perkins (see Knight [7]). Pathwise
uniqueness is a property which concerns all probability spaces. Kallsen [1] has
found a TsireFson-like example of an SDE which enjoys the existence and unique-
uniqueness properties on a particular space but not on all spaces.
Sect. 2. As for many results on SDE's, the results of this section originate with ltd.
Theorem B.1) was proved by Doleans-Dade [1] for general (i.e. non continuous)
semimartingales. Proposition B.3) comes from an unpublished paper of Yoeurp
and Yor.
Theorem B.4) and its corollaries are taken from Neveu [2] and Priouret [1],
but of course, most ideas go back to ltd. Theorem B.4) is the starting point for
the theory of flows of SDE's in which, for instance, one proves, under appropriate

400 Chapter IX. Stochastic Differential Equations
hypothesis, the differentiability in x of the solutions. It also leads to some aspects
of stochastic differential geometry. An introduction to these topics is provided by
the lecture course of Kunita [4].
Exercise B.6) is taken from Jacod [3] and Karandikar [1].
Exercise B.8) is due to Doss [1] and Sussman ([1] and [2]). Exercise B.9)
is from Yoeurp [3] and Exercise B.10) is taken in part from Ikeda-Watanabe [2].
Exercise B.12), inspired by Jeulin-Yor [2] and Yor [10], originates with Ito [6] and
provides a basic example for the theory of enlargements of filtrations. Exercise
B.14) is taken from El Karoui and Chaleyat-Maurel [1]. Exercise B.15) describes
results which are due to Feller; generally speaking, the contribution of Feller to
the theory of diffusions is not sufficiently stressed in these Notes and Comments.
Exercise B.17) is very close to Chitashvili-Toronjadze [1] and is further de-
developed in Jeulin-Yor [4]. It would be interesting to connect the results in this
exercise with those of Carlen [1] and Carlen-Elworthy [1]. For Exercise B.18)
see Bougerol [1] and Alili [1], Alili-Dufresne-Yor [1]; related results, for Levy
processes instead of BM with drift, are found in Carmona et al. [1].
Some explicit solutions to SDE's are exhibited in Kloeden-Platen [1] using
Ito's formula.
The result of Exercise B.19) may be found in Fitzsimmons [1] who studies in
fact a converse to Levy's equivalence.
Sect. 3. Our exposition of Theorem C.5) is based on Le Gall [1] who improved
earlier results of Nakao [1] and Perkins [5]. Problems of stability for solutions of
such one-dimensional SDE's are studied in Kawabata-Yamada [1] and Le Gall [1].
The proof given here of the Tsirel'son example is taken from Stroock-Yor [1]
and is inspired by a proof due to Krylov which is found in Liptser-Shiryayev [ 1 ].
Benes [1] gives another proof, as well as some extensions. The exercises linked to
this example are also mainly from Stroock-Yor [ 1 ] with the exception of Exercise
C.17) which is from Le Gall-Yor [1]. Further general results about Tsirel'son's
equation in discrete time are developed in Yor [19].
For the comparison theorems see Yamada [1], Ikeda-Watanabe ([1] and [2])
and Le Gall [1], but there are actually many other papers, too numerous to be
listed here, devoted to this question.
Exercise C.10) is from Williams [4] and Yor [11] and Exercise C.11) is a
result of Stroock-Yor [2]. Exercise C.16) is taken from Yor [7]; with the notation
of this exercise let us mention the following open
Question 1. If in 3°) the matrix A is no longer supposed to be symmetric, is the
filtration of the martingale still that of a BMr and, in the affirmative, what is r in
terms of A?
A partial answer is found in Auerhan-Lepingle [1]; further progress on this
question has been made by Maine [1].

Chapter X. Additive Functionals of Brownian Motion
§1. General Definitions
Although we want as usual to focus on the case of linear BM, we shall for a while
consider a general Markov process for which we use the notation and results of
Chap. III.
A.1) Definition. An additive functional ofX is a R+-valued, {.^)-adapted process
A = {A,, t > 0} defined on Q and such that
i) it is a.s. non-decreasing, right-continuous, vanishing at zero and such that A, =
A(_ on {f < t};
ii) for each pair (s, t), As+t = A, + As о в, a.s.
A continuous additive functional (abbreviated CAF) is an additive functional such
that the map t —> A, is continuous.
Remark. In ii) the negligible set depends on s and t, but by using the right-
continuity it can be made to depend only on t.
The condition A, = Af_ on {? < t} means that the additive functional does
not increase once the process has left the space. Since by convention f(A) = 0
for any Borel function on E, if Л is a Borel subset of E, this condition is satisfied
by the occupation time of Г, namely A, = J^ \r{Xs)ds, which is a simple but
very important example of a CAF. In particular А, =(л(, which corresponds to
the special case Г = E, is a CAF.
Let X be a Markov process with jumps and for s > 0 put
Te = inf{t >0:d(X,,Xt) > e].
Then Te is an a.s. strictly positive stopping time and if we define inductively a
sequence (Г„) by
the reader will prove that A, = Y1T 1(г„<о is a purely discontinuous additive
functional which counts the jumps of magnitude larger than e occuring up to
time t.
We shall now give the fundamental example of the local time of Brown-
Brownian motion which was already defined in Chap. VI from the stochastic calculus

402 Chapter X. Additive Functionals of Brownian Motion
point of view. Actually, all we are going to say is valid more generally for lin-
linear Markov processes X which are also continuous semimartingales such that
(X, X), = fQ'<t>(Xs)ds for some function ф and may even be extended further by
time-changes (Exercise A.25) Chap. XI). This is in particular the case for the OU
process or for the Bessel processes of dimension d > 1 or the squares of Bessel
processes which we shall study in Chap. XI. The reader may keep track of the
fact that the following discussion extends trivially to these cases.
We now consider the BM as a Markov process, that is to say we shall work
with the canonical space W = C(R+, R) and with the entire family of probability
measures Pa, a e Ш.
With each Pa, we may, by the discussion in Sect. 1, Chap. VI, associate a
process L which is the local time of the martingale В at zero, namely, such that
\Bt\ = \a\+ f
Jo
sgn(Bs)dBs + L, Pa-a.s.
Actually, L may be defined simultaneously for every Pa, since, thanks to Corollary
A.9) in Chap. VI
1 f
L,=hm— l^ek_ek[(Bs)ds a.s.,
* ¦"*: JO
where {e^} is any sequence of real numbers decreasing to zero. The same discussion
applies to the local time at a and yields a process La. By the results in Chap. VI,
the map (a, t) ->¦ Lat is a.s. continuous.
Each of the processes La is an additive functional and even a strong additive
functional which is the content of
A.2) Proposition. If T is a stopping time, then, for every a
LaT+s = LaT + Las(9T) Pb-a.s.
for every b and every positive random variable S.
Proof. Set /(e) =]a - e, a + e[; it follows from Corollary A.9) in Chap. VI that
if Г is a stopping time
1 fs
LaT+s = LaT + Hm — / l/(o (Bu@t))du Pb-a.s.
for every b. By the Strong Markov property of BM
Рь \щвт) = limi- / l/(e) (Bu@t))du]
|_ e|0 Z? Jq J
= Eb [fBr \l°s = Hm 1 jf ll(e)(BH)dMjj = 1.
Consequently,
a.s.

§1. General Definitions 403
A case of particular interest is that of two stopping times T and S. We saw
(Proposition C.3) Chap. Ill) that T + S о вт is again a stopping time and the
preceding result reads
We draw the attention of the reader to the fact that Las о вт is the map ш ->•
L"s(eT(w)) @r(<*>)), whereas the LasFT) in the statement of the proposition is the
mapo>-+ LaS(oj)@T(w)).
The upcoming consequence of the preceding result is very important in Chap.
XII on Excursion theory as well as in other places. As in Chap. VI, we write т,
for the time-change associated with L, = L®.
A.3) Proposition. For every t, there is a negligible set Г, such that for w ? Г,
and for every s > 0,
t,+s((o) = r,(w) + ts (вг,(a))), r(,+j)- (w) = т,(w) + r,- @T, (w)).
Proof. Plainly, r, is a.s. finite and r,+s > r,. Therefore
r,+J = inf [u > 0 : Lu > t + s}
= т, + inf {m > 0 : Ltl+U > t + s].
Using the strong additivity of L and the fact that LXl = t, we have that, almost-
surely, for every s,
r,+s = r, + inf {и > 0 : LTl + Lu{6Xl) > t + s]
— r, + inf {m > 0 : Lu(eTl) > s) = r, + г., о9Ь.
The second claim follows at once from the first. О
Remarks. 1°) The same result is clearly true for La in place of L. It is in fact true
for any finite CAF, since it is true, although outside the scope of this book, that
every additive functional of a Markov process has the strong additivity property.
2°) Since the processes (L,) and (S,) have the same law under Po (Sect. 2
Chap. VI), it follows from Sect. 3 Chap. Ill that the process (r,) is a stable
subordinator of index 1 /2. This may also be proved using the above result and
the strong Markov property as is outlined in Exercise A.11).
3°) Together with Proposition C.3) in Chap. HI, the above result entails that
eTl+< — вь о 0Xs a.s.
Our goal is to extend these properties to all continuous additive functionals
of linear BM. This will be done by showing that those are actually integrals of
local times, thus generalizing what is known for occupation times. To this end, we
will need a few results which we now prove but which, otherwise, will be used
sparingly in the sequel.
Again, we consider a general Markov process. By property i) of the definition
of additive functionals, for every w, we can look upon А,{ш) as the distribution
function of a measure on R+ just as we did for the increasing processes of Sect. 1

404 Chapter X. Additive Functionals of Brownian Motion
Chap. IV. If F, is a process, we shall denote by /0' FsdAs its integral on [0, t] with
respect to this measure provided it is meaningful. Property ii) of the definition is
precisely an invariance property of the measure dAs.
A.4) Proposition. If f is a bounded positive Borel function and A, is finite for
every t, the process (f ¦ A) defined by
(/ • A), = / f(Xs)dAs
Jo
is an additive functional. If A is a CAF, then (/ • A) is a CAF.
Proof. The process X is progressively measurable with respect to 0>O, thus so
is f(X) and as a result, (/ • A), is .^"-measurable for each t. Checking the other
conditions of Definition A.1) is easy and left to the reader. ?
Remarks. 1°) The hypothesis that / is bounded serves only to ensure that t —>
{f ¦ A), is right-continuous. If / is merely positive and (/ • A) is right-continuous,
then it is an additive functional. The example of the uniform motion to the right
and of f{x) = (l/jc)l(jt>0) shows that this is not always the case.
2°) If A, - t A t, and / = lr, then (/ • A), is the occupation time of Г by X.
A.5) Definition. Fora > 0, the function
is called the a-potential of A. We also write UAf(x)for the a-potential of f ¦ A,
in other words
UaAf(x) = Ex \f e-a'f{Xt)dA,] .
For A, — t л?, we have UaAf = Ua f, and, more generally, it is easily checked
that the map (x, Г) —> UAlr(x) is a kernel on (?\ rf) which will be denoted
U%(x, ¦). Moreover, these kernels satisfy a resolvent-type equation.
A.6) Proposition. Fora, $ > 0, ifVaAf{x) and UpAf(x) are finite,
UaAf(x) - Vif(x) = W- a)UaU?Af(x) = 08 - a
Proof. Using the definitions and the Markov property
UaUpAf(x) = Ex
= Ex
dt
¦ лоо /-oo -i
/ e-a'dt / e-*f(X1+l)dA,(e,)\.
Jo Jo J

§1. General Definitions 405
By property ii) in Definition A.1), we consequently have
' POO y»OO
/ e~atdt / e-fisf(Xs+t)dAs+,
.Jo Jo
Uau'Af(x) = Ex
" poo pc
= Ex / e-(a'^dt \
YJo h
and Fubini's theorem yields the desired result. D
We will prove that the map which with A, associates the family of kernels
UA(x, •) is one-to-one. Let us first mention that as usual A = В will mean that A
and В are indistinguishable i.e. in this context, Px [3t : А, Ф B,] — 0 for every x.
Because of the right-continuity, this is equivalent to Px [A, = Bt] = 1 for every
x and t. The following proposition will be used in §2.
A.7) Proposition. If A and В are two additive fimctionals such that for some
a>Q,UaA=UaB<oo and UaAf = UaBf for every f e C+, then A = B.
Proof. The hypothesis extends easily to UAf = Ugf for every / e b%+. In
particular UAUaf = UgUaf. But computations similar to those made in the
preceding proof show that
UaAUaf(x) = Ex \Г e^atf{X,)A,dt\
and likewise for B. As a result,
pOO pOO
/ e-a'Ex[f{X,)At]dt= e~a'Ex[f(X,)Bt]dt.
Jo Jo
By the resolvent equation of Proposition A.6), the same result is true for each
If / is continuous and bounded, the map t ->¦ Ex [f(X,)A,] is finite since
oo > UaA(x) >Ex\f e~asdAs] > e-°"Ex[A,l
Uo J
hence right-continuous and the same is true with В in place of A. The injectivity
of the Laplace transform implies that
Ex[f(X,)A,] = Ex[f(X,)B,]
for every / and for every continuous and bounded /. This extends at once to
/ € bfj+. Finally, an induction argument using the Markov property and the
additivity property of A and В shows that for 0 < t\ < t2 < ... < tn < t arid
fk e b%+,
[П/,(Х„,„]=?,[П Л№,,..,].
Since A, and B, are ,5f-measurable and .3^"is generated by Xs, s < t, it follows
that A — В. а

406 Chapter X. Additive Functional of Brownian Motion
A.8) Exercise. If A and В яте two additive functional such that for every t > 0
and every x e E
Ex[At] = Ex[Bt] < oo,
then M, — A, — B, is a (Jf, /\)-martingale for every x ? E. (Continuation in
Exercise B.22)).
# A.9) Exercise (Extremal process). 1°) For the standard linear BM, set, with the
usual notation, Xa — LTa with a > 0. Prove that the process a -> Xa has
independent (non stationary) increments. Prove further that for x\ < x2 < ... < xp
and a\ < a2 < . ¦. ap,
Po [Xai > x,, Xai > x2, ¦ ¦ ¦, Xaf > xp]
/ X! X2-X, Хр-Хр
пуп I "
\ 2ai
2a2 '" 2a
p
2°) If as usual r, is the time-change associated with L, prove that the process
Y, = ST) is the inverse of the process Xa. Deduce therefrom, that for t\ < t2 <
... < tp and yx <уг < ... <yp,
Yu <y\,Y,2 <y2,...,Ytp <yp]
( h h-h tp-tp-A
= exp I —— ... ———J-— .
V 2y, 2y2 2yp J
Some further information on these processes may be found in Exercise D.11) in
Chap. XII.
A.10) Exercise. For 0 < a < 1, define
f+OO
Z«@ = f
Jo
where т, is the time-change associated with L = L°.
1°) Prove that Za is a stable subordinator of index a.
[Hint: Use the scaling properties of Exercise B.11) in Chap. VI.]
2°) Prove that
lim (Za(t)f = Sr, a.s.
(see the previous exercise for the study of the limit process).
t A.11) Exercise. Let L be the local time of BM1 at zero and r, its inverse.
1°) Deduce from Proposition A.3) and the strong Markov property that r, is
under Po, an increasing Levy process.
2°) From the definition of L in Chap. VI and the integration by parts formula,
deduce that the a-potential (a > 0) of L at 0, namely f/"@) is equal to Ba)~]/2.
This will also follow from a general result in the next section.
3°) For x e R, derive from the equality

/
Jo
/о
that
§1. General Definitions 407
?,[ехр(-ат,)]А,
Ех [exp (-ar,)] = Ex [exp (-ato)] exp (-t-Jla j ,
and conclude that under Pq, the process (т,) is a stable subordinator of index 1/2.
We thus get another proof of this result independent of the equivalence in law of
L, and S,. See also 6°) in Exercise D.9) of Chap. VI.
4°) If ft is an independent BMd, prove that fiXl is a ^-dimensional Cauchy
process, i.e. the Levy process in № such that the increments have Cauchy laws
in Rd (see Exercise C.24) in Chap. III).
A.12) Exercise. Let X be a Markov process on M with continuous paths such
that its t.f. has a density p,(x, y) with respect to the Lebesgue measure which is
continuous in each of the three variables. Assume moreover that for each Px, X
is a semimartingale such that (X, X), = t. If Lx is the family of its local times,
prove that
±Ex[L>] = p,(x,y)
dt L J
for every t, x, у in ]0, oo[xR x R.
The reader is invited to compare this exercise with Exercise D.16) Chap. VII.
# A.13) Exercise. Let X = (X,,.!%, Px) be a Markov process with state space E
and A an additive functional of X. Prove that, for any Px, the pair (X, A) is a
(^)-Markov process on E x R+ with t.f. given by
Q,4>(x,a) = Ех[ф(Х,,а + A,)].
[Hint: The reader may find it useful to use the Markov property of Chap. Ill
Exercise C.19).] A special case was studied in Exercise B.18) of Chap. VI.
A.14) Exercise. 1°) Let La be the family of local times of BM and set L* =
supa L". Prove that for every p e]0, oo[ there are two constants cp and Cp such
that
cpE[T»'2]<E[(L*T)p]<CpE[TP'2]
for every stopping time T.
[Hint: Use Theorems D.10) and D.11) in Chap. IV.]
2°) Extend adequately the results to all continuous local martingales. This
result is proved in Sect. 2 Chap. XI.
# A.15) Exercise. 1°) Let A be an additive functional of the Markov process X and
e an independent exponential r.v. with parameter у. Let
? = inf {t : A, > e).
Prove that the process X defined by

408 Chapter X. Additive Functionals of Brownian Motion
X, = X, on {t < f), X, = A on{r>C)
is a Ma?kov process for the probability measure Px of X. Prove that the semigroup
Q, of X is given by
Q,f(x) = Ex[f(X,)exp(-yA,)].
If X is the linear BM and A its local time at 0, the process X is called the elastic
Brownian motion.
2°) Call Ua the resolvent of the elastic BM. If / is bounded and continuous
prove that и = Uaf satisfies the equations
аи - и" = /,
loosely speaking, this means that the infinitesimal generator A of X is given by
Аи = \u" on the space [u e C2(K) : и'+@) - и'_@) = 2yu@)}.
** A.16) Exercise. (Conditioning with respect to certain random times). Let X
be a continuous Markov process such that there exists a family Px y of probability
measures on (Q, -&Zo) with the following properties
i) for each s > 0, there is a family ^ of .^-measurable r.v.'s, closed under
pointwise multiplication, generating .3^ and such that the map (t,y) ->• E'x [C]
is continuous on ]s, oo[x? for every x e E and Ce^.
ii) for every Г e .Д", the r.v. P'x Х,(Г) is a version of ЯА [\г | X,].
The reader will find in Chap. XI a whole family of processes satisfying these
hypotheses.
1°) Prove that for every additive functional A of X such that EX[A,] < oo for
every t, and every positive process H,
HsdAs] = Ex |7°° ExXs[Hs]dAs] .
2°) Let L be a positive a.s. finite .^-measurable r.v., A. a predictable process
and Л an additive functional such that
for every positive predictable process H (see Exercise D.16) Chap. VII). If Гх —
[(s, y) : Exy[ks] = 0], prove that Px [(L, Xt) e Гх]=0 and that
Я, № | L = s, XL=y} = Exy [HSXS]/Exy[ks].
3°) Assume that in addition, the process X satisfies the hypothesis of Exercise
A.12) and that there is a measure v such that Л, = / L*v{dx). Prove that the law
of the pair (L, XL) under Px has density E'x y[\,]p,{x, y) with respect to dtv(dy).
Give another proof of the final result of Exercise D.16) Chap. VII.

§2. Representation Theorem for Additive Functionals 409
§2. Representation Theorem for Additive Functionals
of Linear Brownian Motion
With each additive functional, we will associate a measure which in the case of
BM, will serve to express the functional as an integral of local times. To this end,
we need the following definitions.
If N is a kernel on (E, <5") and m a positive measure, one sets, for A e <%',
= 1
mN(A) = / m(dx)N(x, A).
We leave to the reader the easy task of showing that the map A —*¦ mN(A) is
a measure on К which we denote by mN.
B.1) Definition. A positive a-finite measure m on (E, ft) is said to be invariant
(resp. excessive) for X, or for its semi-group, if mPt = m (resp. mPt < m) for
every t > 0.
For a positive measure m, we may define a measure Pm on {Q, ,7") by setting
as usual
Рт[П = / РЛПтШ-
Je
This extends what was done in Chap. Ill with a starting probability measure v, but
here Pm is not a probability measure if m(E) ф 1. Saying that m is invariant is
equivalent to saying that Pm is invariant by the maps в, for every t. Moreover, if
m is invariant and bounded and if we normalize m so that m(E) = 1 then, under
Pm, the process is stationary (Sect. 3 Chap. I). For any process with independent
increments on M.d, in particular BMJ, the Lebesgue measure is invariant, as the
reader will easily check. Examples of excessive measures are provided by potential
measures vU = /0°° vP,dt when they are a-finite as is the case for transient
processes such as BM'5' for d > 3.
The definition of invariance could have been given using the resolvent instead
of the semi-group. Plainly, if m is invariant (resp. excessive) then mUx = m (resp.
ml/1 < m) and the converse may also be proved (see Exercise B.21)).
From now on, we assume that there is an excessive measure m which will
be fixed throughout the discussion. For all practical purposes, this assumption is
always in force as, for transient processes, one can take for m a suitable potential
measure whereas, in the recurrent case, it can be shown under mild conditions
that there is an invariant measure. In the sequel, a function on E is said to be
integrable if it is integrable with respect torn.
Let A be an additive functional; under the above assumption, we set, for
/ e bV+,
vA(/) = sup-?„,[(/• A),].
(>0 «

410 Chapter X. Additive Functionals of Brownian Motion
B.2) Proposition. For every f and A,
vA(f) = lim-Em [(/ • A),] = lim a /m(dx)UaAf(x).
г|0 t «->oo J
Moreover, the second limit is an increasing limit.
Proof. Since m is excessive, it follows easily from the Markov property that
t -*¦ Em [(f ¦ A)t] is sub-additive; thus, the first equality follows from the well-
known properties of such functions. The second equality is then a consequence of
the abelian theorem for Laplace transform and finally the limit is increasing as a
consequence of the resolvent equation. ?
For every a > 0, the map / —> a f m(dx)UAf(x) is a positive measure, and
since one can interchange the order of increasing limits, the map / —*¦ vA(f) is a
positive measure which is denoted by vA.
B.3) Definition. The measure vA is called the measure associated with the additive
functional A. IfvA is a bounded (a-finite) measure, A is said to be integrable (a-
integrablej.
It is clear that the measures vA do not charge polar sets. One must also observe
that the correspondence between A and vA depends on m, but there is usually a
canonical choice for m, for instance Lebesgue measure in the case of BM. Let us
further remark that the measure associated with A, = t л f is m itself and that
Vf.A = f ¦ vA. In particular, if A, = f0 f(Xs)ds is an additive functional, then
vA — f ¦ m and A is integrable if and only if / is integrable.
Let us finally stress that if m is invariant then vA is defined by the simpler
formula
= Em \j
As a fundamental example, we now compute the measure associated with the
local time La of linear BM. As we just observed, here m is the Lebesgue measure
which is invariant and is in fact the only excessive measure of BM (see Exercise
C.14)).
B.4) Proposition. The local time La is an integrable additive functional and the
associated measure is the Dirac measure at a.
Proof. Plainly, we can make a = 0 and we write L for L°. Proposition A.3) in
Chap. VI shows that the measure vL has {0} for support and the total mass of vL
is equal to
Л+ОО Г+ОО Г /•+00 "I
/ dxEx[Li]= dxEo[Li] = Eo\l L\dx =
J—oo J—oo LJ—oo J
by the occupation times formula. The proof is complete.

§2. Representation Theorem for Additive Functionals 411
The measure vA will be put to use in a moment, thus it is important to know
how large the class of a -integrable additive functionals is. We shall need the
following
B.5) Lemma. If С is an integrable AF and h e b%'+ and if there is an a > 0
such that UaAh < Щ, then vA{h) < oo.
Proof. For ft >a, the resolvent equation yields
p I (и*- U^ dm = p f (Uac - UaAh) dm-tf-a) f flU? (Uac - UaAh) dm,
and because m is excessive
P f (ll?- U*h) dm > p f (U?- UaAh) dm-ф-а) f (U? - UaAh) dm
= a f (U?- UaAh) dm > 0.
Thus fi f UAh dm < f pU^-dm for every p > a, which entails the desired result.
D
We may now state
B.6) Theorem. Every CAF is a-integrable. Moreover, E is the union of a se-
sequence of universally measurable sets En such that the potentials UA(-, Е„) are
bounded and integrable.
Proof. Let / be a bounded, integrable, strictly positive Borel function and set
e~'f{X,)e-A'dt\.
Plainly, 0 < ф < Il/H; let us compute и\ф. We have
\ = Ex Гу? е-'Ех, Гу? e~sf(Xs)e-A'ds] dAt]
= Ex \J( e"Ex И* e-sf{Xs+t)e-A<o9'ds \<%\ dA
t] .
Using the result in 1°) of Exercise A.13) of Chap. V and Proposition D.7) Chap. 0,
this is further equal to
Г /"С , Л АЛ
Ех / e-'dA, / e-5/(^v+r)e"A'+'eA'«
Uo Jo
= Ex\f e-sf(Xs)e-A'ds Г eA'dAt]
Jo Jo J
= Ex\ f e~sf(Xs) A - e-A<)ds] = (/'/(jc) - 0(jc).
Jo J

412 Chapter X. Additive Functional of Brownian Motion
It remains to set Е„ = {ф > \/n] or {ф > l/и} and to apply the preceding lemma
with C, - /J f(Xs)ds and h = n~x 1 En. ?
Now and for the remainder of this section, we specialize to the case of linear
BM which we denote by B. We recall that the resolvent U" is a convolution kernel
which is given by a continuous and symmetric density. More precisely
Uaf{x) = j ua{x,y)f{y)dy
where ua(x, y) = (\/2a) exp ( — \fla\y — x\ j. Asa result, if m is the Lebesgue
measure, then for two positive functions / and g,
gUafdm= f (Uag)f dm.
Moreover, these kernels have the strong Feller property: if / is a bounded Borel
function, then Ua f is continuous. This is a well-known property of convolution
kernels.
Finally, we will use the following observation: if / is a Borel function on К
and f(B,) is a.s. right-continuous at t = 0, then / is a continuous function; this
follows at once from the law of the iterated logarithm. This observation applies
in particular to the «-potentials of a CAF when they are finite. Indeed, in that
case,
UaA(B,) = Ex\J e-atdAsoel\.^]=^tEx\f e~asdAs\.^\ P,-a.s.,
and this converges to U"(x) when / | 0 thanks to Theorem B.3) in Chap. II.
?
We may now state
B.7) Proposition. In the case of linear BM, for every continuous additive func-
functional A, the measure vA is a Radon measure.
Proof. Thanks to Lemma B.5) it is enough to prove that the sets Е„ = {ф > l/n]
in the proof of Theorem B.6) are open, hence that ф is continuous. Again, this
follows from the right-continuity of ф(В,) at / =0 which is proved as above. We
have Pj-a.s.,
Ф(В,) = EB\j" euf(Bu)e-A"du\
= Ex ПГ A
IT
e~u f(Bu)e-A"du |.^
and this converges P,-a.s. to ф(х) by Corollary B.4) of Chap. II.

§2. Representation Theorem for Additive Functionals 413
We shall now work in the converse direction and show that, to each Radon
measure v, we can associate a CAF A such that v = vA. Although it could be
done in a more general setting, we keep with linear BM.
B.8) Theorem. If A is a CAF, then for every a > 0 and f e %+,
Ulf(x) = f ua(x,y)f(y)vA(dy).
In particular, 1!% = /«"(-, y)vA{dy).
Proof. Since Vf.A = f ¦ vA, it is enough to prove the particular case.
Supppose first that U% is bounded and integrable and let ф be in Cj. By
Proposition B.2),
/ Ua4>(y)vA(dy) = lim pE
J /з->°°
The function s -> е~^иаф(В5) is continuous, hence, is the limit of the sums
Sn =J^exp(-P(k+ 1)/п)иаф(Вк/п)\]к/пЛк+1)/п];
k>0
as a function on Q x K+, the sum Sn is smaller than «"'Ц^Це"^1, which thanks
to the hypothesis that vA is bounded, is integrable with respect to the measure
Pm g) dAs defined on Q x R+ by
Pm®dAs{r) = Em\j \r(oo,s)dAs\.
Consequently, Lebesgue's dominated convergence theorem tells us that
Em Г / e-psU^(Bs)dAA = lirn^ Em\ j Sn (s)dA,A .
Using the Markov property and the fact that m is an invariant measure, this is
further equal to
lim ?]?m[exp (-/?(?+ 1)/п)иаф(Вк/п)ЕВк111[А1/п]]
k>0
= lim (?ехр (_?(*+О/и)) fиаф{у)Еу[АХ1п\т^у)
*>o
= д1|т)ехр(-^/и)A - ехрНб/и)) J ф(у)иа (Е, [АХ/п\) (y)m(dy)
where we lastly used the duality relation (*) above. As for every p we have
(-^/и)/пA - ехр(-0/и)) = 1, it follows that

414 Chapter X. Additive Functionals of Brownian Motion
J Ua<P(y)vA(dy) = In^nj ф(у)иа (Е. [Al/n]) (y)m(dy)
= lim / пе'а'Еф.т[А,+]/п- A,]dt
n [fln - 1) / e~asE^.m{As\ds- n / e~as E^.m{As\ds
J\/n Jo J
= aj e~asЕф.т[А^5 = Еф.т \Г e'asd
where, to obtain the last equality, we used the integration by parts formula. Thus,
we have obtained the equality
Ua(P(y)vA(dy)= I UaA{x)<t>{x)m{dx)
namely
ff ua{x, y)<P(x)m(dx)vA(dy) = f UaA{x)<t>{x)m{dx),
which entails
/
m"(x, y)vA(dy) = UaA{x) m-a.e..
Both sides are continuous, the right one by the observation made before B.7)
and the left one because of the continuity of ua, so this equality holds actually
everywhere and the proof is complete in that case. The general case where A is
cr-integrable follows upon taking increasing limits. D
We now state our main result which generalizes the occupation times formula
(see below Definition B.3)).
B.9) Theorem. If A is a CAF of linear BM, then
/•+00
A, = / LatvA{da).
J — 00
As a result, any CAF of linear BM is a strong additive functional.
Proof. Since vA is a Radon measure and a —> Lat is for each t a continuous
function with compact support, the integral
/•+00
A, = / LatvA{da)
J — ОС
is finite and obviously defines a CAF. Let us compute its associated measure. For
/ 6 %+, and since m is invariant,

§2. Representation Theorem for Additive Functional 415
= f+ vA(da)EmU f{Bs)dL"} = vA(f).
Thus Уд = vA and by Proposition A.7) and the last result, it is easily seen that
A = A (up to equivalence) which completes the proof. ?
Remark. It is true that every additive functional of a Feller process is a strong
additive functional, but the proof of this result lies outside the scope of this book.
We now list a few consequences of the above theorem.
B.10) Corollary. Every CAF of linear BM is finite and the map A -*¦ vA is one-
to-one.
Proof. Left to the reader. ?
Let us mention that for the right-translation on the line there exist non finite
CAF's i.e. such that A, = oo for finite t, for instance /0' f(Xs)ds with / =
1*1 l(*<o)-
B.11) Corollary. For every CAF, there is a modification such that A is 0^"°)-
adapted.
Proof. We recall that (-3^°) is the uncompleted filtration of BM. The property is
clear for local times by what was said in the last section, hence carries over to all
CAF's. ?
We also have the
B.12) Corollary. For any CAF A of linear BM, there exists a convex function f
such that
A, = f(B,) - f(BQ) - [ f'_(Bs)dBs.
Proof. Let / be a convex function whose second derivative is precisely 2vA
(Proposition C.2) in the Appendix). If we apply Tanaka's formula to this function,
we get exactly the formula announced in the statement. ?
B.13) Corollary. If A is a CAF and f a positive Borel function, then
j f(Bs)dAs = J LaJ{a)vA(da).
Proof. Obvious.
B.14) Corollary. For any non zero CAF A of linear BM, Aoo = oo a.s.

416 Chapter X. Additive Functionals of Brownian Motion
Proof. Because of Corollary B.4) Chapter VI and the translation invariance of
BM, L^ = oo for every a simultaneously almost-surely; this obviously entails
the Corollary.
Remark. A more general result will be proved in Proposition C.14).
Let us further mention that, although we are not going to prove it in this book,
there are no additive functionals of BM other than the continuous ones; this is
closely related to Theorem C.4) in Chap. V.
We will now turn to the investigation of the time-change associated with CAF
A, namely
r, = inffs : As > t}, t > 0.
Because A is an additive functional and not merely an increasing process, we have
the
B.15) Proposition. For every t, there is a negligible set outside which
rt+s — xt + rs о 6tt for every *.
Proof. Same as for the local time in Sect. 1. ?
In the sequel, A is fixed and we denote by S(A) the support of the measure vA.
We write TA for 7j(A). We will prove that A, increases only when the Brownian
motion is in S(A). As a result, A is strictly increasing if and only if vA charges
every open set.
B.16) Lemma. The times r0 and TA are a.s. equal.
For notational convenience, we shall write T for TA, throughout the proof.
Proof. Pick a in S(A); it was seen in the proof of Proposition B.5) in Chap. VI
that for / > 0, Lat is > 0 Pa-a.s. By the continuity of L* in x, it follows that Lt
is > 0 on a neighborhood of a /Va.s., which entails
/•+00
A, = / LxtvA{dx) > 0 Pe-a.s.
J -oo
Let now x be an arbitrary point in K; using the strong additivity of A and the
strong Markov property, we get, for t > 0,
Px [M+< > 0] > Ex [PBt [A, > 0]] = 1
since, S(A) being a closed set, BT e S(A) a.s. This shows that r0 < T a.s.
On the other hand, by Corollary B.13),
I h(Ay (Bs)dA, = / LyA(da) = 0
Jo JS(AY
A,= I \S{A)(Bs)dAs
Jo
a.s.,
Jo ч ' Js(ay
hence
a.s.,
which entails that At =0 and r0 > T a.s.

§2. Representation Theorem for Additive Functionals 417
B.17) Proposition. The following three sets are a.s. equal:
i) the support of the measure dA,;
ii) the complement J2a °f Uj ]T* > *>[.'
Hi) the set Г = {t : В, е S(A)}.
Proof. The equality of the sets featured in i) and ii) is proved as in the case of
the local time, hence Ea 's me support of dA,.
Again, because /„' \S(Ay(Bs)dAs — 0, we have Гс С UJts-, t5[ since these
intervals are the intervals of constancy of A; consequently, Ea ^ Г. To prove
the reverse inclusion, we observe that
Ea = {' : A(t + e)- A(t-s)>0 for all e > 0}.
Consequently,
\ш: ГШ <f У,А(о)\ С I \\AS - Ar = 0, r + TA овг < s)
where the union is over all pairs (r, s) of rational numbers such that 0 < r < s.
But for each x,
Px [A, -Ar = 0,r + TAo9r<s] = Ex [PBr [A,-r =0,TA <s- r]]
and this vanishes thanks to the preceding lemma, which completes the proof. D
We now study what becomes of the BM under the time-change associated with
an additive functional A, thus substantiating in part the comment made in Sect. 3
Chap. VII about linear Markov processes on natural scale being time-changed
BM's. We set B, = BTi; the process ? is adapted to the filtration (.3^) = (.3^).
Since t -*¦ z, is right-continuous, the filtration (J^) is right-continuous; it is also
complete for the probability measures (Px) of BM. If Г is a (J%)-stopping time,
then tj i.e. the map ш —> Тт(ш){ы) is a (.J^")-stopping time; indeed for any s
{rT <s}= |J {T <Ч}Г){тч <s}
and since {T < q} e .VXq, each of these sets is in .^T- Moreover,
T,+T — rT + t, о вТт for every t a.s.
as can be deduced from Proposition B.15) using an approximation of T by de-
decreasing, countably valued stopping times. All these facts will be put to use to
prove the
B.18) Theorem. If the support of v^ is an interval I, the process (B,,.^, Px,
x e I) is a continuous regular strong Markov process on I with natural scale and
speed measure 2vA. Its semi-group Q, is given by Qtf{x) = Ex[f(BTt)].

418 Chapter X. Additive Functionals of Brownian Motion
Proof. We first observe that BTi has continuous paths. Indeed, by the foregoing
proposition, if z,- ф zt, then BTt must be a finite end-point of / and В must
be leaving / at time r,-. But then r, is the following time at which В reenters /
and it can obviously reenter it only at the same end-point. As a result BTl — BTi_
which proves our claim.
We now prove that В has the strong Markov property. Let / be^ a positive
Borel function; by the remarks preceding the statement, if Г is a (J^)-stopping
time г —i
Ex [f(Bt+T) I Я\ = Ex [f(Br, овгт | .WTT] = Q,f(BT)
Px-a.s. onJT < oo}. The result follows from Exercise C.16) Chap. III.
That В is regular and on its natural scale is clear. It remains to compute its
speed measure. Let J —]a, b[ be an open sub-interval of /. With the notation of
Sect. 3 Chap. VII we recall that for x e J, and a positive Borel function /,
" \ J [y] j, y)f{y)dy;
\f " f(Bs)ds~\ = J^EX [LyTaATh] f{y)dy =
it follows that Ex [^"ГоЛГй] = 2Gj(x, •) a.e. and, by continuity, everywhere. We
now compute the potential of B. Define Ta = inf [t : B, = a] = inf {t : r, — Ta}
and 7fc similarly. By the time-change formula in Sect. 1 Chap. V
Г Г?аЛТь
Ex j f<Bs)ds
= Ex \j '"*Tb /(Я,)<мЛ = Ex\f ' " f{Bs)dA}
= f Ex [LyT ATb] f(y)vA{dy) =2 f Gj(x, y)f(y)vA(dy)
Jj Jj
which completes the proof. Actually, we ought to consider the particular case of
end-points but the same reasoning applies. ?
Using Corollary B.11), it is easy to check that Q,f is a Borel function if/
is a Borel function. One can also check, using the change of variable formula in
Stieltjes integrals that the resolvent of В is given by
e-aA'f(B,)dA,].
An unsatisfactory aspect of the above result is that if / has a finite end-point,
this point must be a reflecting or slowly reflecting boundary and in particular
belongs to E in the notation of Sect. 3 Chap. VII. We haven't shown how to
obtain a process on ]0, oof as a time-changed BM. We recall from Exercise C.23)
Chap. VII that if the process is on its natural scale, then 0 must be a natural
boundary. We will briefly sketch how such a process may be obtained from BM.

§2. Representation Theorem for Additive Functionals 419
Let v be a Radon measure on ]0, oof such that vQO, e[) = oo for every e > 0;
then the integral
C La,vA(da)
I
no longer defines an additive functional because it is infinite /Va.s. for any t > 0.
But it may be viewed as a CAF of the BM killed at time Tq. The associated
time-changed process will then be a regular process on ]0, oo[ on natural scale
and with 0 as natural boundary. We leave as an exercise for the reader the task
of writing down the proof of these claims; he may also look at the exercises for
particular cases and related results.
It is also important to note that Theorem B.18) has a converse. Namely, starting
with X, one can find a BM В and a CAF of В such that X is the associated time-
changed process (see Exercise B.34)).
B.19) Exercise. If A is a CAF of BM and (r,) the inverse of the local time at 0,
prove that the process ATl has stationary independent increments.
B.20) Exercise. Let В be the BMd, d > 1, and v a unit vector. The local time
at у of the linear BM X = (v, B) is an additive functional of B. Compute its
associated measure.
B.21) Exercise. If m is a ст-finite measure and mUx = m, then m is invariant.
[Hint: If m(A) < oo, prove that amUa(A) = m{A) for each a and use the
properties of Laplace transforms.]
B.22) Exercise (Signed additive functionals). If, in the definition of a CAF, we
replace the requirement that it be increasing by that it merely be of finite variation
on each bounded interval, we get the notion of signed CAF.
1°) Prove that if A is a signed CAF there exist two CAF's A+ and A" such
that A — A+ — A~, this decomposition being minimal.
2°) In the case of BM, extend to signed CAF's the results of this section.
In particular for a signed CAF A, there exists a function / which is locally the
difference of two convex functions, such that f"(dx) = 2vA(dx) and
A, = f(Bt) - /(Bo) - / f'ABs)dBs.
Jo
B.23) Exercise (Semimartingale functions of BM). Г) If /(B,) is for each Px
a continuous semimartingale, prove that / is locally the difference of two convex
functions.
[Hint: Use the preceding exercise and Exercise C.13) in Chap. II.]
As it is known that all additive functionals of BM are continuous, one can
actually remove the hypothesis that /(B,) is continuous and prove that it is so.
** 2°) If f(Bt) is a continuous semimartingale under Pv for one starting measure
v, prove that the result in 1°) still holds.
[Hint: Prove that the hypothesis of 1°) holds.]

420 Chapter X. Additive Functionals of Brownian Motion
B.24) Exercise (Skew Brownian motion). Let 0 < a < 1 and define
ga(x) = (\ -ay2 ifx>0, ga(x)=a-2 if x < 0.
If В is the linear BM, we call Y" the process obtained from В by the time-change
associated with the additive functional
/ ga(Bs)ds.
Jo
Finally we set ra(x) — x/{\ - a) if x > 0, ra(x) ~ x/a if x < 0 and X" =
ra(Y"). The process Xa is called the Skew Brownian Motion with parameter a.
For completeness, the reader may investigate the limit cases a = 0 and a = 1.
1°) Compute the scale function, speed measure and infinitesimal generator of
Xa. As a result X" has the transition density of Exercise A.16), Chap. III. Prove
that P0[X, > 0] = a for every t.
2°) A skew BM is a semimartingale. Prove that for а ф 1/2, its local time is
discontinuous in the space variable at level 0 and compute its jump.
3°) Prove that X°[ - ft, + (la - \)L, where jS is a BM and L the symmetric
local time of Xa at zero given by
L, = lim-!- f l(_e.B](Xf)d.v.
no 2e Jo
4°) Let у be a constant and let X be a solution to the equation X, — /3, + yL,
where L is the symmetric local time of X as defined in 3°). Compute L°(X),
and L°~(X)t as functions of L,. Derive therefrom that a solution X to the above
equation exists if and only if \y\ < 1.
[Hint: Use Exercise B.24) Chapter VI. See also Exercise C.19) in Chapter IX.]
B.25) Exercise (Additive local martingales). 1°) Let Л be a continuous additive
functional of linear BM. Prove that a.s. the measures dA, are absolutely continuous
with rspect to dt if and only if there is a positive Borel function / such that
A, = [ f(B,)ds.
Jo
2°) Prove that M is a continuous process vanishing at 0 and such that
i) it is an (.3^")-local martingale for every Pv,
ii) for every pair (s, t),
Mt+S - M, = Ms о в, a.s.,
if and only if there is a Borel function / such that
M, = / f(Bs)dB, a.s.
./o
[Hint: Use the representation result of Sect. 3 Chap. V and the fact that (A/, B)
is an additive functional.]

§2. Representation Theorem for Additive Functionals 421
B.26) Exercise (Continuation of Exercise A.13)). 1°) Suppose that X is the lin-
linear BM and prove that the pair (X, A) has the strong Markov property for every
(J^")-stopping time.
2°) If A, = /0' f(Xs)ds, then (X, A) is a diffusion in Ж2 with generator
\ lp "*" f(Xl}~ir- In Particular, the process of Exercise A.12) of Chap. Ill has the
generator I ? ?
* B.27) Exercise. P) In the setting of Theorem B.18), check that vA is excessive
for B.
2°) If more generally X is a linear Markov process on a closed interval and is
not necessarily on natural scale, prove that the speed measure is excessive.
3°) Extend this result to the general case.
[Hint: Use Exercise C.18) in Chap. VII.]
B.28) Exercise. In the setting of Theorem B.18) suppose that vA{dx) = V(x)dx
о
with V > 0 on / = Int(/). Prove that the extended infinitesimal generator of
В is given on C2(/) by Af = \V~X f". Check the answer against the result in
Theorem C.12) Chap. VII and compare with Proposition A.13) Chap. IX.
[Hint: Use the characterization of the extended generator in terms of martin-
martingales.]
* B.29) Exercise. 1°) In the setting of Theorem B.18), if A, = /„' \{Bs>G)ds+ XL°t
with 0 < X < oo, prove that 0 is a slowly reflecting boundary for B.
2°) Prove that if В is a Feller process, the domain C/A of the infinitesimal
generator of В is
I / e C2(]0, oo[), /"@) = lim /"(*) exists and /'@) = A/"@) ,
( ^° J
and that Af(x) = \f"{x) for x > 0, A/@) = Я"'/+@).
* B.30) Exercise. Take up the skew BM X" of Exercise B.24) with 0 < a < 1
and let L be its local time at 0.
1°) Prove that there is a BM /в such that
2°) Let a and b be two positive numbers such that a = b/(a + b) and put
a{x) = а1и>0)-Ы(*<0). Set Y, = a {X«)+-b (X«)~ and B, = \Xf\-\L, (\X"\)
and prove that (Y, B) is a solution to eo(a, 0).
3°) By considering another skew BM, say Z", such that \Xa\ = \Za\ (see
Exercise B.16) Chap. XII), prove that pathwise uniqueness does not hold for
eo(<r, 0).

422 Chapter X. Additive Functionals of Brownian Motion
4 B.31) Exercise. Let / be a positive Borel function on R and В be the linear BM.
1°) If / is not locally integrable prove that there exists a point a in R such
that for every t > 0,
/ f(Bs)ds = oo />a-a.s.
2°) Prove that the following three conditions are equivalent
/„' f(Bs)ds <ooVie [0, oo[
/0'/(Дд)</5<ооУ/е[0,оо[
@
(ii)
(iii) / is locally integrable.
— 1 for every x e
B.32) Exercise. 1°) In the situation of Theorem B.18) prove that there is a bi-
continuous family of r.v.'s L" such that for every positive Borel function /,
f{%)ds= / f{a)La,vA{da) a.s.
о Ji
2°) If in addition / = R, then В is a loc. mart.; prove that (#, Я)( = r,.
B.33) Exercise. Construct an example of a continuous regular strong Markov
process X on Ш which "spends all its time on Q", i.e. the set \t : X, e R\Q} has
a.s. Lebesgue measure 0.
B.34) Exercise. 1°) With the notation of Proposition C.5) Chap. VII prove that
m/ (Х,ЛСТ;) + / Л сг/
о
is a uniformly integrable /^-martingale for every x in /.
2°) Assume that X is on natural scale and that E = R. Call m its speed
measure and prove that if В is the DDS Brownian motion of X and r, is the
inverse of (X, X), then
(see also Exercise B.32)). In particular X is a pure loc. mart.
[Hint: The measure m is twice the opposite of the second derivative of the
concave function m/; use Tanaka's formula.]
§3. Ergodic Theorems for Additive Functionals
In Sect. 1 Chap. II and Sect. 2 Chap. V, we proved some recurrence properties of
BM in dimensions 1 and 2. We are now taking this up to prove an ergodic result
for occupation times or more generally additive functionals. Since at no extra cost
we can cover other cases, we will consider in this section a Markov process X
for which we use the notation and results of Chap. III. We assume in addition
a{B)m(da)

§3. Ergodic Theorems for Additive Functionals 423
that the resolvent Ua has the strong Feller property, namely U" f is continuous
for every a and every bounded Borel function / and also that P, 1 = 1 for every
t > 0 which is equivalent to P.[t; = oo] = 1.
Our first definition makes sense for any Markov process and is fundamental
in the description of probabilistic potential theory.
C.1) Definition. A positive universally measurable function f is excessive for the
process X (or for its semi-group) if
i) Ptf < f far every t > 0;
ii) \\m,wP,f — f.
A finite universally measurable function h is said to be invariant if P,h — h for
every t.
C.2) Proposition. If f is excessive, f(X,) is a (,^)-supermartingale for every
Pv. Ifh is invariant, h(Xt) is a martingale.
Proof. By the Markov property, and property i) above
Ev [f №+,) I -Щ = EXl [/№)] = P,f{Xs) < /(X,) /Va.s.
In the case of invariant functions, the inequality is an equality. D
This proposition, which used only property i) in the definition above, does not
say anything about the possibility of getting a good version for the super-martingale
f(X,); the property ii) is precisely what is needed to ensure that f(X,) is a.s. right-
continuous, but we are not going to prove it in this book. We merely observe that,
if / is excessive, then aUa f < f for every a and Yinia^ocOiUaf = f as the
reader will easily show; moreover, the limit is increasing and it follows easily
from the strong Feller property of Ua that an excessive function is lower-semi-
continuous. If h is invariant and bounded, then aUah = h hence h is continuous;
the martingale h(Xt) is then a.s. right-continuous, a fact which we will use below.
Moreover if conversely h is bounded and aUah = h for every a, the continuity
of h hence the right-continuity of P,h in t, entails, by the uniqueness property of
Laplace transform that h is invariant.
C.3) Definition. An event Г of ,^ca is said to be invariant 1/в~1(Г) = Г for
every t. The a-field ^ of invariant events is called the invariant afield and an
^ -measurable r.v. is also called invariant. Two invariant r.v. 's Z and Z' are said
to be equivalent if PX[Z = Z'] = 1 for every x.
Invariant r.v.'s and invariant functions on the state space are related by the
following
C.4) Proposition. The formula h(x) = EX[Z] sets up a one-to-one and onto cor-
correspondence between the bounded invariant functions and the equivalence classes
of bounded invariant r.v. 's. Moreover
Z = lim h(X,) a.s.

424 Chapter X. Additive Functionals of Brownian Motion
Proof. If Z is invariant, a simple application of the Markov property shows that
h(-) = E[Z] is invariant (notice that if we did not have ? = oo a.s., we would
only get P,h < h).
Conversely, since h(X,) is a right-continuous bounded martingale, it converges
a.s. to a bounded r.v. Z which may be chosen invariant. Moreover, by Lebesgue's
dominated convergence theorem, h(x) = EX[Z] for every x in E. The correspon-
correspondence thus obtained is clearly one-to-one. ?
Let A be a Borel set; the set
R(A) = {Шп~,^оо 1 a№) = 1} = П {t + TA о в, < oo)
is the set of paths a> which hit A infinitely often as t —> oo; it is in ,W^ since it is
equal to f]n \n + TA о в„ < oo}. It is then clear that it is an invariant event. The
corresponding invariant function hA ~ P[R(A)] is the probability that A is hit at
arbitrarily large times and lim,^^h^X,) = \r(A) a.s. by the above result.
C.5) Definition. A set A is said to be transient ifhA = 0 and recurrent ifhA = 1.
In general, a set may be neither recurrent nor transient but we have the
C.6) Proposition. The following three statements are equivalent:
i) the bounded invariant functions are constant;
ii) the a-algebra Д is a.s. trivial;
Hi) every set is either recurrent or transient.
Proof. The equivalence of i) and ii) follows immediately from Proposition C.4),
and it is clear that ii) implies iii).
We prove that iii) implies ii). Let fe/ and put A = {x : РХ[Г] > a) for
0 < a < 1. We know that 1 r = Ит,-^^ Рх,[Г] a.s.; if A is recurrent, then Г = Q
a.s. and if A is transient then Г = 0 a.s. ?
Although we are not going to develop the corresponding theory, Markov pro-
processes have roughly two basic behaviors. Either they converge to infinity in which
case they are called transient, or they come back at arbitrarily large times to rela-
relatively small sets, for instance open balls of arbitrarily small radius, in which case
they are called recurrent. After proving a result pertaining to the transient case,
we will essentially study the recurrent case. Let us first observe that because of
the right-continuity of paths, if A is an open set, R(A) = {lim9_i.(X)lA(.Xg) = l}
where q runs through the rational numbers.
The following result applies in particular to BMrf, d > 2, in which case
however it was already proved in Sect. 2 Chap. V.
C.7) Proposition. If for every relatively compact set A, the potential ?/(•, А) м
finite, then the process converges to infinity.

§3. Ergodic Theorems for Additive Functionals 425
Proof. We have, by the Markov property
= Ex \eXi ПГ UiX^dsU = Ex [jf* Ы*,)</*1;
it follows on the one hand that ?/(•, A) is excessive, hence lower-continuous, on
the other hand that lim,-,.,^ P,(JJ(-, A)) = 0. From the first property we deduce
that U(Xq, A) is a positive supermartingale indexed by Q+, and by the second
property and Fatou's lemma, its limit as q —>¦ oo is zero a.s.
Let now Г and Г' be two relatively compact open sets such that T С Г'.
The function ?/(-, Г') is strictly positive on Г', because of the right-continuity of
paths; by the lower-semi-continuity of ?/(¦, Г"), there is a constant a > 0 such
that C/(',_O > a on Г. Thus on the paths which hit Г at infinitely large times,
we have Iim9^oof/(X4, Г') > a. By the first paragraph, the set of these paths is
a.s. empty. Therefore, Г is a.s. not visited from some finite time on, and the proof
is now easily completed. ?
We now study the opposite situation.
C.8) Definition. The process X is said to be Harris-recurrent or merely Harris if
there is an invariant measure m such that m(A) > 0 implies that A is recurrent.
In the sequel, when we deal with Harris processes, we will always assume
that the support of m is the whole space. Indeed, the support of an invariant
measure is an absorbing set, a fact which is proved in the following way. Let Г
be the complement of the support; since Г is open, the right-continuity of paths
entails that the set of points from which the process can reach Г is precisely
Г = {x : Ua{x, Г) > 0} for some a > 0. Clearly Г D Г and since m is
invariant, ат1/а{Г) — т(Г) = 0 which proves that т(Г') = 0; as a result
Г' = Г and Гс is absorbing. Thus, one loses little by assuming that Г is empty
and in fact this is naturally satisfied in most cases. This condition implies that
every open set is recurrent.
Conversely we have the following result which shows that BM^, d = 1,2, the
OU process and many linear Markov processes such as the Bessel processes of
low dimensions are Harris-recurrent.
C.9) Proposition. IfX has an invariant measure and if every open set is recurrent,
then X is Harris.
Proof. If m(A) > 0, since Pm[X, & A] = m(A) for every r, there is a constant
a > 0 such that the set Г = {x : Px [TA < oo] > a] is not empty.
Now the function / = P [TA < oo] is excessive because P,f(x) = Px[t + TAo
в, < oo] < f(x) and one checks that lim^o (t + TA о в,) = ТА which implies that
lim^o P,f(x) = f(x). As a result the set Г is open; furthermore, by Corollary
B.4) in Chap. II,
lim Px, [TA < oo] = lim P. [q + TA о вч < oo | .5^1 = 1R(A) a.s.
q—юс q->oc

426 Chapter X. Additive Functionals of Brownian Motion
and since Г is recurrent, we find that \r{A) > a a.s. hence R(A) = Q a.s. which
completes the proof. D
For a Harris process, the equivalent conditions of Proposition C.6) are in force.
C.10) Proposition. If X is a Harris process, the excessive functions and the
bounded invariant functions are constant.
Proof. If the excessive function / were not constant, we could find two constants
a < b such that the sets J — {/ > b) and J' — {/ < a) are not empty. The set
J is open, hence recurrent, and by Fatou's lemma, for each x e E,
f(x) > lim Pqf(x) > Ex lim f(X4)\ > b
g-+oo [_¦?-> =>0 J
and we get a contradiction.
For a bounded harmonic function h, we apply the result just proved to h + \\h ||.
D
By the occupation times formula together with Corollary B.4) Chapter IV (or
Corollary B.14)), we know that in the case of BM1, if m is the Lebesgue measure,
m(A) > 0 implies
/ lA(Xs)ds = oo a.s.,
Jo
which is apparently stronger than the Harris condition. We will prove that actually
this property is shared by every Harris process, in particular by BM2. We consider a
strong additive functional A which, as we already observed, is in fact no restriction.
C.11) Proposition. lfvA does not vanish, then Аж = oo a.s.
Of course, here vA is computed with respect to the invariant measure m which
is the only invariant measure for X (see Exercise C.14)).
Proof. For e > 0, we set xE — inf{? : A, > e). If vA does not vanish, we may find
e > 0 and a > 0 such that
m {{x : Px[t? < oo] > a}) > 0.
Therefore, Нт^оо/^ [rf < oo] > a a.s. But on the other hand, for x e E,
Px, [rf < oo] = Px [t + rE о в, < oo | .Щ /Va.s.,
and by Corollary B.4) in Chap. II, this converges /\-a.s. to 1[п,{г+т(ой,«х>ц- 'l
follows that P|, {/ + xE о в, < oo} = Q a.s. and a fortiori P [re < oo] = 1.
If we define now inductively the stopping times T" by Tx = rf and T" =
T" + те o9r,-i, a simple application of the strong Markov property shows that
P[T" < oo] — 1. By the strong additivity of A, we have Ar, > ns for every n,
which completes the proof.

§3. Ergodic Theorems for Additive Functionals 427
Remarks. 1°) The function P [тг < oo] can be shown to be excessive, hence
constant and the proof could be based on these facts.
2°) This result shows that for m(A) > 0 we have U(-, A) = oo which is to be
compared with Proposition C.7).
We now turn to the limit-quotient theorem which is the main result of this
section.
C.12) Theorem. IfX is Harris, if A and С are two integrable additive functionals
and if || \>c || > 0, then
r->oo
By the preceding result, the condition ||vcll > 0 ensures that the quotient on
the left is meaningful at least for t sufficiently large.
Proof. By afterwards taking quotients, it is clearly enough to prove the result
when C, = /or f(Xs)ds where / is a bounded, integrable and strictly positive
Borel function.
We will use the Chacon-Ornstein theorem (see Appendix) for the operator
ва, a > 0. Since m is invariant for the process, the measure Pm on (Q,.7~) is
invariant by ва, so that Z -> Zo6a is a positive contraction of Ll(Pm). Moreover,
by the preceding result, we have
oo
^ Ca о 0na — Cx = oo a.s.
which proves that the set D in the Hopfs decomposition of Q with respect to 6a
is empty; in other words, Z —>• Z о ва is conservative. We may therefore apply
the Chacon-Ornstein theorem; by hypothesis, Aa and Ca are in L\Pm) so that the
limit
Xa - lim {Ana/Cna)
n—>oo
exists f,,,-a.s. As lim,^^ C, = oo, it is easily seen that \\mn^xCn + i/Cn — 1;
therefore the inequalities
(A[l/a]a/C[,/a\a) (С[,/а]а/C[t/a]a + l)
< A,/Ct < (A[,/a|a+i/C[,/a]a+i) (C[f/a|H+i/C[,/a|a)
imply that
lim(A,/C,)=Xa Pm-m.s.
(->OO
As a result, there is a r.v. X such that
\im(A,/C,) = X />m-a.s.
(-»OO
and X = X о ва Pm-a.s. for every a > 0. It follows from Propositions C.6) and
C.10) that X is Pm-a.s. equal to a constant. From the Chacon-Ornstein theorem,

428 Chapter X. Additive Functionals of Brownian Motion
it follows that this constant must be Em[Aa]/Em[Ca] for an arbitrary a, that is
II VA H/ll Veil-
Set
F = \eo: lim A,М/С,(a>) = 1М1/1Ы1 .
We have just proved that Pm(Fc) = 0; moreover if ш e F then вь{ш) е F for
every s or, in other words, \F < lf°ft for every s. But since lim,^^ C, = +oo
a.s. if в5(со) e F, then to e F. Thus lf = lf oft a.s. for every s which implies
that P.[FC] is a bounded invariant function, hence a constant function, which has
to be identically zero. The proof is complete.
Remarks. Iе) In the case of BM, the end of the proof could also be based on the
triviality of the asymptotic cr-field (see Exercise B.28) Chap. III). The details are
left to the reader as an exercise.
2°) In the case of BM1, one can give a proof of the above result using only
the law of large numbers (see Exercise C.16)).
3°) If the invariant measure m is bounded, as for instance is the case for the OU
process (see Exercise A.13) in Chap. Ill), then constant functions are integrable
and taking C, = t in the above result we get
lim(A,//) = ||vA||/m(?) a.s.
r->oo
Thus in that case the additive functionals increase like t. When m{E) = oo, it
would be interesting to describe the speed at which additive functionals increase
to infinity. This is tackled for BM2 in the following section and treated in the case
of BM1 in Sect. 2 Chap. XIII. (See also the Notes and Comments).
4°) Some caution must be exercised when applying the above result to occu-
occupation times because there are integrable functions / such that /„' f(Xt)ds is not
an additive functional; one may have /0' f(Xs)ds = oo for every / > 0, Px-a.s.
for x in a polar set (see Exercise C.17)). The reader will find in Exercise B.6) of
Chap. XI how to construct examples of such functions in the case of BM2 (see
also Exercise B.31) above in the case of linear BM). The limit-quotient theorem
will then be true /\-a.s. for x outside a polar set. If / is bounded, the result is
true for /0' f(Xs)ds without qualification.
C.13) Exercise. Under the assumptions of this section, if X is Harris then for
every A e К either U(-, A) == oc or U(:, A) = 0. Prove moreover that all
cooptional times are equal to 0 a.s.
[Hint: See Exercise D.14) in Chap. VII].
C.14) Exercise. Suppose X is Harris with invariant measure m and that the hy-
hypotheses of this section are in force.
1°) Prove that m is equivalent to Ua(x, •) for every a > 0 and x e E.
2°) Prove that m is the unique (up to multiplication by a constant) excessive
measure for X.
[Hint: Prove that an invariant measure is equivalent to m, then use the limit-
quotient theorem.]

§3. Ergodic Theorems for Additive Functionals 429
# C.15) Exercise. Let X be a Harris process, A an integrable additive functional
of X and С a a -integrable but not integrable additive functional, prove that
lim (A,/C,) = 0 a.s.
[Hint: Use the ergodic theorem for A and If С where vc(F) < oc.]
C.16) Exercise. For the linear BM and for a < b define
1°) Let A be a continuous additive functional. Prove that the r.v.'s Zn —
At» — A?*-1, n > 1, are independent and identically distributed under every Px,
x e R. If || уд || < oo, prove that the ZH's are /^-integrable.
[Hint: For this last fact, one can consider the case of local times and use the
results in Sect. 4 Chap. VI.]
2°) Applying the law of large numbers to the variables Zn, prove Theorem
C.12).
[Hint: Prove that A,/ inf[n : T" > t] converges as t goes to infinity, then use
quotients.]
3°) Extend the above pattern of proof to recurrent linear Markov processes.
C.17) Exercise. Let X be Harris and / be positive and /я-integrable. Prove that
/o f(Xs)ds < oo Pf-a.s. for every t > 0 and for every x outside a polar set. That
this result cannot be improved is shown in Exercise B.6) Chap. XI.
* C.18) Exercise. In the setting of Theorem C.12), prove that
for m -almost every x.
[Hint: Prove that for each a > 0, Pa is a conservative contraction of Ll(m)
and apply the Chacon-Ornstein theorem.]
C.19) Exercise. We retain the situation of Exercise B.22) 2) and we put vA —
vA+ - vA-.
1°) If vA is bounded, УдA) = 0 and J \x\ \vA\{dx) < oo, prove that / is
bounded and f'_ is in О П L2 of the Lebesgue measure.
[Hint: This question is solved in Sect. 2 Chap. XII1.]
2°) Under the hypothesis of 1°), prove that there is a constant С such that
\EX[AT]\<C
for every point x and stopping time 7" such that EX[T] < oo.
3°) If A', /' = 1,2, are positive integrable additive functionals of BM1 such
that ||yA. || > 0 and / \x\vA-,(dx) < oo, then for any probability measure /j. on R,
Дт E4A']/EM [A2]= IIvAl H/llvA21|.
The results in 2°) and 3°) are strengthenings of the result in the preceding exercise.

430 Chapter X. Additive Functional of Brownian Motion
*# C.20) Exercise. Iе) Let с be a positive real number. On the Wiener space Wrf
the transformation a> —> ш(с-)/л/с is measurable and leaves the Wiener measure
W invariant. By applying BirkhofFs theorem to this transformation, prove that for
d>3,
lim — / \B,\~2ds = —— W-a.s.
i^oc log/ J, d — 2
[Hint: To prove that the limit provided by Birkhoffs theorem is constant
use the 0 - 1 law for processes with independent increments. The value of the
constant may be computed by elementary means or derived from Exercise D.23)
in Chap. IV.]
2°) Prove the companion central-limit theorem to the above a.s. result, namely,
that, in distribution,
-1 / I o 1-2
(log/) / \Bs\-4s-(d-2)-[ = N
where N is a Gaussian r.v.
[Hint: Use the methods of Exercise D.23) in Chap. IV.]
§4. Asymptotic Results for the Planar Brownian Motion
This section is devoted to some asymptotic results for functionals of BM2. In
particular it gives a partial answer to the question raised in Remark 3°) at the end of
the previous section. We use the skew-product representation of BM2 described in
Theorem B.11) of Chap. V and the notation thereof and work with the probability
measure P- for z ф 0.
D.1) Theorem (Spitzer). As t converges to infinity, 29,/ log? converges in distri-
distribution to a Cauchy variable with parameter 1.
Proof. Because of the geometric and scaling invariance properties of BM, we
may assume that z = 1. For r > 1, define ar = inf (и : |Z,,| = r\ and for a > 0,
Ta — mf{t > 0 : f}, = a}. From the representation theorem recalled above, it
follows that COr — T\ogr. As a result
Be — Ус„г = Ут,,,,,. = (logr)yT-,.
the last equality being a consequence of the independence of p and у and of the
scaling properties of Ta and у (Proposition C.10), Chap. III). Therefore for every
r > 1,
~ear m c,
logr
where С is a Cauchy variable with parameter 1; this can alternatively be written
as

§4. Asymptotic Results for the Planar Brownian Motion 431
W)
1ogr°"
We will be finished if we prove that (в, - 9Пф) / log/ converges to zero in prob-
probability.
We have
Ja rt ?*s
s, 1 + ^
where Zs = Zs - 1 is a BM2@). Setting ZS = Z,s/yft, we get
dZs
9, - 6aj-: = 1m
Let Z' be a BM2@) fixed once and for all. Since
аф = м\и: |1+Z,,| = sTt\ = ппф: |A/V/) + Z'u
we have
в, - ва = \m
where v, = inf {м : |(l/V/) + Z^| = l}.
We leave as an exercise to the reader the task of showing that v, converges
in probability to a[ = infjw : \Z'U\ = \\ as / goes to infinity. It then follows
from Exercise A.25) in Chap. VIII that the last displayed integral converges in
probability to the a.s. finite r.v. j\ dZ's/Z's, which completes the proof. ?
We will now go further by studying the windings of Z according as Z is close
to or far away from 0. More precisely, pick a real number r > 0 and set
0,°= f
We want to study the asymptotic behavior of the pair (#,°, 0,00) and we will actually
treat a more general problem. Let ф be a bounded and positive function on R; we
assume that т\(ф) — fw^(x)dx < oo. We set
A, =
Jo
This may not be an additive functional because A, may be infinite P0-a.s. for
every t > 0, but the equality Д,+! = A, + As о в, holds Pr-a.s. for z ф 0 (see
Remark 4 below Theorem C.12)). It is moreover integrable in the sense that
?т[Д(] = 2лт\(ф) as is easily checked using the skew-product representation.

432 Chapter X. Additive Functionals of Brownian Motion
D.2) Theorem. Under Pz, z ф 0, the Ъ-dimensional family ofr.v. 's
converges in distribution as t converges to oo, to
(fi><0) л, J^ <Д>0) i, I hJ
where (P, y) is a standard planar BM starting at 0, L° is the local time of ft at 0
and Г, = inf\t : P, = \\.
In the following proof as well as in similar questions treated in Chap. XIII we
will make an extensive use of the scaling transformations. If В is a BM and a is
> 0, we will denote by B{a) the BM a~xBai, and anything related to BUl) will
sport the superscript (a); when a — 1, we have B{X) = В and we drop the A).
For instance
Proof. Again we may assume that z — 1 and as in the above proof we look at the
given process at time a,. By the time-changes and properties already used in the
previous proof, we get
®о, — / l(log|Z,|<Logr)^yc, = / '(Aslogr)^Ks = / l(A<logr)^Kt;
./0 ./0 Jo
J 0 J()
setting a — log?, we get
(log/ГЧ0 =
The same computation will yield
We turn to the third term for which we have
Jo Jo Jo
= a2 I 4>(apla))ds.
Jo
Consequently, (logi)'1 (в?, в™, AOi) has the same law as
l(^ <a-4ogr)dys. / 1(Д,>а-' \ogr)dys, a I (f>(ap,)ds I
Jo 'Jo )

§4. Asymptotic Results for the Planar Brownian Motion 433
where ((I, y) is a planar standard BM and T\ = infjf : fi, — 1}. The first two
terms on the right converge in probability thanks to Theorem B.12) in Chap. IV;
as to the third, introducing LXT[, the local time of f5 at x up to time T\, and using
the occupation times formula, it is equal to
a / <f>{ax)UT]dx = / ф(уI}т/''с1у
which converges a.s. to тх(фI^Т[ by dominated convergence.
Thus we have proved that щ- (^, 0?°, ACT,) converges in distribution to
Jo /
/о
Furthermore, as in the preceding proof, we have
P- lim [0, -в„ I = P- lim [в?0 -в1? ) = 0.
/ — ос log/ V »'/ l-*cx) log/ V ¦"/
Also
-?-{A,-Ao) = -?-/' |Z,r20(bg|Z,|)^
log? l/'/ log? JCTy;
2||</>lloo Г'
log? /^
where Z5 = /~I/2Z,V and this converges to zero in probability; indeed, as in the end
of the proof of Theorem D.1), the last integral converges in law to f\ \Z's\~2ds.
a
Remark. It is noteworthy that the limiting expression does not depend on r. If,
in particular, we make r — 1 and if we put together the expressions for #? and
в%° given at the beginning of the proof and the fact that j^-y (#o _ в® r, 6f° — в^г)
converges to zero in probability, we have proved that
T~ в". 0Г) " ( / '°gV? 1</з.<0)^. / ^' \iP<>mdy)
'°g? \Jo Jo /
converges to zero in probability, a fact which will be used in Sect. 3 Chap. XIII.
We now further analyse the foregoing result by computing the law of the limit
which we will denote by (W , W+, Л). This triplet takes its values in R2 x R+.
D.3) Proposition. I/т^ф) — I, for a > 0, (b, с) е К2,
E [exp (-аЛ + ibW~ + icW+)] = /Ba + \b\, c)
where f(u,c) = (cosh с + (f) sinhc) for с ф 0, /(и,0) = A +и)
-i

434 Chapter X. Additive Functional of Brownian Motion
Proof. By conditioning with respect to -i^i, we get
E [exp (-a A + ibW~ + icW+)] = E [exp (-///-,)]
where
b2 f с /"'
H,=aL° + — iiPi<0)ds + — \{fS,>0)ds.
¦Wo ^ Jo
The integrand exp (— //r,) now involves only ft and the idea is to find a function
F such that F(P,)exp(-H,) is a suitable local martingale.
With the help of Tanaka's formula, the problem is reduced to solving the
equation
F" = BaS0 + b21 (x<o) + c21 (A>o,) F
in the sense of distributions. We take
F(x) = ехр(|й|дгI(д<о) + (cosh(cj) H sinh(ct) 1 1ц>0).
Stopped at T\, the local martingale F(/?,)exp(—//,) is bounded. Thus, we can
apply the optional stopping theorem which yields the result.
D.4) Corollary, i) The r.v. Л = LUT is exponentially distributed with parameter
1/2.
ii) Conditionally on A, the r.v. 's W and W+ are independent, W is a Cauchy
variable with parameter Л/2 and the characteristic function of W+ is equal to
(c/sinhc)exp (—y(ccothc — 1)).
Hi) The density of W+ is equal to B cosh(jr*/2))~'.
Proof The proof of i) is straightforward. It is also proved independently in Propo-
Proposition D.6) of Chap. VI.
To prove ii) set fc.bW — {^^) exP(~f (ccothc — 1 + \b\)) and compute
E [exp(—aA)fbx(A)]. Using the law of A found in i), this is easily seen to be
equal to
-r^—A + 2a + ccothc- 1 + \b\)~] = /Ba + \b\, c).
sinn с
As a is an arbitrary positive number, it follows that
fhx(A) = E [exp(iW + icW+) \ л]
which proves ii).
Finally, the proof of iii) is a classical Fourier transform computation (see
Sect. 6 Chap. 0).
Remark. The independence in ii) has the following intuitive meaning. The r.v. A
accounts for the time spent on the boundary of the disk or for the number of times
the process crosses this boundary. Once this is known, what occurs inside the disk
is independent of what occurs outside. Moreover, the larger the number of these
crossings, the larger in absolute value the winding" number tends to be.

§4. Asymptotic Results for the Planar Brownian Motion 435
We finally observe that since we work with z Ф 0, by Remark 4 at the end of
the preceding section, if G is an integrable additive functional and if т.\{ф) > О
we have lim(G,/A,) = \\vc\\/2nm\(<p) Pz-a.s. As a result we may use G instead
of A in the above results and get
D.5) Corollary. IfG is any integrable additive functional.
converges in law under Pz to (W~, W+, Bn)~l \\vg\\A) as ' Soes t0 infinity.
In Theorem D.2) we were interested in the imaginary parts of
2 (f'\ > ^
Ц (|Z-Uf) Z,
For later needs, it is also worth recording the asymptotic behavior of the real parts.
We set
/' dZ f
l(|Z,i<r>V- ^°° = Re /
z
With the same notation as above, we have the
D.6) Proposition. As t converges to infinity, 2(log?)~' (/V,°, /V,00, A,) converges
in distribution to (\Л, ^Л — 1, тх{ф)Лу
Proof. The same pattern of proof as in Theorem D.2) leads to the convergence of
the law of 2(log;)"' (N,0, N^°, A,) towards that of
aT' Г' ° ^i
Jo '"'" ' '" T'J
Thus the result follows immediately from Tanaka's formula.
D.7) Exercise. Deduce Theorem D.1) from Theorem D.2).
D.8) Exercise. With the notation of Theorem D.1) prove that Xu = в (oexp{ll))
and Yu — в (сгехр(-м)) are two Cauchy processes.
* D.9) Exercise. Prove Theorem D.1) as a Corollary to Proposition C.8) in Chap.
VIII.
[Hint: Use the explicit expressions for the density of p, and make the change
of variable p = U\fl\
* D.10) Exercise. Let В be a BM2@) and call в,, t > 0 a continuous determination
of arg(B,), t > 0. Prove that as t converges to 0, the r.v.'s 2#,/logf converge in
distribution to the Cauchy r.v. with parameter 1.
[Hint: By scaling or time-inversion, (#, — в,) = (9\/, — в\).~\

436 Chapter X. Additive Functionals of Brownian Motion
* D.11) Exercise (Another proof of Theorem D.1)). 1°) With the notation of
Theorems D.1) and D.2) prove that
a-2C, = \nf\u : a2 f expBapla))ds > 1
2°) Using the Laplace method prove that for a fixed BM, say B,
a" \
expBaBs)ds I - sup Bs = 0
/ 0<.!<и
holds a.s. for every и (see Exercise A.18) Chap. I).
3°) Prove that for a - log//2,
P-\\m \a~2C, - T}a)\ = 0.
[Hint: The processes /3(u) have all the law of В which shows that the conver-
convergence holds in law.]
4°) Give another proof of Theorem D.1) based on the result in 3°).
D.12) Exercise. Let Z be a BM2A) and в be the continuous determination of
argZ such that в0 = 0. Set Т„ - inf{? : \Z,\ > n).
1°) If т = inf{? : в, > 1} prove that
lim(log«)/)[T > Т„]
n-»oo
exists.
[Hint: Р[т > Т„] = P[Cr > Ст.].]
2°) If т = inf{? :\в,\> 1}, prove that P[r > Tn] = 0A/и).
Notes and Comments
Sect. 1. The basic reference for additive functionals is the book of Blumenthal
and Getoor [1] from which most of our proofs are borrowed. There the reader will
find, for instance, the proof of the strong additivity property of additive functionals
and an account of the history of the subject. Our own exposition is kept to the
minimum which is necessary for the asymptotic results of this chapter and of
Chap. XIII. It gives no inkling of the present-day state of the art for which we
recommend the book of Sharpe [3].
The extremal process of Exercise A.9) is studied in Dwass [1] and Resnick
[1]. It appears as the limit process in some asymptotic results as for instance in
Watanabe [2] where one finds also the matter of Exercise A.10).
Exercise A.11) is actually valid in a much more general context as described
in Chap. V of Blumenthal and Getoor [1]. If X is a general strong Markov process,
and if a point x is regular for itself (i.e. x is regular for [x] as defined in Exercise
B.24) of Chap. Ill), it can be proved that there exists an additive functional A

Notes and Comments 437
such that the measure dA, is a.s. carried by the set {t : X, — x). This additive
functional, which is unique up to multiplication by a constant, is called the local
time of X at x. Thus, for a Markov process which is also a semimartingale, as
is the case for BM, we have two possible definitions of local times. A profound
study of the relationships between Markov processes and semi-martingales was
undertaken by Cinlar et al. [1].
Exercise A.14) is from Barlow-Yor ([1] and [2]), the method hinted at being
from Bass [2] and B. Davis [5].
Exercise A.16) is closely linked to Exercise D.16) of Chap. VII. The interested
reader shall find several applications of both exercises in Jeulin-Yor [3] and Yor
[16]. For an up-date on the subject the reader is referred to Fitzsimmons et al. [1]
who in particular work with less stringent hypotheses.
Sect. 2. This section is based on Revuz [1]. The representation theorem B.9) is
valid for every process having a local time at each point, for instance the linear
Markov processes of Sect. 3 Chap. Vll. It was originally proved in the case of BM
in Tanaka [1]. Some of its corollaries are due to Wang [2]. The proof that all the
additive functionals of BM are continuous may be found in Blumenthal-Getoor [1].
For BM'', d > 1, there is no result as simple as B.9), precisely because for
d > 1, the one-point sets are polar and there are no local times. For what can
nonetheless be said, the reader may consult Brosamler [1] (see also Meyer [7])
and Bass [1].
Exercise B.23) is taken from Cinlar et al. [1]. The skew Brownian motion
of Exercises B.24) and B.30) is studied in Harrison-Shepp [1], Walsh [3] and
Barlow [4]. Walsh's multivariate generalization of the skew Brownian motion is
studied by Barlow et al. [2]. Exercise B.31) is due to Engelbert-Schmidt [1].
Sect. 3. Our exposition is based on Azema et al. [1] A967) and Revuz [2], but
the limit quotient theorem had been known for a long time in the case of BM'
(see Ito-McKean [1]) and BM2 for which it was proved by Maruyama and Tanaka
[1]. For the results of ergodic theory used in this section see for instance Krengel
[1], Neveu [4] or Revuz [3].
Exercise C.18) is from Azema et al. [1] A967) and Exercise C.19) from Revuz
[4]. Incidentally, let us mention the
Question 1. Can the result in Exercise C.19) be extended to all Harris processes?
Exercise C.20) is taken from Yor [17].
Sect. 4. Theorem D.1) was proved by Spitzer [1] as a consequence of his explicit
computation of the distribution of в,. The proof presented here, as well as the
proof of Theorem D.2) is taken from Messulam and Yor [1] who followed an idea
of Williams [4] with an improvement of Pitman-Yor [5]. A variant of this proof
based on Laplace's method is given in Exercise D.11); this variant was used by
Durrett [1] and Le Gall-Yor [2]. The almost-sure asymptotic behavior of winding
numbers has been investigated by Bertoin-Werner ([1], [2]) and Shi [1].

438 Chapter X. Additive Functionals of Brownian Motion
The asymptotic property of additive functionals which is part of Theorem D.2)
was first proved by Kallianpur and Robbins [1]. This kind of result is proved for
BM1 in Sect. 2 Chap. XIII; for more general recurrent Markov processes we refer
to Darling-Kac [1], Bingham [1] and the series of papers by Kasahara ([1], [2]
and [3]).
The formula given in Proposition D.3) may be found in the literature in various
disguises; it is clearly linked to P. Levy's formula for the stochastic area, and the
reader is referred to Williams [5], Azema-Yor [2] and Jeulin-Yor [3]. For more
variations on Levy's formula see Biane-Yor [2] and Duplantier [1] which contains
many references.

Chapter XI. Bessel Processes
and Ray-Knight Theorems
§1. Bessel Processes
In this section, we take up the study of Bessel processes which was begun in
Sect. 3 of Chap. VI and we use the notation thereof. We first make the following
remarks.
If В is a BtvT5 and we set p = \B\, Ito's formula implies that
s. r> . .
l+8t.
For 8 > 1, pi is a.s. > 0 for / > 0 and for 8 — 1 the set {s : ps = 0} has a.s. zero
Lebesgue measure, so that in all cases we may consider the process
* = ?,['№/л) dB't
¦ 1 J 0
which, since (j3, fi), = t, is a linear BM; therefore pf satisfies the SDE
2 2 f
p. — p0 + 2 I psd/3s + 8t.
Jo
For any real 8 > 0, and x > 0, let us consider the SDE
f i
Z,—x+2l J\Zs\dBs+8t.
Jo
Since \yfz- >/?| < y/\z-z'\ for z,z > 0, the results of Sect. 3 in Chap. IX
apply. As a result, for every 8 and x, this equation has a unique strong solution.
Furthermore, as for 8 — x = 0, this solution is Z, = 0, the comparison theorems
ensure that in all cases Z, > 0 a.s. Thus the absolute value in the above SDE may
be discarded.
A.1) Definitions. For every 8 > 0 and x > 0, the unique strong solution of the
equation
'¦ f y/Zsdfi,
Jo
Z,=x+2 / y/Zsdp, + 8t
/o

440 Chapter XI. Bessel Processes and Ray-Knight Theorems
is called the square of <5-dimensional Bessel process started at x and is denoted
by BESQs(x). The number 8 is the dimension ofBESQs. The law ofBESQs(x) on
C(R+, Ш) is denoted by Q\. We will also use the number v — (S/2) - 1 which is
called the index of the corresponding process, and write BESQ{V) instead ofBESQs
if we want to use v instead ofS and likewise Q[v). We will use v and 8 in the same
statements, it being understood that they are related by the above equation.
We have thus defined a one-parameter family of processes which for integer
dimensions, coincides with the squared modulus of BM5. For every t and every
a > 0, the map x —» QSX[X, > a] where X is the coordinate process is increasing,
thanks to the comparison theorems, hence Borel measurable. By the monotone
class theorem, it follows that x —> Q\[Xt e A] is Borel measurable for every
Borel set A. By Theorem A.9) in Chap. IX, these processes are therefore Markov
processes. They are actually Feller processes, which will be a corollary of the
following additivity property of the family BESQ. If P and Q are two probability
measures on C(R+,R), we shall denote by P * Q the convolution of P and Q,
that is, the image of P <g> Q on C(R+, RJ by the map (со, со') ->¦ co + co'. With this
notation, we have the following result which is obvious for integer dimensions.
A.2) Theorem. For every 8, 8' > 0 and x, x > 0,
Proof For two independent linear BM's /3 and /3', call Z and Z' the corresponding
two solutions for (x, S) and (x', S'), and set X = Z + Z'. Then
X, = x + x' + 2 I (Jz<dfa + y/Z^dp'^ + (S + S')t.
Let j6" be a third BM independent of ft and ft'. The process у defined by
Yt= l(x,>0) J? + l(X,=0)<C
./0 V^.s ./0
is a linear BM since (у, у), = t and we have
X, = (x + x') + 2 / /X~sdys + (8 + S')t
Jo
which completes the proof.
Remark. The family Q\ is not the only family with this property as is shown in
Exercise A.13).
A.3) Corollary. If ц is a measure on R+ such that /0°°(l + t)djx(t) < oo, there
exist two numbers Ац and Bfl > 0 such that
Q* IexP ( -
where X is the coordinate process.

§1. Bessel Processes 441
Proof. Let us call ф{х, S) the left-hand side. The hypothesis on ц. entails that
ф(х,8)>ехр(-0^A Xtdn(t)j\=exp(- f (x + 8t)d^(t) J > 0.
Furthermore, from the theorem it follows easily that
ф(х + x\ 8 + 8') = ф{х, 8)ф(х', 8'),
so that
ф(х,8) =ф(х,0)ф@,8).
Each of the functions </>(-, 0) and 0@, •) is multiplicative and equal to 1 at 0.
Moreover, they are monotone, hence measurable. The result follows immediately.
?
By making ц = ks,, we get the Laplace transform of the transition function
of BESQS. We need the corresponding values of Ац and Вц which we compute
by taking 8—1. We then have for к > 0,
Q\ [exp(-AX,)] = Q\ Гехр (-А. f Xs?,(ds)X] - E^ [exp(-XB,2)]
where В is BM'. This is easily computed and found equal to
A +2А?Г1/2ехр(-А.*/A +2А.О).
As a result
Q\ [exp(-kX,)] = A + 2kty exp ( - kx/(l + 2kt)).
By inverting this Laplace transform, we get the
A.4) Corollary. For S > 0, the semi-group ofBESQs has a density in у equal to
( )() t>0,x>0,
where v is the index corresponding to 8 and Iv is the Bessel function of index v.
For x = 0, this density becomes
ys/2-] exp(-y/2t).
The semi-group of BESQ0 is given by
Q°(x, ¦) = s\p(-x/2t)e0 + S,(x, •)
where 3,(x, •) has the sensity
<7,V y) = Bt)-l(y/xy]/2 exp ( - {x + y)/2t)lx (Jxy/t)
(recall that h = 1-х)-

442 Chapter XI. Bessel Processes and Ray-Knight Theorems
A consequence of these results is, as announced, that BESQ* is a Feller pro-
process. This may be seen either by using the value of the density or by observing
that for / e Co([O, oof), Q\[f{Xt)] is continuous in both x and t; this follows
from the special case f(x) — exp(—Xx) and the Stone-Weierstrass theorem. Thus
we may apply to these processes all the results in Chap. III. We proceed to a few
observations on their behavior.
The comparison theorems and the known facts about BM in the lower dimen-
dimensions entail that:
(i) for S > 3, the process BESQ^ is transient and, for S < 2, it is recurrent,
(ii) for 8 > 2, the set {0} is polar and, for S < 1, it is reached a.s. Furthermore
for <5 = 0, {0} is an absorbing point, since the process X = 0 is then clearly
a solution of the SDE of Definitions A.1).
These remarks leave some gaps about the behavior of BESQS for small S. But
if we put
sv{x) = —x~v for v > 0, sq(x) — log;t, sv(x) — x'v for v < 0
and if T is the hitting time of {0}, then by Ito's formula, sv{X)T is a local
martingale under Q\. In the language of Sect. 3 Chap. VII, the function sv is a
scale function for BESQ*, and by the reasonings of Exercise C.21) therein, it
follows that for 0 < 8 < 2 the point 0 is reached a.s.; likewise, the process is
transient for S > 2. It is also clear that the hypotheses of Sect. 3 Chap. VII are in
force for BESQ* with E = [0, oo[ if S < 2 and E =]0, oof if S > 2. In the latter
case, 0 is an entrance boundary; in the former, we have the
A.5) Proposition. For 8=0, the point 0 is absorbing. For 0 < S < 2, the point
0 is instantaneously reflecting.
Proof. The case S — 0 is obvious. For O<<5<2, ifXisa BESQs, it is a semi-
martingale and by Theorem A.7) Chap. VI, we have, since obviously L®~(X) — 0,
LU,(X) = 28 f \{Xl
Jo
On the other hand, since d(X, X), = 4X,dt, the occupation times formula tells us
that
f (' ^, X)s
Jo
= /
Jo
If L°(X) were not = 0, we would have a contradiction. As a result, the time spent
by X in 0 has zero Lebesgue measure and by Corollary C.13) Chap. VII, this
proves our claim.

§1. Bessel Processes 443
Remarks, i) A posteriori, we see that the term /0 \{xs=n)dfi'^ in the proof of The-
Theorem A.2) is actually zero with the exception of the case 8 — 8' = 0.
ii) Of course, a BESQS is a semimartingale and, for S > 2, it is obvious that
L°(A') = 0 since 0 is polar.
This result also tells us that if we call mv the speed measure of BESQ^ then
for S > 0, we have mv({0}) = 0. To find mv on ]0, oo[ let us observe that by
Proposition A.7) of Chap. VII, the infinitesimal generator of BESQ*5 is equal on
C2K ( ]0. oo[ ) to the operator
d2 d
2X—+8-- .
dx1 dx
By Theorem C.14) in Chap. VII, it follows that, for the above choice of the
scale function, mv must be the measure with density with respect to the Lebesgue
measure equal to
jrv/2v for v > 0, 1/2 for v = 0, -x"/2v for v < 0.
The reader can check these formulas by straightforward differentiations or by using
Exercise C.20) in Chap. VII.
Let us now mention the scaling properties of BESQS. Recall that if В is a
standard BM^ and B,c = x + B, then for any real с > 0, the processes B*Sl
and cB^c have the same law. This property will be called the Brownian scaling
property. The processes BESQ have a property of the same ilk.
A.6) Proposition. IfX is a BESQs(x), then for any с > 0, the process c~]Xa is
a BESQs(x/c).
Proof. By a straightforward change of variable in the stochastic integral, one sees
that
л f
c~xXa =c~lx+2 / (c']Xcs)i/2c^/2dBcs +8t
Jo
and since c^x^2Bcl is a BM, the result follows from the uniqueness of the solution
to this SDE. ?
We now go back to Corollary A.3) to show how to compute the constants Ац
and S/(; this will lead to the computation of the exact laws of some Brownian
functionals. Let us recall that if ji is a Radon measure on [0, oo[, the differential
equation (in the distribution sense) ф" = фц has a unique solution ф^ which is
positive, non increasing on [0, oo[ and such that фц{0) = 1. The function ф^ is
convex, so its right-hand side derivative ф'^ exists and is < 0. Moreover, since фц
is non increasing, the limit фц(оо) = \[тх^,осф1,(х) exists and belongs to [0, 1].
In fact, фц(оо) < 1 if we exclude the trivial case \x = 0; indeed, if (/^(oo) = 1,
then фц is identically 1 and ji = 0.
We suppose henceforth that /A + x)d/i(x) < oo; we will see in the proof
below that this entails that </>,, (oc) is > 0. We set

444 Chapter XI. Bessel Processes and Ray-Knight Theorems
^ = Г X,
Jo
dfji(t).
In this setting we get the exact values of the constants At, and Вц of Corollary
A.3).
A.7) Theorem. Under the preceding assumptions,
Proof. The function ф'ц is right-continuous and increasing, hence F/((/) =
<p'fl (?)/</>n(t) is right-continuous and of finite variation. Thus we may apply the
integration by parts formula (see Exercise C.9) Chap. IV) to get
F^t)X, = Ffl@)x + I FpWdXs + I XJF^s);
Jo Jo
but
f f d4>'{s) f </>'(s)
/ XsdF,(s) - / X,-pf--/ Xsg— 4фцE)
Jo Jo Фц(х) Jo Ф1^)
= / Xsdfx{s)- I X.F^sfds.
Jo Jo
As a result, since M, — X, - St is a (^.-continuous local martingale, the process
%(\f Flt(s)dMs\ =expQJ WdM<~\f X>W2ds)
is a continuous local martingale and is equal to
Z<" = exp Ц [F;i(/)Xf - />@)д; - S log^(r)] - l-j X,dfi(s)\ .
Since Ff, is negative and X positive, this local martingale is bounded on [0. a]
and we may write
(t) ?[Z?] = E[Z?]=1.
The theorem will follow upon letting a tend to +00.
Firstly, using Proposition A.6), one easily sees that (X,,/ct) converges in law
as a tends to +00. Secondly,
Ф'11{х) = -(ф11ц)(]х,оо[)
(see Appendix 3), and consequently,
0 >aFlt(a) > f x dfi(x)
J]a.oo[ ч

§1. Bessel Processes 445
which goes to 0 as a tends to +oo. As a result, F^(a)Xa converges in probability
to 0 as a tends to +oo and passing to the limit in (f) yields
-8^ф11(а)))ехр(-хЕ1Л{0))ОЧехр[-{\/2) / X,
1 \ \ Jo
Since Jq° X,dfi{t) < oo a.s., this shows that </>,,(oo) > 0 and completes the proof.
Remarks. 1 ') It can be seen directly that Zf is continuous by computing the jumps
of the various processes involved and observing that they cancel. We leave the
details as an exercise to the reader.
2°) Since aF^{a) converging to 0 was all important in the proof above, let us
observe that, in fact, it is equivalent to -afi{[a, оо[) as a goes to infinity.
This result allows us to give a proof of the Cameron-Martin formula namely
E I exp (-X I B2dsj = (cosh ¦
where В is a standard linear BM. This was first proved by analytical methods but
is obtained by making x = 0 and S = 1 in the following
A.8) Corollary.
04 exp ( / Xsds)\ = (coshbrs/2 exp (--xbtanhb) .
L V 2 Jo /J \ 2 /
Proof. We must compute фц when fx(ds) — b2ds on [0, 1]. It is easily seen that
on [0, 1], we must have фцA) = acoshbt +fi sinhbt and the condition фц@) = 1
forces a = 1. Next, since фA is constant on [1, oo[ and ф' is continuous, we must
have <//A) — 0, namely
bsmhb + fib cosh b — 0
which yields ft = — tanh b. Thus фц (t) — cosh bt — (tanh b) sinhbt on [0. 1 ] which
permits to compute ф^{оо) = </>,,(l) = (coshb) ' and ф'цФ) — —btanhb.
Remark. This corollary may be applied to the stochastic area which was introduced
in Exercises B.19) Chap. V and C.10) Chap. IX.
We have dealt so far only with the squares of Bessel processes; we now turn
to Bessel processes themselves. The function x -*¦ -Jx is a homeomorphism of
R+. Therefore if X is a Markov process on R+, Ух is also a Markov process.
By applying this to the family BESQS, we get the family of Bessel processes.
A.9) Definition. The square root ofBESQs{a2), S > 0, a > 0, is called the Bessel
process of dimension S started at a and is denoted by S?5*(«). Its law will be
denoted by P*.

446 Chapter XI. Bessel Processes and Ray-Knight Theorems
For integer dimensions, these processes were already introduced in Sect. 3 of
Chap. VI: they can be realized as the modulus of the corresponding ВМг.
Some properties of BESQS translate to similar properties for BES*. The Bessel
processes are Feller processes with continuous paths and satisfy the hypothesis of
Sect. 3 Chap. VII. Using Exercise C.20) or Exercise C.18) Chap. VII, it is easily
seen that the scale function of BES may be chosen equal to
-x~2v for v > 0, 2 log* for v = 0, x~2v for v < 0,
and with this choice of the scale function, the speed measure is given by the
densities
x2v+[/v for v > 0, x forv = 0, -x2v+l/v forv<0.
Moreover, for 0 < 8 < 2, the point 0 is instantaneously reflecting and for 8 > 2 it
is polar. Using Theorem C.12) Chap. VII, one can easily compute the infinitesimal
generator of BES*.
The density of the semi-group is also obtained from that of BESQS by a
straightforward change of variable and is found equal, for S > 0, to
p't{x, y) = r\ylx)vy exp (-(x2 + y2)/2t) /„(*>•/?) for x > 0,1 > 0
and
pf(O, >•) = 2-ur("+1)/> + 1ГУУ+1 exp(->-2/20-
We may also observe that, since for S > 2, the point 0 is polar for BESQs(x),
x > 0, we may apply Ito's formula to this process which we denote by X and to
the function jx. We get
/о
where /3 is a BM. In other words, BES5(a), a > 0, is a solution to the SDE
5- 1
A = a +
8-\ f _,
/ pt as.
2 Л '
By Exercise B.10) Chap. IX, it is the only solution to this equation. For 8 < 2
the situation is much less simple; for instance, because of the appearance of the
local time, BES1 is not the solution to an SDE in the sense of Chap. IX.
Finally Proposition A.6) translates to the following result, which, for 8 > 2,
may also be derived from Exercise A.17) Chap. IX.
A.10) Proposition. BES5 has the Brownian scaling property.
We will now study another invariance property of this family of processes. Let
Xs be a family of diffusions solutions to the SDE's
Xf = x + B,
\ + f bs(X*)d
Jo

§1. Bessel Processes 447
where b$ is a family of Borel functions. Let / be a positive strictly increasing
C2-function with inverse /~'. We want to investigate the conditions under which
the process f(X&t) belongs to the same family up to a suitable time change.
Ito's formula yields
/(X?) = /(*) +У f\Xl)dBs+j (f\X&s)b&(X&s)+X-f'\X&s)\ds.
Setting x, = inf [u : /0" f'2(Xss)ds > t} and У/ = /(*,,), the above equation may
be rewritten
Y, = fix) + P, + J llf'h + -/" 1 /f 1 о f-\Y*)ds
where p is a BM. Thus, if we can find 8 such that (f'bs + \f") If'2 - bj о /,
then the process Ys will satisfy a SDE of the given family.
This leads us to the following result where pv is a BES(V).
A.11) Proposition. Let p and q be two conjugate numbers (/?"' +q~l = l)- If
v > —\/q, there is a BES*1"" defined on the same probability space as pv such
that
Proof. For v > 0, the process pv lives on ]0, oo[ and xX!q is twice differentiable
on ]0, oo[. Thus we may apply the above method with f(x) — qxi/4 and since
then, with bv — (u + |) x~\ we have
the result follows from the uniqueness of the solutions satisfied by the Bessel
processes for v > 0.
For v < 0, one can show that qpv (r,), where z, is the time-change associated
with /0 pt7 /p(s)ds, has on ]0. oo[ the same generator as pvq; this is done as in
Exercise B.29) Chap. X. Since the time spent in 0 has zero Lebesgue measure,
the boundary 0 is moreover instantaneously reflecting; as a result the generator of
qpjq(xt) is that of pvq(t) and we use Proposition C.14) Chap. VII to conclude.
The details are left to the reader. ?
This invariance principle can be put to use to give explicit expressions for the
laws of some functional of BM.
A.12) Corollary. In the setting of Proposition A.11), if pv@) — 0 a.s., then
\ 1/

448 Chapter XI. Bessel Processes and Ray-Knight Theorems
Proof. Let C, = /0' Pv2lp(s)ds and r, the associated time-change. Within the
proof of the proposition we saw that qpl/q(rs) = pvq(s); since dx, — pllp(x,)dt,
it follows that
[
Jo
It remains to prove that C\ has the same law as т^1/ч. To this end, we first remark
that {C\ > /} = (r, < 1}; we then use the scaling property of pv to the effect that
Jo
which yields
a
The point of this corollary is that the left-hand side is, for suitable v's, a
Brownian functional whereas the Laplace transform of the right-hand side may be
computed in some cases. For instance, making p — q = 2, we get
and the law of the latter r.v. is known from Corollary A.8).
A.13) Exercise. Iе) Let X*, S > 2, be a family of diffusions which are solutions
to the SDE's
f =x + B
, + ( bs(X*)ds.
Jo
Prove that the laws of the processes (X*J satisfy the additivity property of The-
Theorem A.2) provided that
2xbs(x) + 1 =8 +bx2
for a constant b. This applies in particular to the family of Euclidean norms of
Мг-valued OU processes, namely X, — \Y,\, where
b f
Y,=y + B, + - I Ysds.
1 Jo
(See Exercise C.14) Chap. VIII.)
2~) Prove that the diffusion with infinitesimal generator given on Cx( ]0, oo[ )
by
2x d2/dx2 + {2px + S)d/dx
with ft / 0 and S > 0, may be written
e2f»X {(\ - e~2fit)/2p) or e2tuY {\e'2tu - 1
where X and Y are suitable BESQ* processes.
[Hint: Start with 5 = 1 in which case the process is the square of a OU process,
then use 1°).]

§1. Bessel Processes 449
A.14) Exercise (Infinitely divisible laws on W). 1) For every S > 0 and x > 0,
prove that the law Qsx is infinitely divisible, i.e. for every n, there is a law P{n)
on W = C(M+,M) such that
Q\ = Pw*...*Pm (и terms).
2°) Let В and С be two independent Brownian motions starting from 0, and
R be a BES^(jc) process independent from B. Prove that the laws of
а) ВС; b) /0 (BsdC, - CsdBs); c) /0 RsdB<;
d) J0dR2(p(s), where <p : R+ -» K+ is bounded, Borel
are infinitely divisible.
A.15) Exercise. Compute A^ and B^ in Corollary A.3) when /i is a linear com-
combination of Dirac masses i.e. д = ^A.,e,,.
[Hint: Use the Markov property.]
# A.16) Exercise. (More on the counterexample of Exercise B.13) in Chap. V).
Let X be a BES^(jc), and recall that if S > 2, x > 0, the process X2~s is a local
martingale.
1°) Using the fact that Iv(z) is equivalent to F(v + \)~\z/2)v as z tends to
zero, prove that for a < 5/E — 2) and e > 0,
sup EX,1 ' < +00.
[Hint: Use the scaling properties of BES and the comparison theorems for
SDE's.]
2°) For every p > 1, give an example of a positive continuous local martingale
bounded in Lp and which is not a martingale.
[Hint: Show that E [X2~s] is not constant in t.]
A.17) Exercise. Let M be in ..№ (Exercise C.26) in Chap. IV) and such that
lim,|o M, — +00. Prove that the continuous increasing process
A, = / exp(-2Ms)d{M.M),
is finite-valued and that there exists a BES2, say X, such that e\p(-Mt) = XA,-
*# A.18) Exercise. 1°) If X is a BES"'(O), prove that T\ = inf{/ : X, = 1} and
(supr<] X,) have the same law.
[Hint: Write {T\ > t] = {sup,<, X, < 1} and use the scaling invariance prop-
properties.]
2°) If moreover S > 2, prove that L = sup {/ : X, = 1} and (inf,>i X,) have
the same law.
3°) Let К be a BES^(l) independent of X. Show, using the scaling invariance
of X, that inf,>, X, and X\ inf,>() Y, have the same law.

450 Chapter XI. Bessel Processes and Ray-Knight Theorems
4°) Using the property that Y2~s is a local martingale, show that inf,>0 Y, has
the same law as Ul/S~2, where U is uniformly distributed on [0, 1].
[Hint: Use Exercise C.12) Chap. II.]
5°) Show finally that the law of the r.v. L defined in 2C) above is that of
Z^2 where Z is a BES^~2@). Compare this result with the result obtained for the
distribution of the last passage time of a transient diffusion in Exercise D.16) of
Chap. VII.
A.19) Exercise. For p > 0, let X be a BES2p+3@). Prove that there is a BES3@),
say Y, such that
[Hint: Use Ito's formula and the time-change associated with
A.20) Exercise. For S > 1, prove that BES^ satisfies the various laws of the
iterated logarithm.
[Hint: Use Exercise A.21) in Chap. II and the comparison theorems of Sect.
3, Chap. IX.]
A.21) Exercise. For 0 < S < 1, prove that the set \t : X, — 0} where X is a
BES is a perfect set with empty interior.
*# A.22) Exercise. 1 °) L (.^°) be the filtration generated by the coordinate process
X. For the indexes д, v > 0, T a bounded (-i^)-stopping time, Y any .JZj.1-
measurable positive r.v. and any a > 0, prove that
P{/] [Y exp (-y
[H;nt: Begin with /i = 0, v > 0 and use the fa@)-martingale K(vM) where
M - 1 )g(X/a) as described in Exercise A.34) of Chap. VIII.]
2°) Let Wa be the law of the linear BM started at a and define a probability
measure Ra by
R,, - {Х,лТо/а) ¦ Wa on .Ttu,
where Го = inf{/ : X, = 0). Show that Ra is the law of BES3 i.e. faA/2) in the
notation of this exercise.
[Hint: Use Girsanov's theorem to prove that, under Ra, the process X satisfies
the right SDE.]
3°) Conclude that for every v > 0,
P^ = (XIAro/a)v+V2exp(-((v2-\/4)/2) J X;2dsYwa on .V,\

§1. Bessel Processes 451
# A.23) Exercise. Prove that for v > 0, and b > 0, the law of [XLb-,,t < Lh\
under P0(l>) is the same as the law of {X,,t < To} under P^~v).
[Hint: Use Theorem D.5) Chap. VII.]
In particular Lh under P^v) has the same law as To under Pl~~v). [This gener-
generalizes Corollary D.6) of Chap. VII.] Prove that this common law is that of 2/yv,
for Yv a Gamma (v) variable.
[Hint: Use Exercise A.18).]
A.24) Exercise. Let Z be the planar BM. For a < 2, prove that Z{—alog|Z|)
is not a martingale.
[Hint: Assume that it is a martingale, then follow the scheme described above
Proposition C.8) Chap. VIII and derive a contradiction.]
A.25) Exercise. 1°) Prove that even though BES(v) is not a semimartingale for
v < —1/2, it has nonetheless a bicontinuous family of occupation densities namely
a family If of processes such that
/ f(pv(s))ds= / f(x)l?mv{dx)
Jo Jo
a.s.
Obviously, this formula is also true for v > — 1 /2.
[Hint: Use Proposition A.11) with q — —2v.]
2°) Prove that for v e] — 1, 0[ the inverse г of the local time at 0 is a stable
subordinator with index (—v) (see Sect. 4 Chap. Ill), i.e.
E [exp(-Ar,)] = exp(-c/A~1').
A.26) Exercise. (Bessel processes with dimension 8 in ]0, 1[ ). Let 8 > 0 and
p be a BES5 process.
1°) Prove that for S > 1, p is a semimartingale which can be decomposed as
A) p, = po + p, + ((8-\)/2)fp;lds, if 8>\.
Jo
and
B) A = po + 0,+ 1/,°, if 5=1.
where /" is the local time of p at 0. This will serve as a key step in the study of
p for 8 < 1.
2°) For 0 < a < 1/2 and the standard BM1 B, prove that
\B,
\x-a
f
Jo
where P.V. is denned by

452 Chapter XI. Bessel Processes and Ray-Knight Theorems
P.V. [ \B,r*""ds= Г \b\-]' a(/f -lf)db.
[Hint: For the existence of the principal value, see Exercise A.29) Chap. VI.]
3C) Let now p be a BES1* with 8 e]0, 1[; denote by /" the family of its local
times defined by
/ (p{ps)ds= / (P(x)l}xs-ldx
Jo Jo
in agreement with Exercise A.25) above. Prove that
where к, = P.V. /0' p~lds which, by definition, is equal to /0°°a^2 (/," - If) da.
[Hint: Use 2~) as well as Proposition A.11) with v — —1/2.]
A.27) Exercise. 1°) Prove that Q\ {xX;x) = 1 - exp(-x/2/)-
[Hint: ?-' = /03°exp(-A.^)rfA..]
2°) Comparing with the formulae given in Corollary A.4) for QAt and Qf,
check, using the above result, that Q®(x. R+) = 1.
A.28) Exercise (Lamperti's relation). 1°) Let В be a BM'(O). For v > 0, prove
that there exists a BES(VI, say Riv\ such that
exp(B, + vt) = R{v)
/o
2°) Give an adequate extension of the previous result for any у е R.
3") Prove that, for a e R, а ф 0, and b > 0, the law of f™ ds exp(aBv - bs)
is equal to the law of 2/a2yah/a2U where ya denotes a Gamma (a) variable.
[Hint: Use Exercise A.23).]
A.29) Exercise (An extension of Pitman's theorem to transient Bessel pro-
processes). Let (p,, / > 0) be a BES^O) with S > 2, and set J, = infs>, p,. Prove
that there exists a Brownian motion (y,,t > 0) such that
„ , 8-3 f ds
p, —Yt+ 27, —- / —.
2 Jo Pa
[Hint: Use Proposition A.11).]
A.30) Exercise (On Pitman's theorem). In the situation of Section VI.3, assume
that for every t, conditionally on •^5"fl, the r.v. 5, is uniformly distributed on
[0, 25, - B,\, prove then that the process 25 - В is a BES3@). Compare with the
results in Section VI.3, notably Corollary C.6).
[Hint: Prove that B5, - B,J - 3t is a local martingale.]
A.31) Exercise (Seshadri's identities). In the notation of Sections 2 and 3 in
Chapter VI, write
B5, - B,J = E, - B,J + r2 = 52 + p2.

§1. Bessel Processes 453
1°) For fixed t > 0, prove that the r.v.'s r2 and p2 are exponentially distributed
and that (S, - B,J and r] are independent; likewise S2 and pj are independent.
[Hint: Use the results on Gamma and Beta variables from Sect. 6 Chap. 0.]
T) Prove further that the processes r2 and pf are not BESQ2@), however
tempting it might be to think so.
[Hint: Write down their semimartingale decompositions.]
A.32) Exercise. (Asymptotic distributions for funationals X^ whenever
/0*41 + t)dn(t) = 00). Г) Let X be a BESQs(x) with S > 0; for a < 2,
prove that, as t tends to infinity, ta~2 J[u~aXu du converges in distribution to
/„' u~aYu du, where К is a BES?>5@).
2C) In the same notation, prove that (logf) ft u~2Xu du converges a.s. to
8 = E[Y{]. Prove further that (log/)/2 (<5(logO - /,' u~2Xu du\ converges in
distribution to 2ys where у is a BM'(O).
[Hint: Develop (X,/t) ,t> 1, as a semimartingale.]
A.33) Exercise. ("Square" Bessel processes with negative dimensions). 1°) Let
x, S > 0, and p be a BM1. Prove that there exists a unique strong solution to the
equation
Z, =x+2 f J\z7\dps-8t.
Let Qxs denote the law of this process on W =
2°) Show that Го = inf {f : Z, — 0} < 00 a.s., and identify the process
{-Z7o+,,/>O}.
3°) Prove that the family \QYX. у € R, x > 0} does not satisfy the additivity
property, as presented in Theorem A.2).
A.34) Exercise (Complements to Theorem 1.7). Let д be a positive, diffuse,
Radon measure on R+. Together with 0,,, introduce the function i/,,(t) —
1") Prove that \j/fl is a solution of the Sturm-Liouville equation ф" = цф, and
that, moreover, \jj,t @) = 0, 1// @) = 1.
Note the Wronskian relation
/v- if,,) = Фц$\к - Ф'^-ф,, = 1.
2°) Prove that, for every / > 0, one has
У) Check, by letting t ->¦ 00, that the previous formula agrees with the result
given in Theorem 1.7.
[Hint: If f°° t dji(t) < 00, then: fft(t) -»¦ 1/0,, @0) (> 0).]

454 Chapter XI. Bessel Processes and Ray-Knight Theorems
4°) Let В be a BM'(O). Prove that, if v(ds) is a second Radon measure on
R+, then
Гехр \f BMds) -l-f B^(d
/2expU f du( ( ^r\v{ds)) ~B{t)( f
' [2Jo \Ju 4>(u) ) \Jo
(f @) [
where 0@ = 0; @/2 v;(o-
[Hint: Use the change of probability measure considered in the proof of The-
Theorem C.2) below.]
§2. Ray-Knight Theorems
Let В be the standard linear BM and (L") the family of its local times. The
Ray-Knight theorems stem from the desire to understand more thoroughly the
dependence of (L") in the space variable a. To this end, we will first study the
process
Za=LljT\ 0<fl<l,
where T\ = infff : B, — 1}. We will prove that this is a Markov process and in fact
a BES<32 restricted to the time-interval [0, 1]. We call (~?a)ue[o i]< tne complete
and right-continuous filtration generated by Z. For this filtration, we have a result
which is analogous to that of Sect. 3, Chap. V, for the Brownian filtration.
B.1) Proposition. Any r.v. H of L2 (-?«). 0 < a < 1, may be written
H = E[H]+ /гЛ,В,>1-а^В,
Jo
where h is predictable with respect to the filtration of В and
Г f 1
E\ I h].\{B^\-a)ds < oo.
L/o ' J
Proof. The subspace .Ж of r.v.'s H having such a representation is closed in
L2 (-Za), because
E[H2] = E[H]2 +
and one can argue as in Sect. 3 of Chap. V.
We now consider the set of r.v.'s К which may be written
|- j g(b)Zhdb\

§2. Ray-Knight Theorems 455
with g a positive C'-function with compact support contained in ]0, a[. The vector
space generated by these variables is an algebra of bounded functions which,
thanks to the continuity of Z, generates the ст-field -Za. It follows from the
monotone class theorem that this vector space is dense in L2 (-Za). As a result, it
is enough to prove the representation property for K.
Set U, = expj-/0'g(l - Bs)dsi; thanks to the occupation times formula,
since #A — x) vanishes on ]0, 1 — a[,
K=exp\-j g(\-x)LxTidx\ = UTl.
If F € C2, the semimartingale M, = F(B,)U, may be written
M, = F@) - f F(B,)Usg(\ - Bs)ds+ f UsF'(Bs)dB.s + )- f UsF"(B,)ds.
Jo Jo *¦ Jo
We may pick F so as to have F' = 0 on ] — oo, 1 — a], F(\) Ф 0 and F"(x) —
2g(\ — x)F(x). We then have, since F' = F'l]i_aoc[,
fTl
MTl = F(Q)+ /
fTl
/ U,F\B,)\(Bi>X-a)dBs
Jo
and, as К — F(l) 'Л/г,, the proof is complete.
We may now state what we will call the first Ray-Knight theorem.
B.2) Theorem. The process Za, 0 < a < 1 is a BESB2@) restricted to the time
interval [0, 1].
Proof. From Tanaka's formula
(B,
it follows that
¦,-(l-fl))+= f \iB.>\-a)dBs + \L)-a,
Jo l
Za - 2a = -2 I \{Bl>\-a)dBs.
Jo
It also follows that Za is integrable; indeed, for every t
and passing to the limit yields E[Za] = 2a.
Now, pick b < a and H a bounded -2/,-measurable r.v. Using the representa-
representation of the preceding proposition, we may write
E [(Za - la) H] = E
-2 I hs\{B<>i-b)\{B,>\-a)ds
fTl 1
-2 / hsllBt>\-h)ds
Jo J
= E
= E [(Zh - 2b) H].

456 Chapter XI. Bessel Processes and Ray-Knight Theorems
Therefore, Za-2a is a continuous martingale and by Corollary A.13) of Chap. VI,
its increasing process is equal to 4/0" Zudu. Proposition C.8) of Chap. V then
asserts that there exists a BM J} such that
Za = 2 / Z]/ldpu + 2a;
Л
in other words, Z is a BES?>2@) on [0, 1]. ?
Remarks. 1") This result may be extended to the local times of some diffusions, by
using for example the method of time-substitution as described in Sect. 3 Chap. X
(see Exercise B.5)).
2°) The process Z,, is a positive submartingale, which bears out the intuitive
feeling that L" has a tendency to decrease with a.
3°) In the course of the proof, we had to show that Za is integrable. As it turns
out, Zu has moments of all orders, and is actually an exponential r.v. which was
proved in Sect. 4 Chap. VI.
4°) Using the scaling properties of BM and BES<22, the result may be extended
to any interval [0, c].
We now turn to the second Ray-Knight theorem. For x > 0, we set
xx — inf{/ : L° > x\.
B.3) Theorem. The process LaT>, a > 0, is a BESQ°(x).
Proof. Let g be a positive C1-function with compact support contained in ]0. oof
and Fg the unique positive decreasing solution to the equation F" = gF
such that Fg@) = 1 (See the discussion before Theorem A.7)). If f(k,x) -
exp (-(A/2)F^@)) F^(x), Ito's formula implies that, writing L for 1°,
/ (L,, B+) = 1 + [ /; (Z,,, B+) \{Bt>0)dBs + \ [ f'x {Ls, B+) dLs
Jo ^ Jo
1 f + f
2 Jo Jo
In the integrals with respect to dLs, one can replace Bs+ by 0, and since
j/v'(A,O) + Д'(А.О) = 0, the corresponding terms cancel. Thus, using the in-
integration by parts formula, and by the choice made of Fg, it is easily seen that
f(L,, B,+ ) exp ( — j /0' g(Bs)dsj is a local martingale. This local martingale is
moreover bounded on [0. rr], hence by optional stopping,
g(Bs)ds] = 1.
But BTi = 0 a.s. since rv is an increase time for L, (see Sect. 2 Chap. VI) and of
course LTi = x, so the above formula reads

§2. Ray-Knight Theorems 457
By the occupation times formula, this may also be written
E Гехр (-Ц g(u)LuTduX\ = exp
If we now compare with Theorem A.7), since g is arbitrary, the proof is finished.
Remarks. 1°) The second Ray-Knight theorem could also have been proved by
using the same pattern of proof as for the first (see Exercise B.8)) and vice-versa
(see Exercise B.7)).
2°) The law of the r.v. L" has also been computed in Exercise D.14) Chap.
VI. The present result is much stronger since it gives the law of the process.
We will now use the first Ray-Knight theorem to give a useful BDG-type
inequality for local times, a proof of which has already been hinted at in Exercise
A.14) of Chap. X.
B.4) Theorem. For every p e ]0, oo[, there exist two constants 0 < cp < Cp < oo
such that for every continuous local martingale vanishing at 0,
where L" is the family of local times of M and L* — supasK Lat.
Proof One can of course, thanks to the BDG inequalities, use (M, M)d instead
of M^ in the statement or in its proof. The occupation times formula yields
/•+OC rM"
(M,M)
/•+OC rM"
= L"ooda= L^da <2M*L*.
J-oc J~M-
Therefore, there exists a constant dp such that
if E [(A/^)''] is finite, which can always be achieved by stopping, we may divide
by E \_{М^У\ and get the left-hand side inequality.
We now turn to the right-hand side inequality. If T, is the time-change associ-
associated with (M, M), we know from Sect. 1 Chap. V that В = MT is a BM, possibly
stopped at S = {M. M)x. By a simple application of the occupation times for-
formula, (L'f) is the family of local times, say (I"), of B.aS, and therefore L*x = l*s.
Consequently, it is enough to prove the right-hand side inequality whenever M is
a stopped BM and we now address ourselves to this situation.
We set ?„ = inf{/ : \B,\ = 2"}. For n = 0, we have, since ?0 = T\ л Г_ь
E [LI] = E Г sup l(\ < E Г sup L«l + E \ sup L°T_] .
L-l<u5l J l_0?«?l J l-\<a<0 J

458 Chapter XI. Bessel Processes and Ray-Knight Theorems
By the first Ray-Knight theorem, Lj~" — 2a is a martingale with moments of
every order; thus, by Theorem A.7) in Chap. II the above quantity is finite; in
other words, there is a constant К such that E [Z,|J = К and using the invariance
under scaling of (B,, L,) (Exercise B.11) Chap. VI), E [L*J = 2"K.
We now prove that the right-hand side inequality of the statement holds for
p = 1. We will use the stopping time
T = inf {?„ : ?„ > S)
for which B*s < Bj < 2B*S + 1 (the 1 on the right is necessary when T = ?„).
Plainly, E [L*s] < E [L*r] and we compute this last quantity by means of the
strong Markov property. Let mbea fixed integer; we have
1-1
n=0
Obviously, ?n+i = ?„ + ?n+i о в^ which by the strong additivity of local times
(Proposition A.2) Chap. X), is easily seen to imply that
Therefore,
J<E
«=o
= E
Гт-I
E
\_n=0
Furthermore, if \a\ = 2", then, under the law Pa, %n+\{B) — ?„+2F - a); since
on the other hand L*{B - a) = Lf(B), we get
J < E
= E
m-\
.л=0
"m-1
n=0
By letting w tend to infinity, we finally get E [L*T] < 4KE [B?] and as a result,
By applying this inequality to the Brownian motion с~' Bci, and to the time c~25,
one can check that
E[L*] <8K(E[B*s] + c2).
Letting с tend to zero we get our claim in the case p = 1, namely, going back to
M,
(*)

§2. Ray-Knight Theorems 459
To complete the proof, observe that by considering for a stopping time S, the
loc.mart. Ms+t, / > 0, we get from (*):
E [L*X(M) - Ц(М)] < &КЕ [M^] .
By applying the Garsia-Neveu lemma of Exercise D.29) Chap. IV, with A, — L*,
X = SKM^ and F(k) = Xp we get the result for all p's. П
* B.5) Exercise. 1°) Let I", a > 0, be the family of local times of BES3@). Prove
that the process l"x, a > 0, is a BES02(O).
[Hint: Use the time-reversal result of Corollary D.6) of Chap. VII.]
2°) For p > 0 let k" be the family of local times of BES2p+3(O). Prove that
A.^, a > 0, has the same law as the process Vp(a)| , a > 0, where
Vp(a) = a-" / spdps
Jo
with p a BM2.
[Hint: Use Exercise A.19) in this chapter and Exercise A.23) Chap. VI.]
3°) The result in 2°) may also be expressed: if/" is the family of local times
of BESrf@) with d > 3, then
D,й > 0) Ш {(d - 2rlad-lU(a2">), a > 0)
where U is a BESg2@).
[Hint: Use property iv) in Proposition A.10) Chap. I.]
4°) Let / be a positive Borel function on ]0, oof which vanishes on [b, oo[
for some b > 0 and is bounded on [c. oo[ for every с > 0. If X is the BESd@)
with d > 3, prove that
/f(Xs)ds < oo a.s. iff / rf(r)dr < oo.
Л
[Hint: Apply the following lemma: let ц be a positive Radon measure on ]0, 1];
let (Vr, r € ]0, 1]) be a measurable, strictly positive process such that there exists
a bounded Borel function ф from ]0, 1] to ]0, oo[ for which the law of ф(г)~] Vr,
does not depend on r and admits a moment of order 1. Then
/" /"'
I Vrdix(r) < oo a.s. iff I ф(г)Л/л(г) < oo. ]
Jo Jo
N.B. The case of dimension 2 is treated in the following exercise.
B.6) Exercise. 1°) Let X be a BES2@), Xa the family of its local times and
T\ = inf{/ : X, = 1}. Prove that the process A." , 0 < a < 1, has the same law as
af/-ioga, 0 < a < 1, where U is BESQ2@).
[Hint: Use 1°) of the preceding exercise and the result in Exercise D.12) of
Chap. VII.]

460 Chapter XI. Bessel Processes and Ray-Knight Theorems
2°) With the same hypothesis and by the same device as in 4°) of the preceding
exercise, prove that
/ f(Xs)ds < oo a.s. iff / r\\ogr\f(r)dr < oo.
Jo Jo
Conclude that for the planar BM, there exist functions / such that /J f(Bs)ds =
+oo Po-a.s. for every / > 0 although / is integrable for the two dimensional
Lebesgue measure. The import of this fact was described in Remark 4 after The-
Theorem C.12) Chap. X.
* B.7) Exercise. (Another proof of the first Ray-Knight theorem). 1°) Let Z be
the unique positive solution to the SDE
i)ds, Zo = 0.
, = 2 f yfzsd^s +2 f }(o<s<
Jo Jo
Prove that the stopping time a — inf{f : Z, = 0} is a.s. finite and > 1.
2°) Let g be a positive continuous function on R with compact support and
/ the strictly positive, increasing solution to the equation /" = 2fg such that
/'(-oo) — 0, /@) = 1. With the notation of Theorem B.2) prove that
g(a)LaTda\]=f(\)-].
aTda\]=f
3°) Set v(x) = /A - x) for x > 0; check that
f
Jo
is a local martingale and conclude that
g(\-b)Zbdb\\= f(\y{.
v(a л 1)-' exp (za^p- - f g(\ - b)Zbdb
\ 2v(a) J
4°) Prove that L]T~a, a > 0, has the same law as Za, a > 0, which entails in
particular Theorem B.2).
* B.8) Exercise. (Another proof of the second Ray-Knight theorem). 1°) In the
situation of Theorem B.3), call (^'a) the right-continuous and complete filtration
of Ц, a > 0. Prove that any variable H in L2 (~Z'a) may be written
Я = E[H]+ f ' hsll0<Bt<a)dBs
Jo
for a suitable h.
2°) Prove that L°( — x is a (-2^)-martingale and derive therefrom another
proof of Theorem B.3).

§2. Ray-Knight Theorems 461
* B.9) Exercise. (Proof by means of the filtration of excursions). Let В be the
standard linear BM. For x € Ж, call xf the time-change inverse of /0' \(B,<\\ds
and set #л — а (йт>, / > 0).
1°) Prove that tf, с Ky for x < y.
2°) Prove that, for H e L2(<fv), there exists a (..^-predictable process h,
such that E [f^hj\(Bt<x)ds] < oo and
= ?[#]+
/ hs\{B^x)dB,.
Jo
3C) For x 6 Ш, define .'?,* = Пе>ост (-^+f, #*)• Prove that if У is a (.^>
local martingale, the process JQ \(B,<x)dYs is a local martingale with respect to
the filtration (.'?/).
4°) Let a e R, 7 be a ('?,fl)-stopping time such that BT < a a.s. and L? is
^a-measurable. For a < x < v, set
V, = - (L,v - LI) + у' - x- - (fi, - >-)+ + F, - x)+
and show that
E[(V, V)T |-V]<+oo.
[Hint: For p > 0, set F(z) = cosh v/2p (v - sup(z, x))+ and 2c = JTp
tanhVI/'Cy - -«)• The process U, — F(B,^T)exp(-c(LxT - L*AT))x
exp(—/?/0'A 1(.v<b,<v)^) is a bounded (.'^r-v)-martingale.]
5°) Prove that LyT + 2y~, у > a is an (<5V) >a-continuous martingale with
increasing process 4/a* Lyc/z. Derive therefrom another proof of the Ray-Knight
theorems.
* B.10) Exercise (Points of increase of Brownian motion). Let В be the stan-
standard linear BM.
1°) Let r be a positive number and Г the set of <w's for which there exists a
time U (w) < r such that
B,(co) < BU(to)(w) for 0 < / < U(w), В,(ш) > BU(@)((o) for U(w) < t < r.
Let i be a rational number and set S, = supv<r Bs; prove that a.s. on Г П
R R ~
{Вц < x < 5,}, we have L^'' = LTL > 0 and that consequently Г is negligible.
2C) If / is a continuous function on R+, a point to is called a point of increase
of / if there is an s > 0 such that
/(/) < /Co) for /o - e < / < fo, /(/) > /Co) for г0 < / < ;0 + e.
Prove that almost all Brownian paths have no points of increase. This gives another
proof of the non-differentiability of Brownian paths.
[Hint: Use Exercise B.15) in Chap. VI to replace, in the case of the Brownian
path, the above inequalities by strict inequalities.]

462 Chapter XI. Bessel Processes and Ray-Knight Theorems
3°) Prove that for any T > 0 and any real number x the set
AT = {t < T : B, = x]
is a.s. of one of the following four kinds: 0, a singleton, a perfect set, the union
of a perfect set and an isolated singleton.
* B.11) Exercise. 1°) Derive the first Ray-Knight theorem from the result in Exer-
Exercise D.17) Chap. VI by checking the equality of the relevant Laplace transforms.
2°) Similarly, prove that the process LaT, a < 0, has the law of A - aJ
X(((l -a) - A -от))") where X is a BESQ4 and m a r.v. on ] - oo, 0[
independent of X with density A — x)~2.
[Hint: For the law of m, see Proposition C.13) i) in Chap. VI.]
* B.12) Exercise. Let В be the standard linear BM, L" the family of its local times
and set Y, = L?'.
F) If •'?/" = -Цв Vcr(B,), prove that for each /
-Y, = B,A0- f llBt>B,)dBs
1 Jo
where the stochastic integral is taken in the filtration ('?(") (see Exercise A.39)
Chap. IV).
2°) Prove that for every t,
sup?
< 00
where г = (s,) ranges through the finite subdivisions of [0, t].
[Hint: Use the decomposition of В in the filtration (.'?(") and the BDG in-
inequalities for local martingales and for local times.]
B.13) Exercise. Retain the notation of the preceding exercise and let 5 be a
(•>^~s)-stopping time such that the map x -> Lxs is a semimart.. Prove that if ф is
continuous, then, in the notation of Exercise A.35) Chapter IV,
P- lim W/' 4>(Yu)dLau- - [ ф(Уи)<ИаА =4 / L^mfdx.
By comparison with the result of Exercise A.35) Chap. VI, prove that Y is not a
(
[Hint: Use the result in 3°) Exercise A.33) Chapter IV.]
B.14) Exercise. (Time asymptotics via space asymptotics. Continuation to Ex-
Exercise A.32)) Prove the result of Exercise C.20) Chapter X in the case d = 3, by
considering the expression
(logV") / \BS\ U\<\BA<^]ds-
Jo
[Hint: Use the Ray-Knight theorems for the local times of BES3@) as described
in Exercise B.5).]

§3. Bessel Bridges 463
§3. Bessel Bridges
In this section, which will not be needed in the sequel save for some definitions,
we shall extend some of the results of Sect. 1 to the so-called Bessel Bridges. We
take S > 0 throughout.
For any a > 0, the space Wa — C([0, a],R) endowed with the topology
of uniform convergence is a Polish space and the ст-algebra generated by the
coordinate process X is the Borel ст-algebra (see Sect. 1 in Chap. XIII). As a
result, there is a regular conditional distribution for P^[- \ Xa], namely a family
Pf'y of probability measures on Wa such that for any Borel set Г
X J X
where ixa is the law of Xa under P*. Loosely speaking
Р%[Г] = Р&Х[Г I Xtt =>•].
For fixed x, S and a, these transition probabilities are determined up to sets of
measure 0 in y; but we can choose a version by using the explicit form found
in Sect. 1 for the density pf. For у > 0, we may define Px;° by saying that for
0 < fi < ... < tn < a, the law of (X,x,..., X,n) under P^-" is given by the
density
pi (jc, Jti)/>*_fi(jti, x2) ¦ ¦ ¦ Pa-,,(xn, y)/p*a(x, y)
with respect to dx\dxi ¦ ¦ .dxn. This density is a continuous function of у on
R+\{0). Moreover, since Iv(z) is equivalent for small z to cvzv where cv is a
constant, it is not hard to see that these densities have limits as у -> 0 and that the
limits themselves form a projective family of densities for a probability measure
which we call Px-". From now on, PXMX will always stand for this canonical system
of probability distributions. Notice that the map (x, y) -> Px-" is continuous in
the weak topology on probability measures which is introduced in Chap. XIII.
The same analysis may be carried through with Q\ instead of Px and leads to
a family Qx-ay of probability measures; thus, we lay down the
C.1) Definition. A continuous process, the law of which is equal to Px'°. (resp.
Qx'"y) is called the Bessel Bridge (resp. Squared Bessel Bridge) from x to у over
[0, a] and is denoted by BES^x, >•) (resp. BESQsa(x, y)).
All these processes are inhomogeneous Markov processes; one may also ob-
observe that the square of BES^(x, v) is BESQsa(x2, y2).
Of particular interest in the following chapter is the case of BES3@, 0). In
this case, since we have explicit expressions for the densities of BES3 which
are given in Sect. 3 of Chap. VI, we may compute the densities of BES3@, 0)
without having to refer to the properties of Bessel functions. Let us put /,(>•) =
Bл-г3)/2>>ехр((-(>'2/2ОI(>>о) and call q, the density of the semigroup of BM

464 Chapter XI. Bessel Processes and Ray-Knight Theorems
killed at 0. If 0 < /, < t2 < ... < tn < a, by the results in Sect. 3 Chap. VI, the
density of [Xu X,n) under the law Ро3" is equal to
Letting z converge to zero, we get the corresponding density for Pq'q, namely
2 Bяа3) ' /,,(.vi )<?,,-„ (>'i, V2) ¦ ..?,„-,„.,(>•„_,. yn)la-t,,(yn).
We aim at extending Theorem A.7) to BESQsa(x, y).
C.2) Theorem. Let ц. be a measure with support in [0, 1]. There exist three con-
constants А, А, В depending only on /л, such that
Proof. We retain the notation used in the proof of Theorem A.7) and define a
probability measure R^ on .Ц = a (Xs,s < 1) by /?f" = Zf • Qst. The law
of X, under R^ has a density г,'ц(х. ¦) which we propose to compute. By an
application of Girsanov's theorem, under /?JM, the coordinate process X is a
solution to the SDE
Eq. C.1) X, = x + 2 f yfxldfr +2 \ F^(s)Xsds + 1.
Jo Jo
If H is a solution to
Eq. C.2) H, = и + В, + [ Ffl(s)Hsds,
Jo
then H2 is a solution to Eq. C.1) with x - u2 and p, - /o'(sgn Hs)dBs. But
Eq. C.2) is a linear equation the solution of which is given by Proposition B.3)
Chap. IX. Thus, H, is a Gaussian r.v. with mean um(t) and variance o2(t) where
f
Jo
If we recall that q}(x, ¦) is the density of the square of a Gaussian r.v. centered
at л/х and with variance t, we see that we may write
r]"(x,-)=ql4t)(xm2(t),-).
Furthermore, it follows from Theorem A.2) that
as a result
г!-"{х.-) = я1н„(хт2Ц).-).
We turn to the proof of the theorem. For any Borel" function / > 0,

§3. Bessel Bridges 465
= exp U^(O)jr + 8 log^(l)! **•" [/(X,)]
since <^A) = 0. Consequently, for Lebesgue almost every y,
— exp
= exp
-F
+ 8\оёф1л(\)\г8/(х,у)/с1&1(х,у)
г\\),у)/д\(х,у).
Using the explicit expressions for cr2(l) and m2(\) and the value of q^ found in
Sect. 1, we get the desired result for a.e. у and by continuity for every у. а
In some cases, one can compute the above constants. We thus get
C.3) Corollary. For every b > 0
]
^^)(l - b cothfc)! lv (Ьфсу/ sinhfc) / lv
In particular,
Proof. The proof is patterned after Corollary A.8). The details are left to the
reader. ?
We are now going to extend Corollary A.12) to Bessel Bridges. We will need
the following
C.4) Lemma. Let X be a real-valued Markov process, g a positive Borel function
such that /0°° g(Xs)~lds — сю. If we set
C,= f g(Xsylds, Xl = Xd,
Jo

466 Chapter XI. Bessel Processes and Ray-Knight Theorems
where C, is the time-change associated with C, then dC, = g(X,)dt. Moreover, if
we assume the existence of the following densities
P,(x, y) = PX[X, e dy]/dy. p,(x, y) = PX[X, e dy]/dy,
hx.y(t, u) = PX[C, edu | X, = y]/du, Ад._,.(/, и) = PX[C, edu\X,= y]/du,
then dt dudy-a.e.
Pu(x, y)g(y)hxy(u, t) - p,(x, y)hx,y(t, u).
Proof. The first sentence has already been proved several times. To prove the
second, pick arbitrary positive Borel functions ф, ф and /; we have
Ex\f
= j dt^(t) j p,(x, y)f(y)dy I 0(и)А,._у(Л u)du,
but on the other hand, using the time-change formulas the same expression is
equal to
Ex \J
dt<j>(t) I p,(x,y)g(y)f(y)dy / <)>(u)hx.y(t,u)du.
Interchanging the roles played by t and и in the last expression and comparing
with that given above yields the result.
We may now state
C.5) Theorem. For v > 0, p and q two conjugate numbers > 1 and pv(t), 0 S
t <\ a BES([V)@, 0), we put
j
Then, with ku — BT(v+ 1))"'. for every positive Borel function f
KE [f {xv_p) {q-2xv.py] = KqE[f (?,,„)] -
Proof. We use Lemma C.4) with g(x) = хг>р and X a BESA)@). Using Propo-
Proposition A.11), we see that

§3. Bessel Bridges 467
where RVI/ is a BES(l"". It follows from the first sentence of Lemma C.4) that
C, = f' g(Xs)ds = q-^'P [' Rvq(SJ«l4s.
Jo Jo
By a straightforward change of variable, the density p,@, y) of X, may be deduced
from pl,\0, y) and is found to be equal to
exp ( - {qyV'f/2t).
If we write the equality of Lemma C.4) for и = 1 and x = 0, we obtain
because of the cancellation of the powers of y. If we further make у = 0, we get
\, t) = Xvr(v+^hm(t, 1).
Next, by the definition of hxy and the scaling properties of X seen in Proposition
A.10), one can prove that
this is left to the reader as an exercise. As a result
Now, hooU, 1) is the density of упчр and Aood,/) is that of xvp. Therefore if /
is a positive Borel function
- M"Bv+1) />/(r1/«)r-
Making the change of variable t = u~4, this is further equal to
Xvq~Bv+l) j f(u)uv400(Uu)qdu = Xv j f(u) (u/q2) A00(l, u)du
= KE[f(xv,,)(xv.p/q2)V4]
as was to be proved.
C.6) Exercise. 1°) Prove that under Qsx, the process
A -иJХ„/A_и), 0<м < 1,
has the law Qsxl0.
2°) Prove that
УО * УО И

468 Chapter XI. Bessel Processes and Ray-Knight Theorems
C.7) Exercise. Let P be the law of Xi_,, 0 < / < 1, when X has law P. Prove
that
Qs,[ = eji-
* C.8) Exercise. State and prove a relationship between Bessel Bridges of integer
dimensions and the modulus of Brownian Bridges. As a result, the corresponding
Bessel Bridges are semimartingales.
C.9) Exercise. If p is the BES^@, 0), prove that
C.10) Exercise. For a fixed a > 0 and any x, y, <5, / such that / > a, prove that
P*-[. has a density Z(a'] on the er-algebra .7~a = a(Xs,s < a) with respect to Ръх.
Show that as t -*¦ oo, (Z^°) converges to 1 pointwise and in Z.1 (/**), a fact which
has an obvious intuitive interpretation. The same result could be proved for the
Brownian Bridge.
[Hint: Use the explicit form of the densities.]
* C.11) Exercise. (Stochastic differential equation satisfied by the Bessel
Bridges). In this exercise, we write Л* for />0\'.
F) Prove that for t < 1,
where h(t, x) = limv^0 p\_t{x, y)/p\(Q, >•) = (!- O~'/2 exp (-x2/2A - /)).
[Hint: Compute EPs[F (Xu,u <tL>(X[)] by conditioning with respect to
er(Xi) and with respect to .^.]
2°) Prove that for S > 2, the Bessel Bridge between 0 and 0 over [0, 1] is the
unique solution to the SDE
[Hint: Use Girsanov's theorem.]
3°) Prove that the squared Bessel Bridge is the unique solution to the SDE
C.12) Exercise. 1°) Prove that the Bessel processes satisfy the hypothesis of Ex-
Exercise A.12) Chap. X.
2°) Prove that for S > 2,
P^. = (P^ [• | Ly = a],
where Ly = sup{/ : X, = y} and the notation PL> is defined in Sect. 4 of Chap.
XII.
[Hint: Use Exercise D.16) Chap. VII.]

Notes and Comments 469
Notes and Comments
Sect. 1. The systematic study of Bessel processes was initiated in Me Kean [1];
beside their interest in the study of Brownian motion (see Chaps. VI and VII)
they afford basic examples of diffusions and come in handy for testing general
conjectures. Theorem A.2) is due to Shiga-Watanabe [1]; actually, these authors
characterize the family of diffusion processes which possess the additivity property
of Theorem A.2) (in this connection, see Exercise A.13)). Corollaries A.3) and
A.4) are from Molchanov [1]. An explicit Levy-Khintchine formula for Qsx is
given in Exercise D.21), Chap. XII. Exercise A.14) presents further examples of
infinitely divisible laws on W = C{R+, R). The following question arises naturally
Question 1. Under which condition on the covariance 8(s,t) of a continuous
Gaussian process (X,, t > 0) is the distribution of (Xj, t > 0) infinitely divisible?
Theorem A.7) is taken from Pitman-Yor [3] and Corollary A.8) is in Levy [3].
Proposition A.11) is in Biane-Yor [ I ] where these results are also discussed at the
Ito's excursions level and lead to the computation of some of the laws associated
with the Hilbert transform of Brownian local times, the definition of which was
given in Exercise A.29), Chap. VI.
Exercise A.17) is due to Calais-Genin [1]. Exercise A.22) is from Yor [10]
and Pitman-Yor [1]; Gruet [3] develops similar results for the family of hyperbolic
Bessel processes, mentioned in Chap. VIII, Sect. 3, Case 3.
For Exercise A.25) see Molchanov-Ostrovski [1] who identify the distribution
of the local time at 0 for a BES5, S < 2 as the Mittag-Leffler distribution; their
result may be recovered from the result in question 2C).
Exercise A.26) may serve as a starting point for the deep study of BES6,
0 < S < 1, by Bertoin [6] who shows that @, 0) is regular for the 2-dimensional
process (p.k) and develops the corresponding excursion theory.
Exercise A.28) is a particular example of the relationship between exponentials
of Levy processes and semi-stable processes studied by Lamperti ([1], [2]) who
showed that (powers of) Bessel processes are the only semi-stable one-dimensional
diffusions. Several applications of Lamperti's result to exponential functionals of
Brownian motion are found in Yor ([27], [28]); different applications to Cauchy
principal values are found in Bertoin [9].
The result presented in Exercise A.29) is a particularly striking case of the ex-
extensions of Pitman's theorem to transient diffusions obtained by Saisho-Tanemura
[1]; see also Rauscher [I], Takaoka [1] and Yor [23].
Exercise A.33) is taken from A. Going's thesis [1]. Exercise A.34) is taken
partly from Pitman-Yor [3], partly from Follmer-Wu-Yor [1].
Sect. 2. The Ray-Knight theorems were proved independently in Ray [1] and
Knight [2]. The proof given here of the first Ray-Knight theorem comes from
Jeulin-Yor [1] as well as the second given in Exercise B.8). The proof in Exercise
B.7) is due to McGill [1] and provides the pattern for the proof of the second
Ray-Knight theorem given in the text. Another proof is given in Exercise B.9)
which is based on Walsh [2] (see also Jeulin [3]). Exercise B.11) is from Ray [1].

470 Chapter XI. Bessel Processes and Ray-Knight Theorems
The proof of Theorem B.4) is the original proof of Barlow-Yor [1], its difficulty
stemming from the fact that, with this method, the integrability of L* has to be
taken care of. Actually, Bass [2] and Davis [5] proved that much less was needed
and gave a shorter proof which is found in Exercise A.14) of Chap. X. See Gundy
[1] for some related developments in analysis.
Exercises B.5) and B.6) are from Williams [3] and Le Gall [3], the lemma in
the Hint of B.5) 4°) being from Jeulin ([2] and [4]) and the final result of B.6)
from Pitman-Yor [5]. The result of Exercise B.10) is due to Dvoretsky et al [1]
and Exercise B.12) to Yor [12].
Although it is not discussed in this book, there is another approach, the so-
called Dynkin's isomorphism theorem (Dynkin [3]), to Ray-Knight theorems (see
Sheppard [1]). A number of consequences for local times of Levy processes have
been derived by Marcus and Rosen ([1], [2]).
Exercise B.13) comes from discussions with B. Toth and W. Werner. Another
proof, related to the discussion in Exercise B.12), of the fact that L,' is not a
semimartingale has been given by Barlow [1]. The interest in this question stems
from the desire to extend the Ray-Knight theorems to more general classes of
processes (for instance diffusions) and possibly get processes in the space variables
other than squares of Bessel or OU processes.
Sect. 3. For the results of this section, see Pitman-Yor ([1] and [2]) and Biane-Yor
[1].

Chapter XII. Excursions
§1. Prerequisites on Poisson Point Processes
Throughout this section, we consider a measurable space (U, M) to which is
added a point 8 and we set Us = U U [8), Ms = a( M, {8}).
A.1) Definition. A process e = (e,,t > 0) defined on a probability space
(?2, .7", P) with values in (Us, Ms) is said to be a point process if
i) the map (t, со) -*¦ e,(co) is .S?(]0, oo[) ® .У-measurable;
ii) the set Dw = [t : e,(co) Ф 8} is a.s. countable.
The statement ii) means that the set {со : Dw is not countable} is in .V and
has probability zero.
Given a point process, with each set Г е Ms, we may associate a new point
process er by setting e[(co) — e,(co) if e,(co) e Г, ef (со) — 8 otherwise. The
process er is the trace of e on Г. For a measurable subset Л of ]0, oo[xU, we
also set
In particular, if A — ]0, t] x Г, we will write Nf for NA; likewise
A.2) Definition. A point process is said to be discrete if N^ < oo a.s. for every t.
The process e is a-discrete if there is a sequence (?/„) of sets, the union of which
is U and such that each eu" is discrete.
If the process is a -discrete, one can prove, and we will assume, that all the
N л 's are random variables.
Let us now observe that the Poisson process defined in Exercise A.14) of
Chap. II is the process Nu associated with the point process obtained by making
U — R+ and e,(co) — t if there is an n such that Sn(co) — t, e,(co) — 8 otherwise.
More generally we will set the
A.3) Definition. Let (Q,.W,.^, P) be a filtered probability space. An (.%')-
Poisson process N is a right-continuous adapted process, such that No — 0, and
for every s < t, and к € N,

472 Chapter XII. Excursions
P [N, -Ns=k\ .%\ = c*U~S) exp( - c{t - s))
for some constant с > 0 called the parameter of N. We set AN, — N, — N,-.
We see that in particular, the Poisson process of Exercise A.14) Chap. II is
an (.>^")-Poisson process for its natural filtration (.7^) = (a(Ns, s < t)). We have
moreover the
A.4) Proposition. A right-continuous adapted process is an {.Vt)-Poisson process
if and only if it is a Levy process which increases only by jumps a.s. equal to 1.
Proof Let N be an (.>^)-Poisson process. It is clear from the definition that it is
an integer-valued Levy process and that the paths are increasing. As a result, for
any fixed T, the set of jumps on [0, T] is a.s. finite and consequently,
sup (N, - N,-) = lim max (NkT/n - N(k-i)T/n) a.s.;
0<,<T " !^<"
but
P max (NkT/n - %-1)г/я)
|_1 <k<n
which goes to 1 as я tends to infinity. Hence the jumps of N are a.s. of magnitude
equal to 1.
Conversely, if N is a Levy process which increases only byjumps of magnitude
I, the right-continuity of paths entails that each point is a holding point; moreover
because of the space homogeneity, the times of jumps are independent exponential
r.v.'s with the same parameter, hence the increments N, — Ns are Poisson r.v.'s.
D
Here are a few more properties of the Poisson process, the proofs of which
are left to the reader.
Firstly, recall from Exercise A.14) Chapter II that M, = N, — ct is a (.^")-
martingale and Mj — ct is another one. More generally, the paths of N being
increasing and right-continuous, the path by path Stieltjes integral with respect to
dNs(a)) is meaningful. Thus, if Z is a predictable process such that
K'H
< oo for every /,
L^o J
then
/ ZsdNs -c I Zsds = f ZsdMs
Jo Jo Jo
is an (.^")-martingale. This may be seen by starting with elementary processes
and passing to the limit. The property does not extend to optional processes (see

§1. Prerequisites on Poisson Point Processes 473
Exercise A.14) below). Further, as for BM, we also get exponential martingales,
namely, if / e ЦЖ(Ш.+ ), then
L{ = exp \i J' f(s)dN, - с f (е'/ы - l) с/Л
is a (.^")-martingale. Again, this may be proved by starting with step functions
and passing to the limit. It may also be proved as an application of the "chain
rule" formula for Stieltjes integrals seen in Sect. 4 of Chap. 0.
Finally, because the laws of the jump times are diffuse, it is easily seen that
for any Poisson process and every fixed time /, AN, — N, — N,~ is a.s. zero (this
follows also from the general remark after Theorem B.7) Chapter III).
A.5) Proposition. IfN1 and N2 are two independent Poisson processes, then
= 0 a.s.;
in other words, the two processes almost surely do not jump simultaneously.
Proof. Let Tn, n > 1, be the successive jump times of /V1. Then
a.s..
Since AN? = 0 a.s. for each /, by the independence of /V2 and the Tn's, we get
ANj =0 a.s. for every n, which completes the proof. ?
We now generalize the notion of Poisson process to higher dimensions.
A.6) Definition. A process (Nl,..., Nd) is a rf-dimensional (.^")-Poisson Pro-
Process if each N' is a right-continuous adapted process such that N'o = 0 and if there
exist constants c,- such that for every t > s > 0,
(c{(t - s))k'
By Proposition A.5), no two components N' and N' jump simultaneously.
We now work in the converse direction.
A.7) Proposition. An adapted process N = (Nl Nd) is a d-dimensional
(.V^)-Poisson process if and only if
i) each N' is an (.^)-Poisson process,
ii) no two N' 's jump simultaneously.
Proof. We need only prove the sufficiency, i.e. that the r.v.'s N',—N's,i = 1,..., d,
are independent. For clarity's sake, we suppose that d = 2. For any pair (/i, /2)
of simple functions on E+, the process

474 Chapter XII. Excursions
X, = exp Ii (j Ms)dN} + j Ms)dNj\ j
changes only by jumps so that we may write
0<y<r
0<s<r
Condition ii) implies that, if AN* — 1 then ANJ = 0 and vice-versa; as a result
X, = 1 + J2 x*- WflW ~ 0 ANl + (е'Ш ' 0 AND ¦
0<s<t
The process Xs- is predictable; using integration with respect to the martingales
M\ = N\ -Cjt, we get
E[Xt] = 1 + E U' Xs_ {(e1^ - I)c, + (e1^ -])c2]ds].
But {s : Xs Ф Xs-} is a.s. countable hence Lebesgue negligible, and consequently
E[X,] = 1 + E Г/" Xs {(e'*»> - 1) c, + (e'/2(i) - 1) c2}
[ E[XS] {(eif^ - 1) с, + (е'ЛA) - l) c-2} ds.
As a result,
ВД] = exp ( Г n (e'/:(I) - 1) с/Л exp ( f c2 (e'/jE) - 1) rfsV
which completes the proof. о
We now turn to the most important definition of this section.
A.8) Definition. An (.5^")-Poisson point process (in short: (.7~t)-PPP) is a in-
indiscrete point process (e,), such that
i) the process e is (.^)-adapted, that is, for any Г е 'М, the process Nf is
{.^)-adapted;
ii) for any s and t > 0 and any Г e 'M, the law of N-? t+t, conditioned on ,WS is
the same as the law of N['.
The property ii) may be stated by saying that the process is homogeneous in
time and that the increments are independent of the past. By Propositions A.4)
and A.7) each process /Vr for which Nf < oo a.s. for every /, is a Poisson
process and, if the sets Г, are pairwise disjoint and such that Np < oo a.s.,
the process ( N,r', / = 1, 2,..., dj is a rf-dimensional Poisson process. Moreover,
when Nf < oo a.s., then E [N,r] < oo and the map / -+ E [Nf] is additive,
thus -tE [Nf] does not depend on t > 0.

§1. Prerequisites on Poisson Point Processes 475
A.9) Definition. The a-finite measure n on M defined by
п(Г) = -E[N,r]. t > 0,
is called the characteristic measure of e.
Thus, if п(Г) < oo, п(Г) is the parameter of the Poisson process Nr.
If Л e .S?(M.+) ® Ms, the monotone class theorem implies easily that
E[NA}= f dt f \A(t,u)n{du).
Let us also observe that, if п(Г) < oo, then Ntr — 1п(Г) is an (.>^")-martingale
and, more generally, we have the
A.10) Proposition (Master Formula). Let H be a positive process defined on
(R+ x Q x Us), measurable with respect to .-У{.?^) ® Ms and vanishing at S,
then
s>0
-АГЧ
H(s, со, u)n(du) .
Proof. The set of processes H which satisfy this equality is a cone which is
stable under increasing limits; thus, by the monotone class theorem, it is enough
to check that the equality holds whenever H(s,w,u) = K(s,w)\r(u) where К
is a bounded positive (.3^)-predictable process and Г e M with п(Г) < oo. In
that case, since Nf - tn(F) is a martingale, the left-hand side is equal to
E ?tf(s,u))lr(ej(u))) = E\f K(s,a>)dNsr(a>)\
- п(Г)Е I / K(s,co)ds
Jo
< 00
which completes the proof.
A.11) Corollary. If moreover
Г f f 1
E\ I ds H(s,to.u)n(dii)
for every t, the process
У2 H(s,co,e,(co))- ds H(s,co, u)n(du)
u7f<t Jo J
is a martingale.
Proof. Straightforward.

476 Chapter XII. Excursions
The following result shows that n characterizes e in the sense that two PPP
with the same characteristic measure have the same law.
A.12) Proposition (Exponential formulas). If f is а ./7(Ж+) ® M-measurable
function such that /J* ds f \f(s, u)\n(du) < oo then, for every t < oo,
/ J2 f(s.eA]=exp\j'dsJ(eif{xu)-\)n(du)\.
For / > 0. the following equality also holds,
Texpl- ? /E,*л)}1=ехр|- J'dsj(\-e-n'-u))n(du)
Proof. The process of bounded variation X, = ?0<л</ f(s< e^ 's Pure'y discon-
discontinuous, so that, for g e ./?(M+),
,)- g(X0) =
If we put 0@ = E [exp(/X,)], we consequently have
= \+E
j
-0<л<( I 0<s'<s
which, using the master formula is equal to
/ dsexpli V f(s\es')\ / (e'fUu) - \)n
Jo [ 0<s-<s i J
The function i/f(-) = f (e'l{'-"] — \)n{du) is in Ll(R+) thanks to the hypothesis
made on /, and we have
f
Jo
It follows that ф is continuous, so that
which is the first formula of the statement. The second formula is proved in much
the same way. ?
Remark. In the above proposition one cannot replace the function / by a non
deterministic process H as in Proposition A.10) (see Exercise A.22)). Moreover
Proposition A.12) has a martingale version along the same line as in Corollary
A.11), namely

§ I. Prerequisites on Poisson Point Processes 477
exp
a ? f(s,es)+ f ds f A -eafi'-u))n(du)\
where a is equal to / or — 1, is a martingale.
We close this section with a lemma which will be useful later on. We set
S = inf{r : N? > 0).
A.13) Lemma. Ifn(U) < oo, then S and es are independent and for any Гё //
P[ese Г] = п(Г)/п(и).
Proof. Since Г П Гс = 0, the Poisson processes Nr and Nr are independent.
Let T and Tc be their respective first jump times. For every f,
lS>t\eser} = {t<T< Г]
and consequently, since T and Tc are independent exponential r.v.'s with param-
parameters п(Г) and п(Гс),
P[S>t\eser] = f n(r)e-"insds f и(Г>-"(Г<M ds'
this shows in one stroke the independence of S and es and the formula in the
statement and confirms the fact that S is exponentially distributed with parameter
n(U).
# A.14) Exercise. 1°) Let N be the Poisson process on R+ and M, — N, — ct.
Prove that /0' Ns_dMs is a martingale and that /0' NsdMs is not. Conclude that
the process /V is not predictable with respect to its natural filtration.
2°) If Z is predictable, then under a suitable integrability hypothesis,
is a martingale. This generalizes the result in 3°) of Exercise A.14) Chapter 11.
# A.15) Exercise. 1°) Let (.V,) be the natural filtration of a Poisson process N, and
set M, — N, — ct. Prove that every (.J^)-local martingale X, may be written
Г
Xq + I ZsdMs
Jo
for a .^"-predictable process Z.
[Hint: Use the same pattern of proof as for Theorem C.4) in Chap. V.]
2°) Let B, be a BM independent of N, and (.'?,) the smallest right-continuous
complete filtration for which В and /V are adapted. Give a representation of (•%)-
local martingales as stochastic integrals.

478 Chapter XII. Excursions
3°) Let N' be another (.J^")-Poisson process. Deduce from 1°) that N' — N
a.s. The reader will observe the drastic difference with the Brownian filtration for
which there exist many (.5^)-BM's.
[Hint: Use 2°) of the previous exercise.]
4°) Describe all the (.>^)-Poisson processes, when (.5^") is the natural filtration
of a (/-dimensional Poisson process.
[Hint: The discussion depends on whether the parameters of the components
are equal or different.]
A.16) Exercise. 1°) Let N be a Poisson process and / e L'(R+). What is the
necessary and sufficient condition under which the r.v. N(f) — /0°° f(u)dNu is a
Poisson r.v.?
2°) If g is another function of L'(R+), find the necessary and sufficient con-
condition under which N(f) and N(g) (not necessarily Poisson) are independent.
A.17) Exercise. Let N be an (Jf )-Poisson process and T a (.^-stopping time.
Prove that N't = NTH - NT is a Poisson process. [For the natural filtration of
N, this is a consequence of the Strong Markov property of Sect. 3 Chap. Ill; the
exercise is to prove this with the methods of this section.] Generalize this property
to (.>^")-Poisson point processes with values in (t/, '//>).
A.18) Exercise. Let (?, K) be a measurable space and .V/j the space of non-
negative, possibly infinite, integer-valued measures on (E. K). We endow . /6
with the coarsest ст-field for which the maps \x -> д(А), А е К, are measurable.
An ./Ж-valued r.v. N is called a Poisson random measure if
i) for each A e >6, the r.v. N(A) is a Poisson r.v. (the map which is identically
+oo being considered as a Poisson r.v.);
ii) if (Aj) is a finite sequence of pairwise disjoint sets of <C, the r.v.'s N(Aj) are
independent.
1°) Prove that if N is a Poisson random measure and if we set k(A) =
E[N(A)], then X is a measure on <C. Conversely, given a a-finite measure к on
(E, fC) prove that there exist a probability space and a Poisson random measure
-V on this space such that E[N(A)] = k(A) for every Ae^. The measure к is
called the intensity of N.
2°) Prove that an. /^-valued r.v. N is a Poisson random measure with intensity
k, if and only if, for every / e K+,
- f f(X)N(dx)\\ =exp(-J(\ -exp(-f{X)))k(dx)Y
3°) Let я be a a-finite measure on a measurable space (U', //). Prove that
there exists a PPP on U with characteristic measure n.
[Hint: Use 1°) with dk = dt dn on Ш+ x U.}

§ I. Prerequisites on Poisson Point Processes 479
# A.19) Exercise. Let N be a Poisson random measure on (E, %) with intensity A.
(see the preceding exercise) and / be a uniformly bounded positive function on
]0, oo[x? such that for every s > 0, the measure /Д is a probability measure.
Assume that there exist an exponential r.v. S with parameter 1 and an ii-valued
r.v. X such that, conditionally on S — s
i) N is a Poisson random measure with intensity gsX where gs(x) = f^° f,(x)dt;
ii) X has distribution fsk;
iii) X and N are independent.
Prove that /V* = N + ex is a Poisson random measure with intensity g0X.
[Hint: Use 2°) in the preceding exercise.]
# A.20) Exercise. (Chaotic representation for the Poisson process). 1°) Retain
the notation of Exercise A.15) and let / be a positive locally bounded function
on R+. Prove that
-[ f(s)dNs+cj {\-exp(-f(s)))ds\
is a (.5^)-martingale for which the process Z of Exercise A.15) is equal to
/(exP{-f(s))-\).
2°) Prove that every У е L2(.yx) can be written
dMSl f dMSl...f dMsJn{si,...,sn)
J[om •'[о..ь-il
where (/„) is a sequence of functions such that
„=1 Jo
00.
A.21) Exercise. ln) Let e be a PPP and in the notation of this section, let / be
./? (Ж+) <g> ^-measurable and positive. Assume that for each /, the r.v.
is a.s. finite.
1°) Prove that the following two conditions are equivalent:
i) for each /, the r.v. N; is a Poisson r.v.;
ii) there exists a set Г e ./9(R+) <S> Ms such that / = lr ds <g> dn-a.e.
[Hint: See Exercise A.16).]
2°) Prove that the process (/V,r; t > 0) satisfies the equality
Nf = Na(,)
where a(t) = /„' n(F(s, ))ds and N is a Poisson process. Consequently, Nr is a
Poisson process iff n(F(s, •)) is Lebesgue a.e. constant in s.

480 Chapter XII. Excursions
A.22) Exercise. 1°) Let N be a Poisson process with parameter с and call T\ the
time of its first jump. Prove that the equality
E exp(- j f(s,co)dNs(co)j = ? expf-c / A - exp(-/(s. со))) ds J
fails to be true if f(s, со) — А.1(о<.5<г(ш)}, with T = T\ and 1 > 0.
2°) Prove that if g is not ds (g> rfP-negligible, the above equality cannot be
true for all functions f(s, со) = g(s, co)\\q<s<t{W)\ where T ranges through all the
bounded (.>^~)-stopping times.
§2. The Excursion Process of Brownian Motion
In what follows, we work with the canonical version of BM. We denote by W
the Wiener space, by P the Wiener measure, and by ¦'W the Borel cr-field of W
completed with respect to P. We will use the notation of Sect. 2, Chap. VI.
Henceforth, we apply the results of the preceding section to a space (Us. Ms)
which we now define. For а ш е W, we set
R(w) = inf{? > 0 : w(t) = 0}.
The space U is the set of these functions w such that 0 < R(w) < oo and
w(t) — 0 for every t > R(w). We observe that the graph of these functions lies
entirely above or below the /-axis, and we shall call U+ and LL the corresponding
subsets of U. The point <5 is the function which is identically zero. Finally, № is
the a -algebra generated by the coordinate mappings. Notice that U/, is a subset
of the Wiener space W and that Ms is the trace of the Borel cr-field of W. As
a result, any Borel function on W, as for instance the function R defined above,
may be viewed as a function on U. This will often be used below without further
comment. However, we stress that Us is negligible for P, and that we will define
and study a measure n carried by Us, hence singular with respect to P.
B.1) Definition. The excursion process is the process e = (es, s > 0), defined on
(W,.y\ P) with values in (Us. Ms) by
i) ifzs(w) - ts-(w) > 0, then es(w) is the map
ii) ifrs(w) — гл_(«;) = 0, then es(w) = S.
We will sometimes write es(r, w) or es(r)for the function es(w) taken at time r.
The process e is a point process. Indeed, to check condition i) in Definition
A.1), it is enough to consider the map (t.w) —> Xr(e,(w)) where Xr is a fixed
coordinate mapping on U, and it is easily seen that this map is measurable. More-
Moreover, es is not equal to <5 if, and only if, the local time L has a constant stretch

§2. The Excursion Process of Brownian Motion 481
at level s and es is then that part of the Brownian path which lies between the
times zs- and r, at which В vanishes. It was already observed in Sect. 2 Chap. VI
that there are almost-surely only countably many such times s, which ensures that
condition ii) of Definition A.1) holds.
B.2) Proposition. The process e is о-discrete.
Proof. The sets
Un = {u € U\R(u) > \/n)
are in if/,, and their union is equal to U. The functions
are measurable. Indeed, the process / -»¦ r, is increasing and right-continuous; if
for n € N, we set
7"i =inf{r > 0 : r, -r,_ > 1/n},
then P [7"i > 0] = 1. If we define inductively
Tk - mf{t > 7i_i : t, - r,_
then, the 7^'s are random variables and
is a random variable. Moreover, /v/7" < иг,, as is easily seen, which proves our
claim.
We will also need the following
B.3) Lemma. For every r > 0, almost-surely, the equality
es+r(w) = es (9Tr(w))
holds for all s.
Proof. This is a straightforward consequence of Proposition A.3) in Chap. X.
We may now state the following important result.
B.4) Theorem (ltd). The excursion process (et) is an {.f^^-Poisson point process.
Proof. The variables N[ are plainly .Я^ -measurable. Moreover, by the lemma
and the Strong Markov property of Sect. 3 Chap. Ill, we have, using the notation
of Definition A.8),
= PBrr [/V,r e A] = P [Nf € A] a.s.,
since BTr = 0 P-a.s. The proof is complete. D

482 Chapter XII. Excursions
Starting from B, we have defined the excursion process. Conversely, if the
excursion process is known, we may recover B. More precisely
B.5) Proposition. We have
z,(w) =
and
B,(w) = ^ es (t - г,_(ш), ш)
where L, can be recovered as the inverse о/т,.
Proof. The first two formulas are consequences of the fact that r, =
5Zj<,(tv — rs-). For the third, we observe that if rs_ < t < ts for some .?,
then L, = s, and for any и < L,, eu(t — т„_) — 0; otherwise, t is an increase
point for L, and then eLl(t — rt,) = 0 so that В,(ю) = 0 as it should be.
Remark. The formula for В can also be stated as B, = e, (t — ts-) if т,_ < t < ts.
The characteristic measure of the excursion process is called the ltd measure.
It will be denoted by n and its restrictions to U+ and ?/_ by n+ and и_. It is
carried by the sets of m's such that м@) = О. Our next task is to describe the
measure n and see what consequences may be derived from the formulas of the
foregoing section. We first introduce some notation.
If w ? W, we denc.e by z'o(w) the element и of U such that
if t < R(w), и(О = 0 \U > R(w).
If.? e R+, we may apply this to the path 0s(w). We will put i.,(w) — io(ds(w)).
We observe that R Fs(w)) = R (is(w)).
We will also call Gw the set of strictly positive left ends of the intervals
contiguous to the set Z(w) of zeros of B, in other words, the set of the starting
times of the excursions, or yet the set
(г,_(ш) : г,_(ш) ф г,(ш)} = {ts-(w) : R (er.,_(w)) > 0}.
Let Я be a .S'{.Vt) ® ^-measurable positive process vanishing at S. The
process (s, w, u) -»¦ H (г,_(ш), ш; и) is then .У(.Я^) (8 ^-measurable as is
easily checked. The master formula of Sect. 1 applied to this process yields
E ^H(Ts-(w),w;es(w)) = E I ds I H (rs-{w), w; u)n(du)\ .
The left-hand side may also be written E YlyeG,,, H(Y< w< 'y(w)) and in the
right-hand side, owing to the fact that {s : rt_ ф г,} is countable and that we
integrate with respect to the Lebesgue measure, we may replace rs_ by ts. We
have finally the

§2. The Excursion Process of Brownian Motion 483
B.6) Proposition. If H is as above,
E
\ Y^ и (У' w;iy(w))\ = E \ ds ? H(Ts(w),w;u)n(du)\
= e\ dL,(w) / H{t,w\u)n(du)\.
Proof. Only the second equality remains to be proved, but it is a direct conse-
consequence of the change of variables formula in Stieltjes integrals. D
The above result has a martingale version. Namely, provided the necessary
integrability properties obtain,
Y]H (г,_(ш).ш;/т, (ш)) - I ds I H (is(w),w;u)n(du)
T^! Jo J
is a (.3^)-martingale and
~ / dLs(w) / H(s, w\ u)n(du)
is a (,^)-martingale, the proof of which is left to the reader as an exercise.
The exponential formula yields also some interesting results. In what follows,
we consider an additive functional A, such that va({0}) = 0, or equivalently
/ \zdA = 0. In this setting, we have
B.7) Proposition. The random variable ATl is infinitely divisible and the Laplace
transform of Us law is equal to
ф,(Х) = exp It J (e-Xx - l)mA(dx)\
where m\ is the image of n under Ar, the random variable Ar being defined on
U by restriction from W to U.
Proof. Since dA, does not charge Z, we have
A, = / \Z'(s)dA.,,
Jo
hence, thanks to the strong additivity of A,
The exponential formula of Proposition A.12) in Sect. 1 then implies that
ф,(Х) = Е[ехр{-ХАт,)] =
= exp
which is the announced result.
It f n(du) (e-XA«w - 1I

484 Chapter XII. Excursions
Remark. The process ATl is a Levy process (Exercise B.19) Chap. X) and mA
is its Levy measure; the proof could be based on this observation. Indeed, it is
known in the theory of Levy processes that the Levy measure is the characteristic
measure of the jump process which is a PPP. In the present case, because of the
hypothesis made on vA, the jump process is the process itself, in other words
AZl = J]j<r ^«(e^' as a result
mA(C) = E\ 2_^ lc (^«(ej) = n {Аи е С}.
The following result yields the "law" of R under n.
B.8) Proposition. For every x > 0,
n(R > x) = B/jtx)U2.
Proof. The additive functional A, = t plainly satisfies the hypothesis of the pre-
previous result which thus yields
E [exp (-At,)] = exp \tj mA(dx) (e~Xx - l) .
By Sect. 2 in Chap. VI, the law of xa is that of Ta which was found in Sect. 3 of
Chap. II. It follows that
/ mA{dx)(\ -e~kx) = V2A.
Jo
By the integration by parts formula for Stieltjes integrals, we further have
mA{ ]x, oof )e~"'dx =
Since it is easily checked that
we get mA(]x, oof) = B/л"хI/2; by the definition of тл- the proof is complete.
Remarks. Iе) Another proof is hinted at in Exercise D.13).
2°) Having thus obtained the law of R under n, the description of n will be
complete if we identify the law of и (I), I < R, conditionally on the value taken
by R. This will be done in Sect. 4.
The foregoing proposition says that R (es(w)) is a PPP on R+ with character-
characteristic measure h given by n(]x. oof) = B/ттх)[/2. We will use this to prove another
approximation result for the local time which supplements those in Chap. VI. For
e > 0, let us call r]t(s) the number of excursions with length > e which end at
a time s < t. If N is the counting measure associated with the PPP R(es), one
moment's reflection shows that ri,(s) = N[ , where Nf, = Njf-00*-, and we have
the
f
Jo

§2. The Excursion Process of Brownian Motion 485
B.9) Proposition. P [limfi0 y/^4t(?) = L, for every f] = 1.
Proof. Let Ek = 2/лкг\ then n{[ek. oo[) = к and the sequence {A^1*1 - NF,k) is
a sequence of independent Poisson r.v.'s with parameter /. Thus, for fixed t, the
law of large numbers implies that a.s.
ii n ' « V 2
As Nf increases when e decreases, for e,,+1 < s < e,,,
and plainly
We may find a set E of probability 1 such that for w e E,
for every rational t. Since Nf increases with f, the convergence actually holds for
all ?'s. For each ю е Г, we may replace t by L,(w) which ends the proof.
Remarks. 1°) A remarkable feature of the above result is that rj,(e) depends only
on the set of zeros of в up to t. Thus we have an approximation procedure for
L,, depending only on Z. This generalizes to the local time of regenerative sets
(see Notes and Comments).
2°) The same kind of proof gives the approximation by downcrossings seen
in Chap. VI (see Exercise B.10)).
B.10) Exercise. 1°) Prove that
> x ) - \/x.
[Hint: If Л, = \u : supl<Ru{t) > x}, observe that LT, is the first jump time
of the Poisson process М,Лк and use the law of LTx found in Sect. 4 Chap. VI.]
2") Using 1°) and the method of Proposition B.9), prove, in the case of BM,
the a.s. convergence in the approximation result of Theorem A.11) Chap. VI.
3C) Let a > 0 and set Ma(u) = sup,<?f u(t) where, as usual, grcl is the last
zero of the Brownian path before the time Ta when it first reaches a. Prove that
Ma is uniformly distributed on [0, a]. This is part of Williams' decomposition
theorem (see Sect. 4 Chap. VII).
[Hint: If Ja (resp. /,.) is the first jump of the Poisson process NA- (resp. NAx),
then P [Ma < v] = P [Jy = /„]; use Lemma A.13).]

486 Chapter XII. Excursions
B.11) Exercise. Prove that the process X defined, in the notation of Proposition
B.5), by X,(w) = к (t ~ т,-(ш), w)\ - s, if r,_(w) < / < ts(w), is the BM
в, = / sga(Bs)dBs
Jo
and that Y,(w) = s + \es (t - rs-(w), w)\ if ts_(w) < t < zs(w), is a BES3(O).
B.12) Exercise. 1°) Let A e Ms be such that n(A) < oo. Observe that the
number Cj of excursions belonging to A in the interval [0, d,] (i.e. whose two end
points lie between 0 and d,) is defined unambiguously and prove that E \Cf\ =
n(A)E[L,].
2°) Prove that on {ds < t},
and consequently that C^ — «(A)Lr is a (.5^)-martingale.
[Hint: Use the strong Markov property of BM.]
# B.13) Exercise (Scaling properties). Iе) For any с > 0, define a map sc on W
or Us by
sc(w)(t) = w(ct)/>/c.
Prove that e, (sc(w)) = sc (ets^.(w)) and that for A e '//^
n(s~\A)) =n(A)/y/d.
[Hint: See Exercise B.11) in Chap. VI.]
2°) (Normalized excursions) We say that и € U is normalized if 7?(и) = 1.
Let Ul be the subset of normalized excursions. We define a map v from U to U]
by
V(M) = 5-Л(„)(и).
Prove that for Г С С", the quantity
у (Г) =n+(v-\r)r\(R >c))/n+(R >c)
is independent of с > 0. The probability measure у may be called the law of the
normalized Brownian excursion.
3°) Show that for any Borel subset S of K+,
n+ (у"'(Г) n (/? e S)) = у(Г)п+(Я е S)
which may be seen as displaying the independence between the length of an
excursion and its form.
4°) Let ec be the first positive excursion e such that R(e) > с Prove that
y(D = P[v(ec) € Г].
[Hint: Use Lemma A.13).]

§2. The Excursion Process of Brownian Motion 487
# B.14) Exercise. Let A,(s) be the total length of the excursions with length < s,
strictly contained in [0, /[. Prove that
P Iim./ — A,(e) — L, for every t\ — 1.
no V 2e
[Hint: Л,(е) = — f^ xr\,(dx) where r\t is defined in Proposition B.9).]
# B.15) Exercise. Let S, — supv<, Bs and n,(s) the number of flat stretches of S of
length > e contained in [0, /]. Prove that
P Iim /—n,(e) — S, for every/ —1.
LnoV 2 J
B.16) Exercise (Skew Brownian motion). Let (K,,) be a sequence of indepen-
independent r.v.'s taking the values 1 and —1 with probabilities a and 1 — a @ < a < 1)
and independent of 6. For each w in the set on which В is defined, the set of
excursions e, (w) is countable and may be given the ordering of N. In a manner
similar to Proposition B.5), define a process X" by putting
X? = Yn\es(t-xs-(w),w)\
if ts^(w) < t < ts(w) and es is the и-th excursion in the above ordering. Prove
that the process thus obtained is a Markov process and that it is a skew BM by
showing that its transition function is that of Exercise A.16), Chap. I. Thus, we
see that the skew BM may be obtained from the reflecting BM by changing the
sign of each excursion with probability 1 — a. As a result, a Markov process X is
a skew BM if and only if |X| is a reflecting BM.
# B.17) Exercise. Let Л+ — /0' \{B,>o)ds, A~ - /0' \(B,<o)ds.
1°) Prove that the law of the pair L^2(A+, A~) is independent of/ and that
Л+ and A~ are independent stable r.v.'s of index 1/2.
[Hint: Л+ + A~ — t, which is a stable r.v. of index 1/2.]
2°) Let a+ and a' be two positive real numbers, S an independent exponential
r.v. with parameter 1. Prove that
E [exp (—Ls (a +A^ +a~Aj))~\ = I exp(—<p(s))<p'(s)ds
Jo
where 0(j) = -^ Us2 + a+I/2 +- (^2 +a~)'/2 . Prove that, consequently, the
pair L~2 (A+. A~) has the same law as \(T+, T~) where T+ and T~ are two
independent r.v.'s having the law of T\ and derive therefrom the arcsine law of
Sect. 2 Chap. VI (see also Exercise D.20) below). The reader may wish to compare
this method with that of Exercise B.33) Chap. VI.
B.18) Exercise. Prove that the set of Brownian excursions can almost-surely be
labeled by Q+ in such a way that q < q' entails that eq occurs before e4.
[Hint: Call e\ the excursion straddling 1, then e\/2 the excursion straddling

488 Chapter XII. Excursions
§3. Excursions Straddling a Given Time
From Sect. 2 in Chap. VI, we recall the notation
g, = sup {s < t : Bs = 0}, d, = inf {s > t : Bs = 0}.
and set
A,=t-gt, A,=d,-g,.
Plainly, d, > g, if and only if there is an excursion which straddles t and in
that case, A, is the age of the excursion at time t and Л, is its length. We have
Л, = R(igl).
C.1) Lemma. The map t —*¦ g, is right-continuous.
Proof. Let tn I t; if there exists an n such that ghi < t, then glm = g, for m > n;
if gta > t for every n, then t < gtn < tn for every n, hence t is itself a zero of В
and g, = t = limn g,n.
Fig. 8.
For a positive r.v. S we denote by .>*i the a-algebra generated by the variables
Hs where H ranges through the optional processes. If S is a stopping time, this
coincides with the usual a-field .7~s. Let us observe that in general S < S' does
not entail .7~$ с -Щ\ when S and 5" are not stopping times; one can for instance
find a r.v. S < 1 such that .7~s — -7^.
Before we proceed, let us recall that by Corollary C.3), Chap. V, since we are
working in the Brownian filtration (.Pf), there is no difference between optional
and predictable processes.
C.2) Lemma. The family (.%) = (.7'gi) is a subfiltration of (.Vt) and if T is a
(¦%)-stopping time, then -КЯт С -Щ- С -Щ-.

§3. Excursions Straddling a Given Time 489
Proof. As g, is .^"-measurable, the same reasoning as in Proposition D.9) Chap. I
shows that, for a predictable process Z, the r.v. Zft is .^"-measurable whence
• Pf С -Я" follows.
Now choose и in K+ and set Z\ — Zgl,u\ thanks to Lemma C.1) and to what
we have just proved, Z' is (->f)-optional hence predictable. Pick v > u; if gv < u,
then gu — gv and since gg: = g, for every /,
and if gv > u, then Z' — Zgii. As a result, each of the r.v.'s which generate ./u
is among those which generate -Wv. It follows that .5^ С -Vv, that is, (.if) is a
filtration.
Let now T be a (.^(-stopping time. By definition, the a -algebra '^gT is gen-
generated by the variables ZgT with Z optional; but ZgT = (Zg)r and Zg is -Vgi-
optional because of Lemma C.1) which entails that Zgl is -Щ-measurable, hence
that .Wgl с .^. On the other hand, since .W, с .7i, the time T is also a (->f )-
stopping time from which the inclusion .Щ с -Щ is easily proved. D
We now come to one of the main results of this section which allows us to
compute the laws of some particular excursions when n is known. If F is a positive
^/-measurable function on U, for s > 0, we set
q(s, F) = n(R > s)~l F dn=n(F \ R> s).
J[R>s}
We recall that 0 < n(R > s) < oo for every s > 0.
C.3) Proposition. For every fixed t > 0,
F) a.s..
and for a (.^-stopping time T,
=q(AT, F) a.s. on the set {0 < gT < T].
Proof. We know that, a.s., / is not a zero of В hence 0 < g, < t and q(A,, F) is
defined; also, if s e Gu, and л < /, we have ,v = gt if and only if s + R о 0s > I.
As a result, g, is the only л e Gv, such that s < t and s + R о 0, > /. If Z is a
positive (.yf )-predictable process, we consequently have
E [Zgl F (ig: jj — E\ y^ ZsF(i.5)l{flofl,>/-.s>o) .
We may replace /?o0, by /?(i.,) and then apply the master formula to the right-hand
side which yields
ds / Z^(w)F(u)\{Riu)>,-TAw)>0}n(du) \.

490 Chapter XII. Excursions
Since by Proposition B.8), for every x > 0, we have n(R > x) > 0, the right-hand
side of the last displayed equality may be written
El I ds ZTK(w)n(R > t - zs(w))q{t ~ г,(ш), F)\.
And, using the master formula in the reverse direction, this is equal to
zsq(t-s,F)llRol)t>l-s>o}] = E[Zg,q(t-g,,F)]
.veGu.
which yields the first formula in the statement.
To get the second one, we consider a sequence of countably valued (.Pi)-
stopping times Т„ decreasing to T. The formula is true for Т„ since it is true for
constant times. Moreover, on {0 < gr < T], one has {grn = gr} from some и0
onwards and limn l[gTn<T,,} — 1; therefore, for bounded F,
= \\mE[q(ATn,F)]{gTn<Tn] \Jti\=
because Нт„ Ат„ = At, the function q(-. F) is continuous and At is ,Яг-
measurable. The extension to an unbounded F is easy. D
The foregoing result gives the conditional expectation of a function of the
excursion straddling / with respect to the past of BM at the time when the ex-
excursion begins. This may be made still more precise by conditioning with respect
to the length of this excursion as well. In the sequel, we write E[- \ .Pf, A,]
for ?[• | .Vt\/ a {A,)]. Furthermore, we denote by v(-; F) a function such that
v(R; F) is a version of the conditional expectation n(F \ R). This is well de-
defined since n is a -finite on the cr-field generated by R and by Proposition B.8),
the function r —> v(r; F) is unique up to Lebesgue equivalence. We may now
state
C.4) Proposition. With the same hypothesis and notation as in the last proposition,
E[F(igl) |.ЯГ, A^ = v(A,;F),
and
E [f (iKT) | .Pi, At] = v(AT; F) on {0 < gT < T).
Proof. Let ф be a positive Borel function on R+; making use of the preceding
result, we may write
E[ZHi4>(A,)F(igi)] = E[Zgi<p(R(iH:))F(igl)]
= E[ZH,q(At,<P(R)F)].

§3. Excursions Straddling a Given Time 491
But, looking back at the definition of q, we have
R\ F)),
so that using again the last proposition, but in the reverse direction, we get
E [ZHi4>(A,)F (igi)] = E [Zgi4>{A,)v(At\ F)]
which is the desired result. The generalization to stopping times is performed as
in the preceding proof. D
We now prove an independence property between some particular excursions
and the past of the BM up to the times when these excursions begin. A (-^°)-
stopping time T is said to be terminal if T = t + T о в, a.s. on the set [T > t};
hitting times, for instance, are terminal times. For such a time, T = gT + To9gT a.s.
on [gj < T). A time T may be viewed as defined on U by setting for и = io(w),
T(u) = T(w) if R(w) > T(w), T(u) = +oo otherwise.
By Galmarino's test of Exercise D.21) in Chap. I, this definition is unambiguous.
If T(u) < oo, the length Aj of the excursion straddling T may then also be
viewed as defined on U. Thanks to these conventions, the expressions in the next
proposition make sense.
C.5) Proposition. IfT is a terminal (¦^^-stopping time, then on {0 < gj < T),
E[F(igl) \.PgT]=n(Fl{R^T))/n(R> T)=n(F\R> T)
and
E [F (igr) | j?, AT] = v {AT- F\(R>T)) /v (At; l(«>r>) .
Proof. For a positive predictable process Z, the same arguments as in Proposition
C.3) show that
= E
= E
= E
= e
л»
/ ds ZT>l(Ti<r) / ^(иI(№)>г(йця((/и)
J ds ZT,l(Tl<T)n(R > T)]n(F\iR>T))/n(R > T)
0<gT<T)] n (F\(R>T)) /n{R > T)
which proves the first half of the statement. To prove the second half we use the
first one and use the same pattern as in Proposition C.4).

492 Chapter XII. Excursions
Remark. As the right-hand side of the first equality in the statement is a constant,
it follows that any excursion which straddles a terminal time T is independent
of the past of the BM up to time gj. This is the independence property we had
announced. We may observe that, by Proposition C.3), this property does not hold
with a fixed time t in lieu of the terminal time T (see however Exercise C.11)).
We close this section with an interesting application of the above results which
will be used in the following section. As usual, we set
TE{w) = inf{? > 0 : w(t) > e].
On U, we have {Te < oo} = [Te < R}, and moreover
C.6) Proposition, n (sups<Sw u{s) > s) = n(Tc < oo) = \/2e.
Proof. Let 0 < x < y. The time Tx is a terminal time to which we may apply the
preceding proposition with F = \(T,<R)', it follows that
P [Ty (egu) < oo] = n(T, < oo)/n{Tx < oo).
The left-hand side of this equality is also equal to Px [7\. < To] = x/y; as a result,
n(Te < oo) — c/s for a constant с which we now determine.
Proposition B.6) applied to H(s, •; u) = \{jt<oc)("H(.?<<) yields
n(Te <oo)?[L,] =
.seGu.n[O./]
which, in the notation of Sect. 1 Chap. VI, implies that
cE[L,] = E[e(dc(t)±\)];
letting e tend to 0, by Theorem A.10) of Chap. VI, we get с = 1/2.
Remark. This was also proved in Exercise B.10).
C.7) Exercise. Use the results in this and the preceding section to give the con-
conditional law of d\ with respect to g\. Deduce therefrom another proof of 4°) and
5°) in Exercise C.20) of Chap. III.
*# C.8) Exercise. 1°) Prove that conditionally on g\ — u, the process (B,, t < u) is
a Brownian Bridge over [0, и]. Derive therefrom that B,gJ\/gi, 0 < / < 1 is a
Brownian Bridge over [0, 1] which is independent of gi and of [Bgl+U, и > 0}
and that the law of g\ is the arcsine law. See also Exercise B.30) Chap. VI.
2°) The Brownian Bridge over [0, 1] is a semimartingale. Let /" be the family
of its local times up to time 1. Prove that L" has the same law as ^fgxl"^^ where
g\ is independent of the Bridge. In particular, L° = *Jgll°; derive therefrom that
/° has the same law as V2e, where e is an exponential r.v. with parameter 1.
[Hint: See Sect. 6 Chap. 0.]

§4. Descriptions of Ito's Measure and Applications 493
3C) Prove that the process Mu = \Bgl+u([__gl)\ /V - gu 0 < и < 1, is in-
independent of the cr-algebra .7{\ M is called the Brownian Meander of length 1.
Prove that M\ has the law of \/2e just as /° above.
[Hint: Use the scaling properties of и described in Exercise B.13).]
4°) Prove that the joint law of (gt. Lt, Bt) is
()ds dl dx-
[Hint: (g\,L\,B\) — (g\, y/g\l°, VI - g\M\) where gu 1°, M\ are indepen-
independent.]
C.9) Exercise. We retain the notation of the preceding exercise and put moreover
A, = JQ \(B,>o)ds and U = JQ \{pt>0)ds where /3 is a Brownian Bridge. We recall
from Sect. 2 Chap. VI that the law of A\ is the Arcsine law; we aim at proving
that U is uniformly distributed on [0, 1].
1°) Let T be an exponential r.v. with parameter 1 independent of B. Prove
that AgT and (Ay- — AgT) are independent. As a result,
where e is a Bernoulli r.v. and T, g\, U, s are independent.
2°) Using Laplace transform, deduce from the above result that
where N is a centered Gaussian r.v. with variance 1 which is assumed to be
independent of U on one hand, and of V on the other hand, and where V is
uniformly distributed on [0, 1]. Prove that this entails the desired result.
C.10) Exercise. Prove that the natural filtration (->^s) of the process g is strictly
coarser than the filtration \-7Л and is equal to (¦>^L)-
C.11) Exercise. For a > 0 let T = inf {/ : t — g, = a). Prove the independence
between the excursion which straddles T and the past of the BM up to time gj.
§4. Descriptions of Ito's Measure and Applications
In Sects. 2 and 3, we have defined the Ito measure n and shown how it can be
used in the statements or proofs of many results. In this section, we shall give
several precise descriptions of n which will lead to other applications.
Let us first observe that when a cr-finite measure is given on a function space,
as is the case for n and U, a property of the measure is a property of the "law" of
the coordinate process when governed by this measure. Moreover, the measure is

494 Chapter XII. Excursions
the unique extension of its restriction to the semi-algebra of measurable rectangles;
in other words, the measure is known as soon as are known the finite-dimensional
distributions of the coordinate process.
Furthermore, the notion of homogeneous Markov process makes perfect sense
if the time set is ]0. oo[ instead of [0, oo[ as in Chap. III. The only differ-
difference is that we cannot speak of "initial measures" any longer and if, given the
transition semi-group P,, we want to write down finite-dimensional distributions
P [Xt] G A\,..., Xh G Au\ for &-uples 0 < t\ < ... < tk, we have to know the
measures A, = X,(P). The above distribution is then equal to
Xtl(dx{)
I P,^h(x^dx2)... I
JA2 JAt
The family of measures A, is known as the entrance law. To be an entrance law,
(A,) has to satisfy the equality X, Ps — X,+s for every s and / > 0. Conversely,
given (A,) and a t.f. (P,) satisfying this equation, one can construct a measure on
the canonical space such that the coordinate process has the above marginals and
therefore is a Markov process. Notice that the A,'s may be cr-finite measures and
that everything still makes sense; if ц. is an invariant measure for P,, the family
X, = ц for every / is an entrance law. In the situation of Chap. Ill, if the process
is governed by Pv, the entrance law is (vP,). Finally, we may observe that in this
situation, the semi-group needs to be defined only for / > 0.
We now recall some notation from Chap. III. We denote by Q, the semi-group
of the BM killed when it reaches 0 (see Exercises A.15) and C.29) in Chap. III).
We recall that it is given by the density
qt(x,y) = B7ztyU2(exp(- — (y - x)A - exp (- — (>• + xJ JJ \{xy>0}-
We will denote by X,(dy) the measure on K\{0} which has the density
m,(y) = Bя/3)~1/2|2
with respect to the Lebesgue measure d\. For fixed y, this is the density in / of
the hitting time Ty as was shown in Sect. 3 Chap. III.
Let us observe that on ]0, oo[
m, = --1 = lim —q,(x, ¦).
ду л-->о 2х
Our first result deals with the coordinate process w restricted to the interval
]0, /?[, or to use the devices of Chap. Ill we will consider - in this first result only
- that w(t) is equal to the fictitious point <5 on [R, oo[. With this convention we
may now state
D.1) Theorem. Under n, the coordinate process w(t), t > 0, is a homogeneous
strong Markov process with Q, as transition semi-group and X,, t > 0, as entrance
law.

§4. Descriptions of Ito's Measure and Applications 495
Proof. Everything being symmetric with respect to 0, it is enough to prove the
result for n+ or n-. We will prove it for n+, but will keep n in the notation for
the sake of simplicity.
The space (Us, Ms) may serve as the canonical space for the homogeneous
Markov process (in the sense of Chap. Ill) associated with Q,, in other words
Brownian motion killed when it first hits {0}; we call (Qx) the corresponding
probability measures on (Us, 'Ms)- As usual, we call вг the shift operators on U&.
Our first task will be to prove the equality
eq. D.1) n ((u(r) e А)Пв^(Г)) = n {\А(и(г))<2и(г)(Г))
for Г e M, A e .J$(R+ - {0}) and r > 0. Suppose that n(u(r) e A) > 0 failing
which the equality is plainly true. For r > 0, we have {u(r) € А) С {г < /?}
hence n(u(r) e A) < oo and the expressions we are about to write will make
sense. Using Lemma A.13) for the process e{u{r)(iA\ we get
n (\A(u(r)) (\говг)) /n(u(r) eA) =
where P is the Wiener measure.
The time 5 which is the first jump time of the process е'"<г)бД' is a (-3^,)-
stopping time; the times т$_ and rj are therefore (J^")-stopping times. We set
T = tS- + r. The last displayed expression may be rewritten
P[{BT e А}П{В.овт e Г}]
where В stands for the BM killed when it hits {0}. By the strong Markov property
for the (¦7~t)- stopping time T, this is equal to
As a result
n ((u(r) e А) П в~\Г)) = n(u(r) e A) I y(dx)Qx[r]
where у is the law of Bj under the restriction of P to {Вт е A). For a Borel
subset С of K, make Г — \u@) e C} in the above formula, which, since then
<2х[Г] = \с(х), becomes
n(u(r) G А П C) = n(u(r) e A)y(C);
it follows that y(-) — n(u(r) e А П -)/n(u(r) e A) which proves eq. D.1).
Let now 0 < t\ < t2 < ... < tk <? be real numbers and let /,..... fk, f be
positive Borel functions on Ш. Since B, the BM killed at 0, is a Markov process
we have, for every x
eq. D.2) QA f\ f, (w(t,)) f(w(t))
L=i
.'=1

496 Chapter XII. Excursions
Set F = П*_2 ft ^(Ь ~ 'i))- ^У rePeated applications of equations D.1) and
D.2), we may now write
f\fi (w('i))) fW)) = n [/i (Mt\)) (F ¦ f(w(t - ГО)) о 6>„]
= n[A (w(t])) QwUl)[F ¦ f (w(t - t^))]]
= n [/, (w(ti)) ?„.</,> [FQt_tJ({w(tk - /,))]]
= n[/, {w{tx))Fo9hQ,_hf{w(tk))}
к
— n
which shows that the coordinate process w is, under л, a homogeneous Markov
process with transition semi-group Q,. By what we have seen in Chap. Ill, В has
the strong Markov property; using this in eq. D.2) instead of the ordinary Markov
property, we get the analogous property for л; we leave the details as an exercise
for the reader.
The entrance law is given by X,(A) = n(u(t) e A) and it remains to prove
that those measures have the density announced in the statement. It is enough to
compute Я, ([j, oo[ ) for у > 0. For 0 < e < v,
*,([>-, oo[) = n(u(t) >y) = n(u(i) > y: Te < t)
where T? — inf{? > 0 : u(t) > e). Using the strong Markov property for n just
alluded to, we get
A.,([y,oo[) = n(Tc <t;QuiTi}(u(t-Tt)>y))
= n(Te <t\Q,_Tl{e.[y.oo[)).
Applying Proposition C.5) with F(u) = \(T,(u)<nQi-T,u<)(?' [>', oo[) yields
A,([y. oof ) = E [l(f,<MG,_f,(e. [.v. oo[ )}n(TF < R)
where TF = TE (i^ ). Using Proposition C.6) and the known value of Qu this is
further equal to
with Ф,(у) = f^ocgi(z)dz. If we let e tend to zero, then Te converges to zero
P-a.s. and we get k,([y, oof ) = g,(y) which completes the proof.
Remarks. 1°) That (X,) is an entrance law for Q, can be checked by elementary
computations but is of course a consequence of the above proof.
2°) Another derivation of the value of (X,) is given in Exercise D.9).
The above result permits to give ltd's description of n which was hinted at in
the remark after Proposition B.8). Let us recall that according to this proposition

§4. Descriptions of Ito's Measure and Applications 497
the density of R under n+ is Bх/2яг3)~]. In the following result, we deal with
the law of the Bessel Bridge of dimension 3 over [0, r] namely P030r which we
will abbreviate to яг. The following result shows in particular that the law of the
normalized excursion (Exercise B.13)) is the probability measure л\.
D.2) Theorem. Under n+, and conditionally on R = r, the coordinate process w
has the law лг. In other words, if Г е 'M?,
п+(Г)= f яг(ГП{/?=г))- dr
Jo 2
Proof. The result of Theorem D.1) may be stated by saying that for 0 < t\ <
ti <...</„ and Borel sets Aj С ] 0, oo[, if we set
then
п + (Г) - mh(xx)dx\ / q,1.ll(xux2)dx2... 4t,,-t,,^ (x,,-\- x,,)dxn.
Ja, Ja2 J а„
On the other hand, using the explicit value for nr given in Sect. 3 of Chap. XI, and
taking into account the fact that Г П {/?</„} = 0, the formula in the statement
reads
fx dr f I f
п+(Г) - \ —== / 2\/2яг3т„(.*1 )</•*! / qh-,,(x\,
Jt,, 2\/2nri JAi J A;
••• / qtn~t,,^(xn-\^xn)>nr^tn{xn)dxn.
t
r
/
Jt,,
т==г,,„(х„) 1
2V27TT1
as was seen already several times. Thus the two expressions for п + (Г) are equal
and the proof is complete.
This result has the following important
D.3) Corollary. The measure n is invariant under time-reversal; in other words,
it is invariant under the map и —> и where
u(t) = u(R(u) - i)l(R(u)>t).
Proof. By Exercise C.7) of Chap. XI, this follows at once from the previous
result.
This can be used to give another proof of the time-reversal result of Corollary
D.6) in Chap. VII.

498 Chapter XII. Excursions
D.4) Corollary. If В is a BM@) and for a > 0, Ta = inf{/ : B, = a), if Z is a
BES3@) and aa — sup{/ : Z, = a), then the processes Y, = a — Bjt_t, t < Ta and
Z,, / < aa are equivalent.
Proof. We retain the notation of Proposition B.5) and set /3, = L, — \B,\ =
s - \es(t - zs-)\ if rs_ < t < zs. We know from Sect. 2 Chap. VI that p is a
standard BM.
If we set Z, = L, + \B,\ — s + \es(t — rs_)| if т^_ < / < гл. Pitman's theorem
(see Corollary C.8) of Chap. VI) asserts that Z is a BES3@).
For a > 0 it is easily seen that
та — \nf{t : L, — a] — inf{t : f$, — a}\
moreover
za = sup{? : Z, =a)
since ZIa = LXa + |BrJ = a and for t > za one has L, > a.
We now define another Poisson point process with values in (U, M) by setting
~es — es if s > a.
es{t) = ea-s(R(ea-,)-t), 0 < t < R(ea.t), ifs<a.
In other words, for s < a, ~e5 = ea-s in the notation of Corollary D.3). Thus, for
a positive .s9(R+) x ^-measurable function /,
2^ f(s,es)= 22 f(a-s,es),
0<i<a 0<s<a
and the master formula yields
E
by Corollary D.3), this is further equal to
У] f(s. es)\ = E\ I ds I f(a - s, u)n(du) ;
o<s<a J L^o J J
e\J ds f f(s,u)n(du)\.
This shows that the PPP ~e has the same characteristic measure, hence the same
law as e. Consequently the process Z defined by
Z, =s + \es(t -Fs_)| if rs- <t <xs,
has the same law as Z. Moreover, one moment's reflection shows that
Z, = a - Р(та - t) for 0 < t < та
which ends the proof.

§4. Descriptions of Ito's Measure and Applications 499
Let us recall that in Sect. 4 of Chap. X we have derived Williams' path decom-
decomposition theorem from the above corollary (and the reversal result in Proposition
D.8) of Chap. VII). We will now use this decomposition theorem to give another
description of и and several applications to BM. We denote by M the maximum
of positive excursions, in other words M is a r.v. defined on (// by
M(u) — sup u(s).
s<R(u)
The law of M under n+ has been found in Exercise B.10) and Proposition C.6)
and is given by n+(M > x) = \/2x.
We now give Williams' description of л. Pick two independent BES3@) pro-
processes p and p and call Tc and Tc the corresponding hitting times of с > 0. We
define a process Z' by setting
0<t<Tc,
Te<t<Tc + fc,
t>Tc + fc.
For Г e (¦>//, we put N(c, Г) = P [Zc e Г]. The map N is a kernel; indeed
(Z,c)f>0 = (cZJ/c2) , thanks to the scaling properties of BES3@), so that N
maps continuous functions into continuous functions on R and the result follows
by a monotone class argument. By Proposition D.8) in Chap. VII, the second part
of Zc might as well have been taken equal to p (Tc + Tc — t), Tc < t < Tc + Tc.
D.5) Theorem. For any Г е Л^+
п + (Г) =l- f N(x, Г)хЧх.
* Jo
In other words, conditionally on its height being equal to c, the Brownian excursion
has the law of Zc.
Proof. Let Uc = {u : M(u) > c); by Lemma A.13), for Г е Щ
п + (Г П Uc) = n + (Uc)P[ec € Г] = ~P[ec e Г],
2c
where ec is the first excursion the height of which is > с The law of this excursion
is the law of the excursion which straddles 7"o i.e. the law of the process
Y, = %,+„ 0<t<dz-gTi.
By applying the strong Markov property to В at time Tc, we see that the process
Y may be split up into two independent parts Yx and Y2, with
Г,1 = BgT +/, 0 < t < Tc - gTr; Y2 = BTi+l, 0 < t < dT, - Tv.
By the strong Markov property again, the part Y2 has the law of B,, 0 < / < To,
where В is a BM(c). Thus by Proposition C.13) in Chap. VI, Y2 may be described

480 Chapter XII. Excursions
A.22) Exercise. 1°) Let N be a Poisson process with parameter с and call Г, the
time of its first jump. Prove that the equality
E exp (- j f(s, a>)dNs(<o)X\ = E Гexp (-c J A - exp(-/(s, ш))) ds\\
fails to be true if f(s, w) — Я1@<.,<г(а,)|, with T = T\ and A. > 0.
2°) Prove that if g is not rfs ® rff-negligible, the above equality cannot be
true for all functions f(s, ш) = g(s, &>I(о<«<7-м) where T ranges through all the
bounded (.^")-stopping times.
§2. The Excursion Process of Brownian Motion
In what follows, we work with the canonical version of BM. We denote by W
the Wiener space, by P the Wiener measure, and by .W the Borel a -field of W
completed with respect to P. We will use the notation of Sect. 2, Chap. VI.
Henceforth, we apply the results of the preceding section to a space (Us, Ms)
which we now define. For a w e W, we set
/?(u;) = inf{/ > 0: w(t) = 0}.
The space U is the set of these functions w such that 0 < R(w) < oo and
w(t) = 0 for every t > R(w). We observe that the graph of these functions lies
entirely above or below the f-axis, and we shall call U+ and t/_ the corresponding
subsets of U. The point 8 is the function which is identically zero. Finally, M is
the a -algebra generated by the coordinate mappings. Notice that Us is a subset
of the Wiener space W and that M& is the trace of the Borel ст-field of W. As
a result, any Borel function on W, as for instance the function R defined above,
may be viewed as a function on U. This will often be used below without further
comment. However, we stress that Us is negligible for P, and that we will define
and study a measure n carried by Us, hence singular with respect to P.
B.1) Definition. The excursion process is the process e = (e5, s > 0), defined on
(W, .W, P) with values in (Us, М&) by
i) ifrs(w) — xs_(w) > 0, then es(w) is the map
ii) ifrs(w) - г,_(ш) = 0, then es(w) = S.
We will sometimes write es(r, w) or es(r)for the function es(w) taken at time r.
The process e is a point process. Indeed, to check condition i) in Definition
A.1), it is enough to consider the map (/, w) —> Xr(et(w)) where Xr is a fixed
coordinate mapping on U, and it is easily seen that this map is measurable. More-
Moreover, es is not equal to 5 if, and only if, the local time L has a constant stretch

§2. The Excursion Process of Brownian Motion 481
at level s and e.v is then that part of the Brownian path which lies between the
times rs_ and r, at which В vanishes. It was already observed in Sect. 2 Chap. VI
that there are almost-surely only countably many such times s, which ensures that
condition ii) of Definition A.1) holds.
B.2) Proposition. The process e is a-discrete.
Proof. The sets
Un = {u e U; R(u) > \/n]
are in /Z, and their union is equal to U. The functions
are measurable. Indeed, the process t —*¦ r, is increasing and right-continuous; if
for n e N, we set
Tx =inf{f > 0 : r, - r,_ > \/n],
then P [T\ > 0] — 1. If we define inductively
7i = inf{? > 7i-i : r, - r,_ > \/n]
then, the 7^'s are random variables and
is a random variable. Moreover, N, " < яг,, as is easily seen, which proves our
claim.
We will also need the following
B.3) Lemma. For every r > 0, almost-surely, the equality
es+r(w) = es (eTr(w))
holds for all s.
Proof. This is a straightforward consequence of Proposition A.3) in Chap. X.
We may now state the following important result.
B.4) Theorem (ltd). The excursion process (e,) is an ^W^-Poisson point process.
Proof. The variables N[ are plainly .^-measurable. Moreover, by the lemma
and the Strong Markov property of Sect. 3 Chap. Ill, we have, using the notation
of Definition A.8),
P [N[rl+r] eA|.^r] - P [< о вТг € A | ,Цг]
= PBrr [/V,r eA] = P [N[ e A] a.s.,
since BTr = 0 P-a.s. The proof is complete. D

482 Chapter XII. Excursions
Starting from B, we have defined the excursion process. Conversely, if the
excursion process is known, we may recover B. More precisely
B.5) Proposition. We have
r,(u;) = ]T R (es(w)), r,_(u>) = ]T R (es(w))
and
B,(w) = ]ГеЛ'-т*_(ш), w)
s<L,
where L, can be recovered as the inverse oft,.
Proof. The first two formulas are consequences of the fact that r, =
5Z,</(rs ~ xs-)- For the third, we observe that if rs_ < t < rs for some s,
then L, = s, and for any и < Lt, eu(t — ти_) = 0; otherwise, / is an increase
point for L, and then e^(t ~ rLl) = 0 so that В,(ш) = О as it should be.
Remark. The formula for В can also be stated as B, = es (r — rv_) if rs < t < rs.
The characteristic measure of the excursion process is called the ho measure.
It will be denoted by n and its restrictions to U+ and U'_ by n+ and n_. It is
carried by the sets of w's such that м@) = О. Our next task is to describe the
measure n and see what consequences may be derived from the formulas of the
foregoing section. We first introduce some notation.
If w G W, we dence Ьу г'о(ш) the element и of U such that
u(t) = w(t) ift<R(w), m(/)=0 if t>R(w).
If s g R+, we may apply this to the path es(w). We will put is(w) = io(9s(w)).
We observe that R @s(w)) = R (is(w)).
We will also call Gw the set of strictly positive left ends of the intervals
contiguous to the set Z(w) of zeros of B, in other words, the set of the starting
times of the excursions, or yet the set
{ts-(w) : т,_(|и) ф rs(w)} = {r,_(w) : R (eTl_(w)) > 0}.
Let Я be a .-?°(.&f) ® ^-measurable positive process vanishing at S. The
process (s,w,u) —>¦ H (ts_(w),w;u) is then rjP (.3%,) ® ^-measurable as is
easily checked. The master formula of Sect. 1 applied to this process yields
1 Г f I
H (rs-(w), w\e,(w)) \=E\\ ds H (г,_(ш), ш; u)n(du)
1
J
The left-hand side may also be written E ^уес„ H(Y,w,iY(w))\ and in the
right-hand side, owing to the fact that {s : rs- ф rs) is countable and that we
integrate with respect to the Lebesgue measure, we may replace r^_ by r^. We
have finally the

§2. The Excursion Process of Brownian Motion 483
B.6) Proposition. IfH is as above,
E\ Л H(y,w;iy(w))~\ = e\J ds f H(Ts{w),w;u)n(du)\
= E\ I dL,(w) ? H(t, w; u)n(du) .
Proof. Only the second equality remains to be proved, but it is a direct conse-
consequence of the change of variables formula in Stieltjes integrals. D
The above result has a martingale version. Namely, provided the necessary
integrability properties obtain,
У^Я (т.5_(ш), ш; iTs (ш)) - I ds I H (xs{w),w\u)n(du)
7<1 Jo J
is a (J^)-martingale and
]T H (y, w\ iY(w)) - / dLs(w) / H(s, w; u)n(du)
yeC,,n[0.t]
is a (,3^)-martingale, the proof of which is left to the reader as an exercise.
The exponential formula yields also some interesting results. In what follows,
we consider an additive functional A, such that im({0}) = 0, or equivalently
/ \zdA = 0. In this setting, we have
B.7) Proposition. The random variable ATl is infinitely divisible and the Laplace
transform of its law is equal to
ф,(X) = exp jt j (e~kx -])mA(dx)
where m\ is the image ofn under Ar, the random variable Ar being defined on
U by restriction from W to U.
Proof. Since dA, does not charge Z, we have
A,= f \
Jo
hence, thanks to the strong additivity of A,
The exponential formula of Proposition A.12) in Sect. 1 then implies that
</>,(*) = ? [exp (-А/Ц)] -
which is the announced result.
= exp !r fn(du)(e-kA'iH)-l)\

484 Chapter XII. Excursions
Remark. The process ATl is a Levy process (Exercise B.19) Chap. X) and mA
is its Levy measure; the proof could be based on this observation. Indeed, it is
known in the theory of Levy processes that the Levy measure is the characteristic
measure of the jump process which is a PPP. In the present case, because of the
hypothesis made on vA, the jump process is the process itself, in other words
Аъ = Ls<, AR(es); as a result
mA(C) = E\ у lc (AR(es)) = n [AR e C).
¦v<l
The following result yields the "law" of R under n.
B.8) Proposition. For every x > 0,
n(R > x) = B/ttx)u2.
Proof. The additive functional A, = t plainly satisfies the hypothesis of the pre-
previous result which thus yields
E [exp (-Яг,)] = exp j t j mA(dx) (e~u - l)
By Sect. 2 in Chap. VI, the law of xa is that of Ta which was found in Sect. 3 of
Chap. II. It follows that
(•00
f
Jo
f
Jo
By the integration by parts formula for Stieltjes integrals, we further have
/¦00
X / mA( ]x, oo[ )e~Xxdx = -Лк.
Jo
Since it is easily checked that
/•00
V2A = A / e~kj<B/7zx)]/2dx,
Jo
we get mA(]x, oo[) = B/яхI/2; by the definition of mA, the proof is complete.
Remarks. Г) Another proof is hinted at in Exercise D.13).
2°) Having thus obtained the law of R under и, the description of n will be
complete if we identify the law of u(t), t < R, conditionally on the value taken
by R. This will be done in Sect. 4.
The foregoing proposition says that R (ex(w)) is a PPP on K+ with character-
characteristic measure h given by n(]x, oo[) = B/лхI^2. We will use this to prove another
approximation result for the local time which supplements those in Chap. VI. For
e > 0, let us call r),(e) the number of excursions with length > e which end at
a time s < t. If N is the counting measure associated with the PPP R(es), one
moment's reflection shows that rj,(e) = N[, where N^ = Л^?-°°1, and we have
the

§2. The Excursion Process of Brownian Motion 485
B.9) Proposition. P [!imn0 у[Щп,^) = L, for every t] = \.
Proof. Let sk - 2/nk2; then n([sk, oof) = к and the sequence {N^*1 - N?"\ is
a sequence of independent Poisson r.v.'s with parameter t. Thus, for fixed t, the
law of large numbers implies that a.s.
n n ' и V 2
As Nf increases when s decreases, for sn+\ < s < е„,
and plainly
LV 2 '
We may find a set E of probability 1 such that for w e E,
for every rational t. Since Nf increases with t, the convergence actually holds for
all t's. For each w e E, we may replace t by L,(w) which ends the proof.
Remarks. 1°) A remarkable feature of the above result is that r], (e) depends only
on the set of zeros of В up to /. Thus we have an approximation procedure for
Lt, depending only on Z. This generalizes to the local time of regenerative sets
(see Notes and Comments).
2°) The same kind of proof gives the approximation by downcrossings seen
in Chap. VI (see Exercise B.10)).
B.10) Exercise. Г) Prove that
n I sup \u(
\i<-R
[Hint: If Лх = \u : sup,</? u(t) > x}, observe that Z.^ is the first jump time
of the Poisson process Nt ' and use the law of Ljx found in Sect. 4 Chap. VI.]
2°) Using 1°) and the method of Proposition B.9), prove, in the case of BM,
the a.s. convergence in the approximation result of Theorem A.11) Chap. VI.
3°) Let a > 0 and set Ma(u) = sup;<gr u{t) where, as usual, gTa is the last
zero of the Brownian path before the time Ta when it first reaches a. Prove that
Ma is uniformly distributed on [0, a]. This is part of Williams' decomposition
theorem (see Sect. 4 Chap. VII).
[Hint: If Ja (resp. Jy) is the first jump of the Poisson process NAa (resp. NA"),
then P [Ma < y] = P [Jy = Ja]; use Lemma A.13).]

486 Chapter XII. Excursions
B.11) Exercise. Prove that the process X denned, in the notation of Proposition
B.5), by X,(w) — \es (t - r,_(u>), w)\ - s, if rs-(w) < t < rs(w), is the BM
/
Jo
A = / sgn(Bs)dBs
Jo
and that Y,(w) =s + \es(t- r,_(w), w)\ if rs^(w) < t < rs(w), is a BES3@).
B.12) Exercise. 1°) Let A e Ms be such that n(A) < oo. Observe that the
number CjJ of excursions belonging to A in the interval [0, d,] (i.e. whose two end
points lie between 0 and d,) is defined unambiguously and prove that E [C1^] =
n(A)E[L,].
2°) Prove that on {ds < t],
Cd, - Ci = CAdt_di о ed,, Ld, - Ldl = Lit-i, о edi,
and consequently that Cj — n(A)L, is a (.3^")-martingale.
[Hint: Use the strong Markov property of BM.]
# B.13) Exercise (Scaling properties). 1°) For any с > 0, define a map sc on W
or Us by
sc(w)(t) = w{ct)/yfc.
Prove that e, (sc(w)) = sc (et^.(w)) and that for A e %
[Hint: See Exercise B.11) in Chap. VI.]
2°) (Normalized excursions) We say that и e f/ is normalized if /?(w) = 1.
Let t/1 be the subset of normalized excursions. We define a map v from f/ to (/'
by
Prove that for Г С U\ the quantity
= n+ (v-'(^) П (Л > c)) /«+(/? > c)
is independent of с > 0. The probability measure / may be called the law of the
normalized Bmwnian excursion.
3°) Show that for any Borel subset 5 of R+,
n+ (и~'(Г) П (R e 5)) = y(r)n+(R e 5)
which may be seen as displaying the independence between the length of an
excursion and its form.
4°) Let ec be the first positive excursion e such that R(e) > с Prove that
[Hint: Use Lemma A.13).]

§2. The Excursion Process of Brownian Motion 487
# B.14) Exercise. Let A,(e) be the total length of the excursions with length < e,
strictly contained in [0, t[. Prove that
p\limJ—A,(e) = L, for every t = 1.
|_e|0 V 2s J
[Hint: A,(e) = - Jq xr],(dx) where rj, is defined in Proposition B.9).]
# B.15) Exercise. Let S, = sups</ Bs and nt(s) the number of flat stretches of S of
length > s contained in [0, t]. Prove that
lim / —-n,(e) — S, for every / = 1.
|_n° V 2 J
B.16) Exercise (Skew Brownian motion). Let (Yn) be a sequence of indepen-
independent r.v.'s taking the values 1 and —1 with probabilities a and 1 -a @ < a < 1)
and independent of B. For each w in the set on which В is defined, the set of
excursions e,{w) is countable and may be given the ordering of N. In a manner
similar to Proposition B.5), define a process Xa by putting
X? = Yn\es(t-Ts.(w),w)\
if ts-(w) < t < rs(w) and es is the и-th excursion in the above ordering. Prove
that the process thus obtained is a Markov process and that it is a skew BM by
showing that its transition function is that of Exercise A.16), Chap. I. Thus, we
see that the skew BM may be obtained from the reflecting BM by changing the
sign of each excursion with probability 1 —a. As a result, a Markov process X is
a skew BM if and only if |.Y| is a reflecting BM.
B.17) Exercise. Let A+ = /„' l(S,>0)^ A~ = /„' l(Sj<0)^-
1°) Prove that the law of the pair L~2(A,+ , A~) is independent of t and that
Л+ and A~ are independent stable r.v.'s of index 1/2.
[Hint: A+ + A~ — r, which is a stable r.v. of index 1/2.]
2°) Let a+ and a~ be two positive real numbers, S an independent exponential
r.v. with parameter 1. Prove that
/»o
E[exp(-L-2(a+A+s+a-A-s))]= /
Jo
where </>(s) = -W \(s2 + a+) / + (s2 + a~) \. Prove that, consequently, the
pair L~2 (Л+, A~) has the same law as |(Г+, Т~) where T+ and T~ are two
independent r.v.'s having the law of T\ and derive therefrom the arcsine law. of
Sect. 2 Chap. VI (see also Exercise D.20) below). The reader may wish to compare
this method with that of Exercise B.33) Chap. VI.
B.18) Exercise. Prove that the set of Brownian excursions can almost-surely be
labeled by Q+ in such a way that q < q' entails that eq occurs before eq>.
[Hint: Call e\ the excursion straddling 1, then e1/2 the excursion straddling

488 Chapter XII. Excursions
§3. Excursions Straddling a Given Time
From Sect. 2 in Chap. VI, we recall the notation
g, — sup {s < t : Bs — 0}, d, = inf{s > t : Bs = 0},
and set
At =t ~ gt, Л, -d, -g,.
Plainly, d, > g, if and only if there is an excursion which straddles t and in
that case, A, is the age of the excursion at time / and A, is its length. We have
A, = R(igl).
C.1) Lemma. The map t —*¦ g, is right-continuous.
Proof. Let tn 4- t; if there exists an n such that gUi < t, then glm — g, for m > n\
if gin > t for every n, then t < gln < tn for every и, hence / is itself a zero of В
and g, =t -\\mngtri.
Fig. 8.
For a positive r.v. S we denote by .Щ the cr-algebra generated by the variables
Hs where H ranges through the optional processes. If S is a stopping time, this
coincides with the usual a-field .Щ. Let us observe that in general S < S' does
not entail.% С -Щ', when S and S' are not stopping times; one can for instance
find a r.v. S < 1 such that .Я? = .F^.
Before we proceed, let us recall that by Corollary C.3), Chap. V, since we are
working in the Brownian filtration (-Ю, there is no difference between optional
and predictable processes.
C.2) Lemma. The family (J^) = (.3^,) is a subfiltration of '(.>f) and ifT is a
(.^-stopping time, then .VgT С .5*r С -Щ.

§3. Excursions Straddling a Given Time 489
Proof. As g, is J^-measurable, the same reasoning as in Proposition D.9) Chap. I
shows that, for a predictable process Z, the r.v. Zgl is .^"-measurable whence
¦К С .5f follows.
Now choose и in R+ and set Z\ = Zgltyu; thanks to Lemma C.1) and to what
we have just proved, Z' is (.3^)-optional hence predictable. Pick v > u; if gv < u,
then gu = gv and since ggl = g, for every t,
7' — 7 —7
and if gv > u, then Z'g> = Zgu. As a result, each of the r.v.'s which generate J^
is among those which generate ,WV. It follows that -Wu С ,WV, that is, (J^) is a
filtration.
Let now Г be a (,3^)-stopping time. By definition, the a-algebra &gT is gen-
generated by the variables ZgT with Z optional; but ZgT = (Zg)T and Zg is ,W%l-
optional because of Lemma C.1) which entails that ZgT is .^-measurable, hence
that ,WgT с .Щ. On the other hand, since .'Щ с .^, the time Г is also a Out-
Outstepping time from which the inclusion .Щ С -Щ is easily proved. D
We now come to one of the main results of this section which allows us to
compute the laws of some particular excursions when n is known. If F is a positive
^-measurable function on U, for s > 0, we set
q(s,F)=n(R>s)~i F dn = n(F | R> s).
J{R>i)
We recall that 0 < n(R > s) < oo for every s > 0.
C.3) Proposition. For every fixed t > 0,
— q(A,, F) a.s.,
and for a (-^-stopping time T,
E\F(igT) \-Щ\ =q(AT, F) a.s. on the set {0 < gT < T).
Proof. We know that, a.s., t is not a zero of В hence 0 < g, < t and q(A,, F) is
defined; also, if s € Gw and s < t, we have s = g, if and only if s + R о #, > t.
As a result, g, is the only 5 € Gu, such that 5 < г and s + R о 9S > t. If Z is a
positive (.j^")-predictable process, we consequently have
We may replace Ro0s by R(is) and then apply the master formula to the right-hand
side which yields
Г f°° f 1
E[ZglF(igl)] = E\ / ds ZZi(w)F(u)l{RM>t_Tt(W)>0)n(du)\.
Uo J J

490 Chapter XII. Excursions
Since by Proposition B.8), for every x > 0, we have n(R > x) > 0, the right-hand
side of the last displayed equality may be written
e\J ds ZTs(w)n(R > t - zs(w))q(t - r,(w), F)] .
And, using the master formula in the reverse direction, this is equal to
Zsq(t - s, F)l1Jjofll>,_,>o}] = E [Zgrq(t - g,, F)]
which yields the first formula in the statement.
To get the second one, we consider a sequence of countably valued (-if)-
stopping times Tn decreasing to T. The formula is true for Tn since it is true for
constant times. Moreover, on {0 < gj < T}, one has {#7; = gT] from some n0
onwards and limn 1\8п<т„) = 1; therefore, for bounded F,
E [F (igT) I Щ = UrnE [F (igTn) llgTn<Tn) I Л | Л]
= lim? [q (At-,,, F) l{gTn<Tn\ I Щ = q(AT, F)
because limn ATn — At, the function q(, F) is continuous and Aj is .im-
.immeasurable. The extension to an unbounded F is easy. D
The foregoing result gives the conditional expectation of a function of the
excursion straddling t with respect to the past of BM at the time when the ex-
excursion begins. This may be made still more precise by conditioning with respect
to the length of this excursion as well. In the sequel, we write E[- \ .^Г, Л,]
for E[- I .J^"v а(Л,)]. Furthermore, we denote by v(-; F) a function such that
v(R; F) is a version of the conditional expectation n(F | R). This is well de-
defined since n is a-finite on the cr-field generated by R and by Proposition B.8),
the function r —> v(r; F) is unique up to Lebesgue equivalence. We may now
state
C.4) Proposition. With the same hypothesis and notation as in the last proposition,
and
E \F (igr) I-Щ, AT\ = v(Ar; F) on {0 < gT < T].
Proof. Let ф be a positive Borel function on R+; making use of the preceding
result, we may write
= E[Zglq(A,,<P(R)F)].

§3. Excursions Straddling a Given Time 491
But, looking back at the definition of q, we have
so that using again the last proposition, but in the reverse direction, we get
E [Zgl</>(A,)F (igl)] = E [Zg^{A,)v(At\ F)]
which is the desired result. The generalization to stopping times is performed as
in the preceding proof. ?
We now prove an independence property between some particular excursions
and the past of the BM up to the times when these excursions begin. A (•3^~0)-
stopping time T is said to be terminal if T — t + T о 9, a.s. on the set {T > ?};
hitting times, for instance, are terminal times. For such a time, T = gj + T o6gT a.s.
on {gr < T]. A time T may be viewed as defined on U by setting for и = io(w),
T(u) - T(w) if R(w) > T(w), T{u) = +oo otherwise.
By Galmarino's test of Exercise D.21) in Chap. I, this definition is unambiguous.
If T{u) < oo, the length Лт of the excursion straddling T may then also be
viewed as defined on U. Thanks to these conventions, the expressions in the next
proposition make sense.
C.5) Proposition. IfT is a terminal (,^~°)-stopping time, then on {0 < gj < T),
E [F (igT) | JTr] = „ (F\(R>T)) /n(R >T) = n(F\R>T)
and
E [F (igT) | ,JT , Лт] = v (лг; F\(R>T)) /v (AT; \{r>t)) ¦
Proof. For a positive predictable process Z, the same arguments as in Proposition
C.3) show that
E[ZgTF (igT) l(o<gr<n]
= E
= E
7 F(i \\ • ¦ I
(*(.,»7-(,,»j
/ ds ZzJ{Tt<T) / F{u)\(R(U)>T(U))n{du)\
j ds ZTtliTt<T)n(R> T)]n(FliR>T))/n(R> T)
= E
= E[ ZgT\@<gT<T)]n(F\(R>T)) /n(R > T)
which proves the first half of the statement. To prove the second half we use the
first one and use the same pattern as in Proposition C.4).

492 Chapter XII. Excursions
Remark. As the right-hand side of the first equality in the statement is a constant,
it follows that any excursion which straddles a terminal time T is independent
of the past of the BM up to time gj. This is the independence property we had
announced. We may observe that, by Proposition C.3), this property does not hold
with a fixed time / in lieu of the terminal time T (see however Exercise C.11)).
We close this section with an interesting application of the above results which
will be used in the following section. As usual, we set
Te(w) = inf{r > 0 : w(t) >?}.
On U, we have {Te < oo} = {Te < R], and moreover
C.6) Proposition, и (sups5fi(M) u(s) > e) = n(Te < oo) = l/2e.
Proof. Let 0 < x < y. The time Tx is a terminal time to which we may apply the
preceding proposition with F = \(Tv<ru it follows that
P [Ty (9gTx) <oo] = n(Ty < oo)/n{Tx < oo).
The left-hand side of this equality is also equal to Px \Ty < 7*o] — x/y; as a result,
n (Te < oo) = c/e for a constant с which we now determine.
Proposition B.6) applied to H(s, •; u) = \(T,<oc)(u)hs<t) yields
n(Te <oo)E[L,] = ]
5€С„.П[0.Г]
which, in the notation of Sect. 1 Chap. VI, implies that
cE[L,] = E[s(de(t)±\)];
letting e tend to 0, by Theorem A.10) of Chap. VI, we get с = 1/2.
Remark. This was also proved in Exercise B.10).
C.7) Exercise. Use the results in this and the preceding section to give the con-
conditional law of d\ with respect to g\. Deduce therefrom another proof of 4°) and
5°) in Exercise C.20) of Chap. III.
C.8) Exercise. 1°) Prove that conditionally on g\ — u, the process (B,,t < u) is
a Brownian Bridge over [0, и]. Derive therefrom that B,gJ\/g\, 0 < f < 1 is a
Brownian Bridge over [0, 1] which is independent of g\ and of \Bg[+u, и > 0}
and that the law of g\ is the arcsine law. See also Exercise B.30) Chap. VI.
2°) The Brownian Bridge over [0, 1] is a semimartingale. Let I" be the family
of its local times up to time 1. Prove that Lagx has the same law as y/g\la/^ where
g\ is independent of the Bridge. In particular, L\ = y/g\l°; derive therefrom that
1° has the same law as V2e, where e is an exponential r.v. with parameter 1.
[Hint: See Sect. 6 Chap. 0.]

§4. Descriptions of Ito's Measure and Applications 493
3°) Prove that the process Mu = \Bgl+uo~g,)\ /VI - gi, 0 < и < 1, is in-
independent of the a-algebra ,Ф\; М is called the Brownian Meander of length 1.
Prove that Mx has the law of ¦s/le just as 1° above.
[Hint: Use the scaling properties of и described in Exercise B.13).]
4°) Prove that the joint law of (g,, Lr, B,) is
2(t-s)
dl dx.
[Hint: (gi, L\, B\) = (gu ,/g\l°, \Л - g\M\) where gu 1°, Mx are indepen-
indepent.]
[
dent.]
C.9) Exercise. We retain the notation of the preceding exercise and put moreover
A-t = /„' 1,bs>o)^ and U = /J \{pi>0)ds where fi is a Brownian Bridge. We recall
from Sect. 2 Chap. VI that the law of A \ is the Arcsine law; we aim at proving
that U is uniformly distributed on [0, 1].
1°) Let T be an exponential r.v. with parameter 1 independent of B. Prove
that AgT and (AT — AgT) are independent. As a result,
where e is a Bernoulli r.v. and T, g\,U,e are independent.
2°) Using Laplace transform, deduce from the above result that
Ш ]-N2V
where N is a centered Gaussian r.v. with variance 1 which is assumed to be
independent of U on one hand, and of V on the other hand, and where V is
uniformly distributed on [0, 1]. Prove that this entails the desired result.
C.10) Exercise. Prove that the natural filtration (Jf *) of the process g is strictly
coarser than the filtration (--^П and is equal to {.WtL}.
C.11) Exercise. For a > 0 let T — inf {t :t~g,=a). Prove the independence
between the excursion which straddles T and the past of the BM up to time gr-
§4. Descriptions of Ito's Measure and Applications
In Sects. 2 and 3, we have defined the Ito measure и and shown how it can be
used in the statements or proofs of many results. In this section, we shall give
several precise descriptions of n which will lead to other applications.
Let us first observe that when a a-finite measure is given on a function space,
as is the case for и and U, a property of the measure is a property of the "law" of
the coordinate process when governed by this measure. Moreover, the measure is

494 Chapter XII. Excursions
the unique extension of its restriction to the semi-algebra of measurable rectangles;
in other words, the measure is known as soon as are known the finite-dimensional
distributions of the coordinate process.
Furthermore, the notion of homogeneous Markov process makes perfect sense
if the time set is ]0, oo[ instead of [0, oo[ as in Chap. III. The only differ-
difference is that we cannot speak of "initial measures" any longer and if, given the
transition semi-group Pt, we want to write down finite-dimensional distributions
P \Xh e A\,... ,Xh € Ak] for &-uples 0 < t\ < ... < ft, we have to know the
measures X, = X,(P). The above distribution is then equal to
/ k,^{dxx) / P,2-,](x\,dx2)... I
JA, JA2 JAk
The family of measures k, is known as the entrance law. To be an entrance law,
(k,) has to satisfy the equality k, Ps — kl+s for every s and t > 0. Conversely,
given (k,) and a t.f. (Pt) satisfying this equation, one can construct a measure on
the canonical space such that the coordinate process has the above marginals and
therefore is a Markov process. Notice that the k, 's may be a-finite measures and
that everything still makes sense; if ц is an invariant measure for P,, the family
k, = /j. for every t is an entrance law. In the situation of Chap. Ill, if the process
is governed by Pv, the entrance law is (vP,). Finally, we may observe that in this
situation, the semi-group needs to be defined only for t > 0.
We now recall some notation from Chap. III. We denote by Q, the semi-group
of the BM killed when it reaches 0 (see Exercises A.15) and C.29) in Chap. III).
We recall that it is given by the density
4t(x,y) = (lnt)~42 (expf- —(y-xJ j -expf- — (y + xJ\ \ \{xy>0).
We will denote by k,(dy) the measure on K\{0} which has the density
m,(y) - B7Г73) ' |y|exp(-y2/2O
with respect to the Lebesgue measure dy. For fixed y, this is the density in t of
the hitting time Ty as was shown in Sect. 3 Chap. III.
Let us observe that on ]0, oo[
dg, 1
m, = -— = lim —q,(x, ¦).
dy x^o 2x
Our first result deals with the coordinate process w restricted to the interval
]0, R[, or to use the devices of Chap. Ill we will consider - in this first result only
- that w(t) is equal to the fictitious point S on [R, oo[. With this convention we
may now state
D.1) Theorem. Under n, the coordinate process w(t), t > 0, is a homogeneous
strong Markov process with Q, as transition semi-group and k,, t > 0, as entrance
law.

§4. Descriptions of Ito's Measure and Applications 495
Proof. Everything being symmetric with respect to 0, it is enough to prove the
result for и+ or «-. We will prove it for n+, but will keep n in the notation for
the sake of simplicity.
The space (Us, JMS) may serve as the canonical space for the homogeneous
Markov process (in the sense of Chap. Ill) associated with Q,, in other words
Brownian motion killed when it first hits {0}; we call (Qx) the corresponding
probability measures on (Us, Ш&). As usual, we call 9r the shift operators on Us.
Our first task will be to prove the equality
eq. D.1) n ((и(г) € А)Пв~\Г)) = n (\A{u{r))Qu(r){r))
for Г € 'U, A € .J?(R+ - {0}) and r > 0. Suppose that n(u(r) € A) > 0 failing
which the equality is plainly true. For r > 0, we have {u(r) € А] С {r < R]
hence n(u(r) € A) < oo and the expressions we are about to write will make
sense. Using Lemma A.13) for the process elu<r>eA), we get
n(\A(u(r))(lro9r))/n(u(r) €A) =
where P is the Wiener measure.
The time 5 which is the first jump time of the process e{u(-r)eA] is a (
stopping time; the times zS- and r$ are therefore (.5^")-stopping times. We set
T = r$- + r. The last displayed expression may be rewritten
P[{BT € А)С\{В.овт € Г}]
where В stands for the BM killed when it hits {0}. By the strong Markov property
for the (.^-stopping time T, this is equal to
As a result
n((u(r) e А) П в;1 (Г)) = n(u(r) eA)jy(dx)Qx[n
where у is the law of Вт under the restriction of P to {Вт € A]. For a Borel
subset С of R, make Г — {м@) € С} in the above formula, which, since then
Qx[H= 1 с(x), becomes
n(u(r) € А П C) = n(u(r) e A)y(C);
it follows that y(-) = n(u(r) eAfl -)/n(u(r) € A) which proves eq. D.1).
Let now 0 < t\ < t2 < .. ¦ < tk < t be real numbers and let /i,..., /*, / be
positive Borel functions on R. Since B, the BM killed at 0, is a Markov process
we have, for every x
eq. D.2) Q
1=1
П-Л-(««CO)
= QAY\fi^{ti))Ql.lJ(w(tk)) .

496 Chapter XII. Excursions
Set F = П/=2 fi (w(// ~ fi))- By repeated applications of equations D.1) and
D.2), we may now write
which shows that the coordinate process w is, under n, a homogeneous Markov
process with transition semi-group Q,. By what we have seen in Chap. Ill, В has
the strong Markov property; using this in eq. D.2) instead of the ordinary Markov
property, we get the analogous property for и; we leave the details as an exercise
for the reader.
The entrance law is given by k,(A) = n(u(t) € A) and it remains to prove
that those measures have the density announced in the statement. It is enough to
compute k,([y, oof) for у > 0. For 0 < e < y,
K([y, oof) = n(u(t) >y)= n(u(t) >y;Te< t)
where Ts — inf{f > 0 : u(t) > e). Using the strong Markov property for n just
alluded to, we get
X,([y,oo[) = n(Te<t;Qu{Tf)(u(t-Te)>y))
Applying Proposition C.5) with F(u) = l(r,(u)<f)Gf-r,(n)(e, [У. «>[) yields
k,([y, oo[) = E [l(f1<r)G,-f, (e. [y. oo[ )]n(Te < R)
where Te = Te (igTr). Using Proposition C.6) and the known value of Q,, this is
further equal to
E [[(f,<n (ф'-т,(У + Ю - Ф,-%{У - е)) /2e]
with Ф,(у) — fixgt{z)dz- If we let e tend to zero, then Ts converges to zero
P-a.s. and we get k,([y, oo[) = g,(y) which completes the proof.
Remarks. 1°) That (k,) is an entrance law for Q, can be checked by elementary
computations but is of course a consequence of the above proof.
2°) Another derivation of the value of (kt) is given in Exercise D.9).
The above result permits to give Ito's description ofn which was hinted at in
the remark after Proposition B.8). Let us recall that according to this proposition

§4. Descriptions of Ito's Measure and Applications 497
the density of R under n+ is B\/27Гг3)"'. In the following result, we deal with
the law of the Bessel Bridge of dimension 3 over [0, r] namely Pqq which we
will abbreviate to жг. The following result shows in particular that the law of the
normalized excursion (Exercise B.13)) is the probability measure ii\.
D.2) Theorem. Under n+, and conditionally on R — r, the coordinate process w
has the law nr. In other words, if Г е M%,
f dr
п+(Г) (m{R })
= nr(
Jo
2V2jrr3
Proof. The result of Theorem D.1) may be stated by saying that for 0 < tx <
ti < ... < tn and Borel sets А, с ] 0, oof, if we set
n
Г = f]{u(ti)eAi],
i = l
then
и+(Г) = / mll(xi)dx\ I ql2-n(xux2)dx2... q^-^ixn-^x^dxn-
Ja, Ja2 Jа„
On the other hand, using the explicit value for itr given in Sect. 3 of Chap. XI, and
taking into account the fact that Г C\{R < tn] = 0, the formula in the statement
reads
f°° dr f I Г
п+{Г) = J — I 2V2:7rr3w,1 (X\)dx\ I qh
Л„ 2v27rr3 Ja, Ja2
...I q,H_,n , (xn^uxn)mr-,n(xn)dxn.
JAn
But
r°° dr
/ - /--^2j27trimr^i(xn) =
Jta 2
as was seen already several times. Thus the two expressions for п+{Г) are equal
and the proof is complete.
This result has the following important
D.3) Corollary. The measure n is invariant under time-reversal; in other words,
it is invariant under the map и —*¦ и where
u(t) = u(R(u) - t)\Wu)>t).
Proof. By Exercise C.7) of Chap. XI, this follows at once from the previous
result.
This can be used to give another proof of the time-reversal result of Corollary
D.6) in Chap. VII.

498 Chapter XII. Excursions
D.4) Corollary. If В is a BM@) and for a > 0, Ta = inf{; : B, = a), ifZ is a
BES3@) andoa — sup{f : Z, = a), then the processes Y, = a — Bju-{, t < Ta and
Z,, t <oa are equivalent.
Proof. We retain the notation of Proposition B.5) and set в, = Lt — \B,\ =
s - \es(t - ts-)\ if Tj_ < t < ts. We know from Sect. 2 Chap. VI that В is a
standard BM.
If we set Z, = L, + \B, \ = s + \es{t — rs_)| if ts_ <t<xs, Pitman's theorem
(see Corollary C.8) of Chap. VI) asserts that Z is a BES3@).
For a > 0 it is easily seen that
za = inf{t : L, = a] = inf{f : R, = a}\
moreover
za = sup{f : Z, = a]
since ZXa — LXa + \BTa\ = a and for t > za one has L, > a.
We now define another Poisson point process with values in (U, ¦?/>) by setting
es = es if 5 > a,
es(t) = ea_s (R(ea-S) - t), 0 < t < R(ea^s), \fs<a.
In other words, for s < a,7s = ea^s in the notation of Corollary D.3). Thus, for
a positive .J?(R+) x ^-measurable function /,
f(s,es) = 2__ f(a-s,es),
and the master formula yields
E
\jt<s<a
by Corollary D.3), this is further equal to
= E \ f ds f f(a-s, u)n{du) ;
E \ j ds I f(s, u)n{du) .
This shows that the PPP ? has the same characteristic measure, hence the same
law as e. Consequently the process Z defined by
Z, =s + \es(t -?,_)| ifT,_<r<T5,
has the same law as Z. Moreover, one moment's reflection shows that
Z,=a-P(Ta-t) forO<t<rfl
which ends the proof. Q

§4. Descriptions of Ito's Measure and Applications 499
Let us recall that in Sect. 4 of Chap. X we have derived Williams' path decom-
decomposition theorem from the above corollary (and the reversal result in Proposition
D.8) of Chap. VII). We will now use mis decomposition theorem to give another
description of и and several applications to BM. We denote by M the maximum
of positive excursions, in other words M is a r.v. defined on Uf by
M(u) = sup u(s).
s<R(u)
The law of M under n+ has been found in Exercise B.10) and Proposition C.6)
and is given by n+(M > x) = \/2x.
We now give Williams' description of и. Pick two independent BES3@) pro-
processes p and p and call Tc and Tc the corresponding hitting times of с > 0. We
define a process Zc by setting
A,
с -
0,
Pit
-Tc),
0
Те
t ;
< t
< t
>TC
<
<
Tc.
Tc
Z4=\ С- p{t - Tc), Tc<t<Tc + Tc,
I 0, t>Tc + Tc.
For Г е ^+, we put N(c, Г) — P [Zc e Г]. The map Л' is a kernel; indeed
(Z,c)(>0 = (cZ* 2J , thanks to the scaling properties of BES3@), so that Л'
maps continuous functions into continuous functions on R and the result follows
by a monotone class argument. By Proposition D.8) in Chap. VII, the second part
of Zc might as well have been taken equal to p (Tc + Tc - t), Tc < t < Tc + Tc.
D.5) Theorem. For any Г е $6%
п+(Г) = - / N(x, r)x~2dx.
2 Jo
In other words, conditionally on its height being equal to c, the Brownian excursion
has the law ofZc.
Proof. Let [/, = {u : M(u) > c); by Lemma A.13), for Г е ^+
n+(rnUc) = n+(Uc)P[ec e Г] = ^^ e Г]'
where ec is the first excursion the height of which is > c. The law of this excursion
is the law of the excursion which straddles Tc, i.e. the law of the process
Y, = BgTc+t, 0 <t <dT< -gT,-
By applying the strong Markov property to В at time Tc, we see that the process
Y may be split up into two independent parts Yx and Y2, with
Yt] = BgTr+t, 0 < t < Tc - gTi; Y2 = Bz+l, 0 < t < dTr - Tc.
By the strong Markov property again, the part Y2 has the law of B,, 0 < t < To,
where В is a BM(c). Thus by Proposition C.13) in Chap. VI, Y2 may be described

500 Chapter XII. Excursions
as follows: conditionally on the value M of the maximum, it may be further split
up into two independent parts V1 and V2, with
V,l=BTc+t, 0<t<TM-Tc, V,2=BTu+l, 0<t<dTc-TM.
Moreover V' is a BES3 (c) run until it first hits M and V2 has the law of M - p,
where p is a BES3@) run until it hits M.
Furthermore, by Williams decomposition theorem (Theorem D.9) Chap. VII),
the process K1 is a BES3@) run until it hits c. By the strong Markov property for
BES3, if we piece together У1 and V1, the process we obtain, namely
BgTi+l, o<t<TM-gTr,
is a BES3@) ran until it hits M.
As a result, we see that the law of ec conditional on the value M of the
maximum is that of ZM. Since the law of this maximum has the density c/M2 on
[c, oof as was seen in Proposition C.13) of Chap. VI we get
/•OO
P[ec e Г] = с x~2N(x, F)dx
which by the first sentence in the proof, is the desired result. D
To state and prove our next result, we will introduce some new notation. We
will call .Jf? the space of real-valued continuous functions со defined on an interval
[0, C(u))] С [0, oof. We endow it with the usual a -fields .^"° and .V? generated
by the coordinates. The Ito measure n and the law of BM restricted to a compact
interval may be seen as measures on (.?', .^).
If ix and /x' are two such measures, we define д о д' as the image of д® д'
under the map {со, со') -> со о со' where
Sicooco') = $ (<»)+ ?(<»'),
cooco'(s) = co(s), if 0 < s <?¦(&>)
= ft>(f(ft>))+ft>'(.s-C(<u))-ft/(O) if СИ <s <<(w) + <(«').
V
We denote by v/w the image of д under the time-reversal map со -> со where
И co(s) = co{t;(co) -s), 0 < s < СИ.
Finally, if Г is a measurable map from ,JK' to [0, oo], we denote by цт the image
of д by the map со -*¦ kj (со) where
S(kT(co)) = t;(co) лТ(со), kT{co)(s) = co(s) \{0 < s < $(со) л Т(to).
We also define, as usual:
Ta(co) = inf{? : to(t) = a], La(co) = sup{? : co(t) = a].

§4. Descriptions of lto's Measure and Applications 501
Although the law Pa of BM(a) cannot be considered as a measure on ¦'?', we will
use the notation Pj° for the law of BM(a) killed at 0 which we may consider as
a measure on .Jf/ carried by the set of paths w such that t, (со) = To(co). If S3 is
the law of BES3@), we may likewise consider S^" and the time-reversal result of
Corollary D.6) in Chap. VII then reads
In the same way, the last result may be stated
The space U of excursions is contained in ,jf/ and carries n; on this subspace,
we will write R instead of t, in keeping with the notation used so far and also use
w or со indifferently. We may now state
D.6) Proposition.
/00 /-+OO v
п"(-П{и <R})du= / (Pj')da.
J-00
Proof. Let (в,) be the usual translation operators and as above put
g,(co) = sup{s < t : co(s) = 0}, d,(co) — inf{.s > t : co(s) = 0},
We denote by ?0 the expectation with respect to the Wiener measure.
The equality in the statement will be obtained by comparing two expressions
of
J = Em Г f l[7-0<,]e-"ft Y о k,-gl о Ogldt\
where Pm — f*™ Pada, У is a positive .3^-measurable function and к is > 0.
From Proposition B.6) it follows that
Г f°°
Eo\ / e~Xs'Y ok,-gl oeg
U
\ / e~Xs'Y ok,-gl oegldt\
kudu\.
Using the strong Markov property for Pa we get
J = f+°°daEa [e-kT°] Eo \j e~XsdbA n(f* Y okudu\ .
Yokt-Soes(co)dt
(f Yoku

502 Chapter Xll. Excursions
But Ea [e-kT<>] = e-^^- so that /_+~daEa [e-XT<>] = *J2/X and by the results
in Sect. 2 of Chap. X, Eo [/0°° e~ksdLs] = \/y/2k. As a result
/•OO
J = X~l I n(Y о kul(U<R)du.
Jo
On the other hand, because t, (k,-gl о 6gt} = t,
/.0
= /
JO
v
and since obviously (/^) = Ргт>
/•CO
J= E'm[l[To<l]e-^Z]dt
Jo
where Z(co) — (Y okTo) (со); indeed t — g, is the hitting time of zero for the
process reversed at t. Now the integrand under E'm depends only on what occurs
before t and therefore we may replace E'm by Em. Consequently
J = EmJ 1То<1]
л+00
/
J —
Ea[Z]da.
I— OO
Comparing the two values found for J ends the proof.
Remark. Of course this result has a one-sided version, namely
/OO /4» V
nu+(-П {u < R})du = / (P/0)*/*.
./0
We may now turn to Bismut's description of и.
D.7) Theorem. Let n+ be the measure defined on R+ x Us by
ii + (dt,du) — l(o<t<R(u))dt n+(du).
Then, under n+ the law of the r.v. (t, u) -* u(t) is the Lebesgue measure da and
conditionally on u(t) = a, the processes [u(s), 0 < s < t} and {u(R(u) — s),t <
s < R(u)} are two independent BES3 processes run until they last hit a.
The above result may be seen, by looking upon Us as a subset of .JK', as an
equality between two measures on K+ x ,JK\ By the monotone class theorem, it
is enough to prove the equality for the functions of the form f(t)H(u>) where H
belongs to a class stable under pointwise multiplication and generating the a -field
on .¦&'. Such a class is provided by the r.v.'s which may be written

§4. Descriptions of Ito's Measure and Applications 503
И /-ОО
e'XiS fi((o(s))ds
"-Ш
where X, is a positive real number and fi a bounded Borel function on R. It is clear
that for each t, these r.v.'s may be written Z, • Y, о в, where Z, is .^""-measurable.
We will use this in the following
Proof. Using the Markov description of и proved in Theorem D.1), we have
/ f(t)Zr(u)Yt(et(u))n+(dt,du)
Jr+xUs
= f dt f(t) f \[l<R(u)]Zt(u)Y,(e,(u))n+{du)
Jo Jus
= f dt f(t) f \u<R{u)]Z,(u)EZ°JY,]n+(du)
Jo Ju$
= f dt f{t) f ZtWElllYWn1^ П (t < R))
Jo Jut
where ETX" is the expectation taken with respect to the law of BM(i) killed at 0.
Using the one-sided version of Proposition D.6) (see the remark after it), this is
further equal to
da E\
V
But for Pj\ we have t, = La a.s. hence in particular <w(f) = a a.s. and for Pj",
we have f = To as- s0 that we finally get
r°° v
/ da E?[f(La)ZLa]E?[YTo].
Jo
Using the time-reversal result of Corollary D.4), this is precisely what was to be
proved.
# D.8) Exercise. (Another proof of the explicit value of X, in Theorem D.1)).
Let / be a positive Borel function on K+ and Up the resolvent of BM. Using the
formula of Proposition B.6), prove that
Upf@) = E J
Compute Xu(f) from the explicit form of Up and the law of rs.
* D.9) Exercise. Conditionally on (R = r), prove that the Brownian excursion is a
semimartingale over [0, r] and, as such, has a family I", a > 0, of local times up
to time r for which the occupation times formula obtains.

504 Chapter XII. Excursions
D.10) Exercise. For x e R+, let sx be the time such that eSx is the first excursion
for which R(es) > x. Let L be the length of the longest excursion eu, и < sx.
Prove that
P[L <y] ui
D.11) Exercise. (Watanabe's process and Knight's identity). 1°) Retaining the
usual notation, prove that the process Y, = STl already studied in Exercise A.9) of
Chap. X, is a homogeneous Markov process on [0, oo[ with semi-group T, given
by
Го = /, T,f(x) = e-t/2xf(x) + J e-41y^-1
[Hint: Use the description of BM by means of the excursion process given in
Proposition B.5)]. In particular,
P [SZl <a] = exp(-//2a).
Check the answers given in Exercise A.27) Chap. VII.
2°) More generally, prove that
E [exp (-X2r,+/2) l(sr,<a)] = exp(-A.f coth(aA)/2)
where r,+ = ft \{Bs>Q)ds.
3°) Deduce therefrom Knight's identity, i.e.
E [exp (-X2r,+ /2Sl)] = 2X/ sinhB1).
Prove that consequently,
z,+ /Sl =' inf{s : Us = 2},
where U is a BES3@).
[Hint: Prove and use the formula
f (\-exp(-R/2)\(M<x))dn+ = (cothx)/2.
where M = supr<ft w(t).]
4°) Give another proof using time reversal.
D.12) Exercise. (Continuation of Exercise B.13) on normalized excursions).
Let p be the density of the law of M (see Exercise D.13)) under y. Prove that
/•00
/ xp(x)dx = s/n/2,
Jo
that is: the mean height of the normalized excursion is ^n/2.
[Hint: Use 3°) in Exercise B.13) to write down the joint law of R and M
under и as a function of p, then compare the marginal distribution of R with the
distribution given in Proposition B.8).]

§4. Descriptions of Ito's Measure and Applications 505
D.13) Exercise. 1°) Set M(w) = supf<« w(t)\ using the description of и given
in Theorem D.1), prove that
n+(M > x) = lim ( f Xs(dy)Qy [Tx <T0]+ f Xs(dy))
s^° \Jo Jx /
and derive anew the law of M under n+, which was already found in Exercise
B.10) and Proposition C.6).
2°) Prove that Mx — sup [B,,t < grx} is uniformly distributed on [0, jc] (a part
of Williams' decomposition theorem).
[Hint: If Mx is less than y, for у < x, the first excursion which goes over x is
also the first to go over y.]
3°) By the same method as in 1°), give another proof of Proposition B.8).
# D.14) Exercise. (An excursion approach to Skorokhod problem). We use the
notation of Sect. 5 Chap. VI; we suppose that ^ is continuous and strictly in-
increasing and call ф its inverse.
1°) The stopping times
T = inf {t : S, > ^(B,)} and Г = inf{? : \Bt\ > L, - ф(Ь,)}
have the same law.
2°) Prove that, in the notation of this chapter, the process {s, es] is a PPP with
values in M.+ x Us and characteristic measure ds dn(u).
3°) Let Гх = {{s, u) e R+ x Us : 0 < s < x and M(u) >s- ^E)} and Nx =
Ел 1гД^.е,). Prove that P [LT > x] = P [Nx = 0] and derive therefrom that
0(Sr) = BT has the law /x.
4°) Extend the method to the case where r/r^ is merely right-continuous.
D.15) Exercise. If A = Lz, z > 0, prove that the Levy measure тпа of AXl defined
in Proposition B.7) is given by
ffiA(]JC,oo[) = Bz)-'exp(-jc/2z), x > 0.
* D.16) Exercise. 1°) Using Proposition C.3) prove, in the notation thereof, that if
/ is a function such that f(\B,\) is integrable,
f(\Bt\) I .Ъ\ = A;' J exP(-y2/2A,)yf(y)dy.
[Hint: Write B, = Bgl+,.gl = BA, (ig,).]
2°) By applying 1°) to the functions f(y) = y2 and \y\, prove that t — 2g, and
j(t - gi) - Li are (.5f)-martingales.
3°) If / is a bounded function with a bounded first derivative /' then
V
is a (,i^)-martingale.

506 Chapter XII. Excursions
D.17) Exercise. Let ra = inf{f : L, = a]. Prove that the processes {B,,t < xa]
and [BZa-,,t < ra} are equivalent.
[Hint: Use Proposition B.5) and Corollary D.3).]
--# D.18) Exercise. In the notation of Proposition D.6) and Theorem D.7), for Pq-
almost every path, we may define the local time at 0 and its inverse r,; thus Por'
makes sense and is a probability measure on .JK'.
1°) Prove that
P^dt = / PTu'ds о / и"(. П (м < R))du.
Jo Jo
This formula in another guise may also be derived without using excursion theory
as may be seen in Exercise D.26). We recall that it was proved in Exercise B.29)
of Chap. VI that
JO JO
where Qu is the law of the Brownian Bridge over the interval [0, и].
2°) Call M' the law of the Brownian meander of length t defined by scaling
from the meander of length 1 (Exercise C.8)) and prove that
n'(-r\(t < R)) = M'/Vbrt.
As a result
/00 J» /-0O
M'—= = / S^da.
yjint Jo
0 \l LTtt JO
3°) Derive from (*) lmhof s relation, i.e., for every t
(+) M' =
where X,(u>) — co(t) is the coordinate process.
[Hint: In the left-hand side of (*) use the conditioning given La = t, then use
the result of Exercise C.2) Chap. XL]
By writing down the law of (f, Xc) under the two sides of (*), one also finds
the law of X, under M' which was already given in Exercise C.8).
4°) Prove that (+) is equivalent to the following property: for any bounded
continuous functional F on C([0, 1], K),
Ml(F) = \im(n/2)l/2Er [Fl[ro>,,]/r
where Pr is the probability measure of BM(r) and Го is the first hitting time of
0. This question is not needed for the sequel.
[Hint: Use Exercise A.22) Chap. XL]
5°) On C([0, 1], Ш), set .&, = o(Xs, s < t). Prove that for 0 < / < 1,
S\

§4. Descriptions of lto's Measure and Applications 507
where ф{а) = J° exp(—y2/2)dy. Observe that this shows, in the fundamental
counterexample of Exercise B.13) Chap. V, how much \/Xt differs from a mar-
martingale.
[Hint: Use the Markov property of BES3.]
6°) Prove that, under Af', there is a Brownian motion /J such that
-s
which shows that the meander is a semimartingale and gives its decomposition in
its natural filtration.
[Hint: Apply Girsanov's theorem with the martingale of 5°).]
** D.19) Exercise. If Bs = 0, call D(s) the length of the longest excursion which
occured before time s. The aim of this exercise is to find the law of D(g,) for a
fixed t. For /6 > 0, we set
cp= A-е-")Bзгг3Г1/2А, dp(x)= e^'{2nP
Jo Jx
and
<t>s{x, P) = E [l(D(r,)> ]
1°) If Lp(x) = E [/0°°exp(-^f)l(D(g,)>x)A], prove that
/•O
Jo
4>s{x,0)ds.
2°) By writing
prove that ф satisfies the equation
ф,(х, P) = - (cp + dp(x)) I ф5(х, P)ds + dp(x) I e~c>>sds.
Jo Jo
3°) Prove that
PLp(x) = dp(x)/(cp+dp(x)).
[Hint: {?>(r,) >x} = {D(r.v_) >x}D [xs - rs_ > x}.]
4°) Solve the same problem with D(dt) in lieu of D(g,).
5°) Use the scaling property of BM to compute the Laplace transforms of
)"' and (/>(</,))-'.
** D.20) Exercise. Let A be an additive functional of BM with associated measure
ix and Se an independent exponential r.v. with parameter в2/2.

508 Chapter XII. Excursions
1°) Use Exercise D.18) 1°) to prove that for X > 0,
E0[exp(-XASlt)]
02 poo г / В1 \~\ Г+°° Г / д2 \~|
= —] E0\exp\-XATi-—TA\dsJ Ea\exp(-XATo-—To\ \da.
2°) If ф and i/r are suitable solutions of the Sturm-Liouville equation ф" =
д + в-р\ ф, then
= {в2/2ф'@+)) /
J -
3°) With the notation ofTheorem B.7) Chap. VI find the explicit values of the
expressions in Г) for A, = A+ and derive therefrom another proof of the arcsine
law. This question is independent of 2°).
[Hint: Use the independence of A+ and A~, the fact that rv = A++ A~ and
the results in Propositions B.7) and B.8).]
** D.21) Exercise (Levy-Khintchine formula for BESQ?). If/a is a family of local
times of the Brownian excursion (see Exercise D.9)). call M the image of n+
under the map w -> laR(w). The measure M is a measure on W+ = С (К+, R+).
If / € W+ and X is a process we set
f°°
X/ = / f(t)X,dt.
Jo
Г) With the notation of Sect. 1 Chap. XI prove that for x > 0
Q°x [exp(-X/)] = exp \-xf(\- exp(-(/,
where (f,4>) = f
[Hint: Use the second Ray-Knight theorem and Proposition A.12).]
2°) For ф € W+ call 0s the function defined by ф*(г) - ф ((t - s)+) and put
N — f™ Msds where Ms is the image of M by the map ф —> ф\ Prove that
Q20 [exp(-X/)] = exp 1-2^A - exp(-{/, ф}))N(dф)\.
[Hint: Use Г) in Exercise B.7) of Chap. XI and the fact that for a BM the
process \B\ + L is a BES3@).]
The reader is warned that Ms has nothing to do with the law M" of the meander
in Exercise D.18).
3°) Conclude that
Qsx [exp(-X/)] = exp { - jA - exp(-{/, ф)))(хМ

§4. Descriptions of Ito's Measure and Applications 509
4°) Likewise prove a similar Levy-Khintchine representation for the laws 6*_>0
of the squares of Bessel bridges ending at 0; denote by Mq and No the correspond-
corresponding measures, which are now defined on C( [0, 1]; K+).
5°) For a subinterval / of Ш+, and x, у e /, with x < y, let PXjy be the
probability distribution on C+(I) of a process Xxy which vanishes off the interval
(x, y), and on (jc, y), is a BESQ^O, 0) that is
Xx,y(v) = (y- x)Z (V—^-) \(x<v<y) {v e /)
\y-xj
where Z has distribution Qo'o.
Prove that the Levy measures encountered above may be represented by the
following integrals:
J /-OO 1 /4X1 /-OO
= - y-2P0,ydy; N = - dx {y-x)-2Px,y
? Jo L h Л
dy
1 /"' -2 1 /"' /"' -2
Mo = - / >' •f'o.vd.y; No = - I dx I {y - x) Px,ydy.
2 Jo '" 2 Jo Jx
# D.22) Exercise. Let ф and / be positive Borel functions on the appropriate
spaces. Prove that
/pR(e) /¦ pR(e)
n + (de) I ф(s)f(es)ds = 2 I п+№е)ф{Н(е)) I fBes)ds.
Jo J Jo
[Hint: Compute the left member with the help of Exercise D.17) 2°) and the
right one by using Theorem D.1).]
* D.23) Exercise. Prove that Theorem D.7) is equivalent to the following result.
Let ? be the measure on Ш+ x W x W given by
$(dt,dw,dw') = \(,>0)dt S3(d
and set L(w) = sup{s : w(s) = t]. If we define an {/-valued variable e by
Iw{s) ifO<s<L(w)
w'(L(w) + L(w') - s) if L(w) < s < L(w) + L(w')
0 \fs>L(w) + L(w'),
then the law of (L, e) under ? is equal to n + .
** D.24) Exercise (Chung-Jacobi-Riemann identity). Let В be the standard BM
and T an exponential r.v. with parameter 1/2, independent of B.
1° Prove that for every positive measurable functional F,
E[F (?„; u<gT)\LT=s]= esE [F (Bu; и < т,)ехр (-
and consequently that

510 Chapter XII. Excursions
E[F(Bu-u <gT)]= / E[F{Bu\u <xs
J
= / E[
Jo
2°) Let S°, /° and 1° denote respectively the supremum, the opposite of the in-
fimum and the local time at 0 of the standard Brownian bridge (b(t); t < 1). Given
a .'V{0, 1) Gaussian r.v. N independent of b, prove the three variate formula
P [\N\S° < x\ |N|/° < y\ \N\P> e dl] = exp(-/(coth;t + coth;y)/2)J/.
3°) Prove as a result that
/>[|W|S° <x\ \N\I° <y] = 2/(coth;t + coth>')
and that, if M° = sup{|6(.y)|; s < 1},
P[\N\M° <x] =tanh^r.
Prove Csaki's formula:
P{S°/S° + 1° < v) = A - u)(l - 7rucot(jru)) @ < v < 1)
[Hint: Use the identity:
2 f°° /sinh(i;A)\2
2v2 dX[ ——I =l-jrvcot(nv). ].
4°) Prove the Chung-Jacobi-Riemann identity:
E°+/0J(=)(^°J + (^0J
where M° is an independent copy of M°.
5°) Characterize the pairs (S, /) of positive r.v.'s such that
i) P[\N\S < x; \N\I <y] = 2/{h(x) + h{y)) for a certain function h,
ii) (S + lf^M2 + M2,
where M and M are two independent copies of S V /.
D.25) Exercise. (Brownian meander and Brownian bridges). Let net, and
let Па be the law of the Brownian bridge (B,,t < 1), with Bo = 0 and B\ — a.
Prove that, under Па, both processes BSt - B,,t < 1) and (\B,\ + L,,t < 1)
have the same distribution as the Brownian meander (m,,t < 1) conditioned on
(mi > \a\).
[Hint: Use the relation (+) in Exercise D.18) together with Exercise C.20) in
Chap. VI.]
In particular, the preceding description for a = 0 shows that, if (bt, t < 1) is
a standard Brownian bridge, with a, — sups<, bs, and (/,, t < 1) its local time at
0, then
(mt,t < \) = {2a,-b,,t< l)-(|b,I+/,,?< 1).
Prove that under the probability measure (cti/c) ¦ Щ (resp. (l\/c) ¦ Щ) the process
Ba, -b,,t < 1) (resp. (\b,\+l,,l < 1)) is a BES3.

Notes and Comments 511
D.26) Exercise. 1°) With the notation of this section set J = /0°° Po' dt and prove
that
ЛЮ /.OO /-00 /-00 0
У = / Ai / /»or"<fa = / da (p? о Por« ) ds.
J-oo J0 J-oo JO ч '
[Hint: Use the generalized occupation times formula of Exercise A.13)
Chapter VI.]
2°) Define a map со -^ cb on .jfj' by
?(й) = С(ш) and w(f) = w@) + ««(«)) -ш@,
and call Д the image by this map of the measure д. Prove that J = J and that
(д о д') = Д' о Д
for any pair (д, д') of measures on ,л*Г.
3°) Prove that Рог° = (р?\ and conclude that
[Hint: See Exercise B.29) Chapter VI.]
Notes and Comments
Sect. 1. This section is taken mainly from Ito [5] and Meyer [4].
Exercise A.19) comes from Pitman-Yor [8].
Sect. 2. The first breakthrough in the description of Brownian motion in terms of
excursions and Poisson point processes was the paper of Ito [5]. Although some
ideas were already, at an intuitive level, in the work of Levy, it was Ito who put
the subject on a firm mathematical basis, thus supplying another cornerstone to
Probability Theory. Admittedly, once the characteristic measure is known all sorts
of computations can be carried through as, we hope, is clear from the exercises of
the following sections. For the results of this section we also refer to Maisonneuve
[6] and Pitman [4].
The approximation results such as Proposition B.9), Exercise B.14) and those
already given in Chap. VI were proved or conjectured by Levy. The proofs were
given and gradually simplified in Ito-McKean [1], Williams [6], Chung-Durrett
[1] and Maisonneuve [4].
Exercise B.17) may be extended to the computation of the distribution of the
multidimensional time spent in the different rays by a Walsh Brownian motion
(see Barlow et al. [1] A989)).
Sect. 3. In this section, it is shown how the global excursion theory, presented
in Section 2, can be applied to describe the laws of individual excursions, i.e.
excursions straddling a given random time T. We have presented the discussion

512 Chapter XII. Excursions
only for stopping times T w.r.t. the filtration (.5f) = (.J^,), and terminal
stopping times. See Maisonneuve [7] for a general discussion. The canevas for
this section is Getoor-Sharpe [5] which is actually written in a much more general
setting. We also refer to Chung [1]. The filtration (J^,) was introduced and studied
in Maisonneuve [6].
The Brownian Meander of Exercise C.8) has recently been much studied (see
Imhof ([1] and [2]), Durrett et al [1], Denisov [1] and Biane-Yor [3]). It has found
many applications in the study of Azema's martingale (see Exercise D.16) taken
from Azema-Yor [3]).
Sect. 4. Theorems D.1) and D.2) are fundamental results of Ito [5]. The proof of
Corollary D.4) is taken from Ikeda-Watanabe [2].
Williams' description of the Ito measure is found in Williams [7] and Rogers-
Williams [1] (see also Rogers [1]) and Bismut's description appeared in Bismut
[3]. The formalism used in the proof of the latter as well as in Exercise D.18)
was first used in Biane-Yor [1]. The paper of Bismut contains further information
which was used by Biane [1] to investigate the relationship between the Brownian
Bridge and the Brownian excursion and complement the result of Vervaat [1].
Exercise D.8) is due to Rogers [3]. Knight's identity (Knight [8]) derived in
Exercise D.11) has been explained in Biane [2] and Vallois [3] using a pathwise
decomposition of the pseudo-Brownian bridge (cf. Exercise B.29) Chap. VI);
generalizations to Bessel processes (resp. perturbed Brownian motions) have been
given by Pitman-Yor [9] (resp. [23]). The Watanabe process appears in Watanabe
[2]. Exercise D.14) is from Rogers [1]. Exercise D.16) originates with Azema
[2] and Exercise D.17) with Biane et al. [1]. Exercise D.18) is taken partly from
Azema-Yor [3] and partly from Biane-Yor ([1] and [3]) and Exercise D.19) from
Knight [6]. Exercise D.20) is in Biane-Yor [4] and Exercise D.21) in Pitman-Yor
[2]; further results connecting the Brownian bridge, excursion and meander are
presented in Bertoin-Pitman [1].
With the help of the explicit Levy-Khintchine representation of Q\ obtained
in Exercise D.21), Le Gall-Yor [5] extend the Ray-Knight theorems on Brownian
local times by showing that, for any S > 0, QSQ is the law of certain local times
processes in the space variable. In the same Exercise D.21), the integral represen-
representations of M, N, Mq and No in terms of squares of BES4 bridges are taken from
Pitman [5]. Exercise D.22) is in Azema-Yor [3], and Exercise D.23) originates
from Bismut [3].
The joint law of the supremum, infimum and local time of the Brownian bridge
is characterized in Exercise D.24), taken from work in progress by Pitman and
Yor. The presentation which involves an independent Gaussian random variable,
differs from classical formulae, in terms of theta functions, found in the literature
(see e.g. Borodin and Salminen [1]). Csaki's formula in question 3°) comes from
Csaki [1] and is further discussed in Pitman-Yor [13]. Chung's identity of question
4°) remains rather mysterious, although Biane-Yor [1] and Williams [9] explain
partly its relation to the functional equation of the Riemann zeta function. See also

Notes and Comments 513
Smith and Diaconis [1] for a random walk approach to the functional equation,
and Biane-Pitman-Yor [1] for further developments.
Exercise D.25) is a development and an improvement of the corresponding
result found in Biane-Yor [3] for a = 0, and of the remark following Theorem 4.3
in Bertoin-Pitman [1]. The simple proof of Exercise D.26) is taken from Leuridan
[1].

Chapter XIII. Limit Theorems in Distribution
§1. Convergence in Distribution
In this section, we will specialize the notions of Sect. 5 Chap. 0 to the Wiener
space Wd. This space is a Polish space when endowed with the topology of
uniform convergence on compact subsets of R+. This topology is associated with
the metric
~ sup,<n | w@ - co'(t)\
d(co,со') = > 2~" = .
Y l+suplSn\w(t)-w'(t)\
The relatively compact subsets in this topology are given by Ascoli's theorem.
Let
VN(w,S) = sup{\w(t)-co(t')\; \t-t'\ <S and t, t' < N\.
With this notation, we have
A.1) Proposition. A subset Г ofWd is relatively compact if and only if
(i) the set {<u@), со е Г] is bounded in W;
(ii) for every N,
limsup VN{co,S) = 0.
HO „er
In Sect. 5 Chap. 0, we have defined a notion of weak convergence for prob-
probability measures on the Borel a-algebra of Wd; the latter is described in the
following
A.2) Proposition. The Borel a-algebra on Wrf is equal to the a-algebra .W gen-
generated by the coordinate mappings.
Proof. The coordinate mappings are clearly continuous, hence .7" is contained
in the Borel a-algebra. To prove the reverse inclusion, we observe that by the
definition of d, the map w —»• d(a>, со') where со' is fixed, is .^-measurable. As a
result, every ball, hence every Borel set, is in .V.
Before we proceed, let us observe that the same notions take on a simpler
form when the time range is reduced to a compact interval, but we will generally
work with the whole half-line.

516 Chapter XIII. Limit Theorems in Distribution
A.3) Definition. A sequence {X") of Rd-valued continuous processes defined on
probability spaces (??", J^", P") is said to converge in distribution to a process
X if the sequence (X"(P")) of their laws converges weakly on Wd to the law of
X. We will write X" -^> X.
In this definition, we have considered processes globally as W'-valued random
variables. If we consider processes taken at some fixed times, we get a weaker
notion of convergence.
A.4) Definition. A sequence (X") of (not necessarily continuous) M.d-valued pro-
processes is said to converge to the process X in the sense of finite distributions z/
for any finite collection (t\,..., rt) of times, the Rdk-valued r.v. 's (X^, ..., X")
converge in law to (X,,,..., Xtk). We will write X" —* X.
Since the map со —> (co(t\),..., to{tk)) is continuous on W^, it is easy to see
that, if Х„ —>• X, then Х„ —± X. The converse is not true, and in fact contin-
continuous processes may converge in the sense of finite distributions to discontinuous
processes as was seen in Sect. 4 of Chap. X and will be seen again in Sect. 3.
The above notions make sense for multi-indexed processes or in other words
for С ((К+)', W) in lieu of the Wiener space. We leave to the reader the task of
writing down the extensions to this case (see Exercise A.12)).
Convergence in distribution of a sequence of probability measures on Wrf is
fairly often obtained in two steps:
i) the sequence is proved to be weakly relatively compact;
ii) all the limit points are shown to have the same set of finite-dimensional dis-
distributions.
In many cases, one gets ii) by showing directly that the finite dimensional
distributions converge, or in other words that there is convergence in the sense
of finite distributions. To prove the first step above, it is usually necessary to use
Prokhorov's criterion which we will now translate in the present context. Let us
first observe that the function VN(-,8) is a random variable on Wd.
A.5) Proposition. A sequence (Р„) of probability measures on W'' is weakly rel-
relatively compact if and only if the following two conditions hold:
i) for every ? > 0, there exist a number A and an integer no such that
Р„[ | <u@)| > A] < ?, for every n > no\
ii) for every n, ? > 0 and N e N, there exist a number 8 and an integer no
such that
P,,[VN(-,8) > r]]<? for every n >n0.
Remark. We will see in the course of the proof that we can actually take «o = 0-
Proof. The necessity with n0 = 0 follows readily from Proposition A.1) and
Prokhorov's criterion of Sect. 5 Chap. 0.

§ I. Convergence in Distribution 517
Let us turn to the sufficiency. We assume that conditions i) and ii) hold. For
every n0, the finite family (Р„)п<„0 is tight, hence satisfies i) and ii) for numbers
A' and 8'. Therefore, by replacing A by A v A' and 8 by S A 8', we may as well
assume that conditions i) and ii) hold with no = 0. This being so, for e > 0 and
N e N, let us pick ANe and 8Nik,e such that
supPH[\io@)\>AN,e] < 2-N~ls,
П
supPn[VN{-,SN.k.e)> \/k] < 2-"-*-'?,
П
and set KNtE = \w : | w@)| < AN,e, VN{to,8N,k,e) < \/k for every к > l}. By
Proposition A.1), the set Ke = f]N KN<e is relatively compact in Wd and we have
Р„ (A'f) < J^N Pn {Kcn e) < e, which completes the proof. n
We will use the following
A.6) Corollary. IfX" — {X"x Xnd) is a sequence of d-dimensional continuous
processes, the set (X"(P")) of their laws is weakly relatively compact if and only
if, for each j, the set of laws X"(P") is weakly relatively compact.
Hereafter, we will need a condition which is slightly stronger than condition ii)
in Proposition A.5).
A.7) Lemma. Condition ii) in Proposition A.5) is implied by the following con-
condition: for any N and E, ц > 0. there exist a number 8, 0 < 8 < 1, and an integer
no, such that
S~lPn\\eo: sup | co(s) - co(t)\ > r\ \ < E for n > n0 and for all t < N.
LI t<s<t+S JJ
Proof. Let N be fixed, pick e, r\ > 0 and let n0 and S be such that the condition
in the statement holds. For every integer / such that 0 < i < N8~\ define
A,: = { sup \eo{i8)-co(s)\>ri\.
As is easily seen {VN(-, 8) < 3?^} D П, А% ап^ consequently for every n > no,
we get
Pn[VN.{;S)>3ri]< Pn
which proves our claim. ?
The following result is very useful.
A.8) Theorem (Kolmogorov's criterion for weak compactness). Let {X") be a
sequence of "Rd-valued continuous processes such that

518 Chapter XIII. Limit Theorems in Distribution
i) the family [X(Pn)\ of initial laws is tight in Rd,
ii) there exist three strictly positive constants a, f), у such that for every s,/et+
and every n,
then, the set {X"(P")} of the laws of the Х„ 's is weakly relatively compact.
Proof. Condition i) implies condition i) of Proposition A.5), while condition ii)
of Proposition A.5) follows at once from Markov inequality and the result of
Theorem B.1) (or its extension in Exercise B.10)) of Chap. I. ?
We now turn to a first application to Brownian motion. We will see that the
Wiener measure is the weak limit of the laws of suitably interpolated random
walks. Let us mention that the existence of Wiener measure itself can be proved
by a simple application of the above ideas.
In what follows, we consider a sequence of independent and identically dis-
distributed, centered random variables ?* such that E [?t2] = a2 < oo. We set So = 0,
Sn = Yl"=i ?ь If M denotes the integer part of the real number x, we define the
continuous process X" by
Xя = (ojn) (S[nt] + (nt -
A.9) Theorem (Donsker). The processes X" converge in distribution to the stan-
standard linear Brownian motion.
Proof We first prove the convergence of finite-dimensional distributions. Let t\ <
ti < ... < tk', by the classical central limit theorem and the fact that [nt]/n con-
converges to t as n goes to +00, it is easily seen that (X^, Х"г — X",..., Х"к — X"h J
converges in law to (#,,, Bl2 — Btl,..., B,t — #,,._,) where В is a standard linear
BM. The convergence of finite-dimensional distributions follows readily.
Therefore, it is sufficient to prove that the set of the laws of the Xn's is weakly
relatively compact. Condition i) of Proposition A.5) being obviously in force, it
is enough to show that the condition of Lemma A.7) is satisfied.
Assume first that the ^'s are bounded. The sequence |S^|4 is a submartingale
and therefore for fixed n
P [max \Si\ > XaVn\ < E [|5„|4] (ка^п)~4 .
One computes easily that E [S4] = nE [?4] + 3n(n — l)a4. As a result, there is
a constant К independent of the law of ?t such that
Г _1 ,
lim P max IS, I > XaJn\ < KX 4.
»oo
By truncating and passing to the limit, it may be proved that this is still true if
we remove the assumption that ?* is bounded. For every к > 1, the sequence

§1. Convergence in Distribution 519
[Sn+k — Sk) has the same law as the sequence {?„} so that finally, there exists an
integer n\ such that
P max \Si+k - Sk\ > Xojn < KX 4
L 's" J
for every к > 1 and n > гц. Pick e and r\ such that 0 < e, rj < 1 and then choose
X such that KX~2 < це1; set further S — e2X'2 and choose n0 > n\S~l.\f n > n0,
then [и<5] > пи and the last displayed inequality may be rewritten as
P max \Sl+k - Sk\ > XaJhiS] < rie2X'2.
|_i<M] J
Since AV[«<5] < e^/n, we get
5 P max \Si+k - Sk\ > ecrVn < ц
\}<{п&\ J
for every к > 1 and n > no. Because the Xn's are linear interpolations of the
random walk E„), it is now easy to see that the condition in Lemma A.7) is
satisfied for every TV and we are done. D
To illustrate the use of weak convergence as a tool to prove existence results,
we will close this section with a result on solutions to martingale problems. At
no extra cost, we will do it in the setting of Ito processes (Definition B.5), Chap.
VII).
We consider functions a and b defined on R+ x Wrf with values respectively
in the sets of symmetric non-negative d x ^-matrices and Rd -vectors. We assume
these functions to be progressively measurable with respect to the filtration (J^"°)
generated by the coordinate mappings co(t). The reader is referred to the beginning
of Sect. 1 Chap. IX. With the notation of Sect. 2 Chap. VII, we may state
A.10) Theorem. If a and b are continuous on R+ x W*, then for any probability
measure /i on Rd, there exists a probability measure P on Wrf such that
i) P[w@) e A] =
ii) for any f 6 C2K, the process f(co(t)) - /(w@)) - /„' Lsf{w(s))ds is a
(.^"°, P)-martingale, where
Lsf(co(s)) = i
0(Jfft,)(
Proof. For each integer n, we define functions а„ and bn by
an(t, со) = a([nt]/n, w), bn(t, со) = b{[nt]/n, со).
These functions are obviously progressively measurable and we call L" the cor-
corresponding differential operators.

520 Chapter XIII. Limit Theorems in Distribution
Pick a probability space (?2, &~, P) on which a r.v. Xo of law fx. and a BMJ@)
independent of Xo, say B, are defined. Let an be a square root of an. We define
inductively a process X" in the following way; we set Хд = Xo and if X" is
defined up to time k/n, we set for k/n < t < (к + 1)/и,
X," = Xnk/n + an{k/n, X")(Bt - Bk/n) + bn(k/n, X")(t - k/n).
Plainly, X" satisfies the SDE
on(s,X")dBs+ / bn(s,X")ds
Jo
and if we call P" the law of X" on Wrf, then Pn[co@) e A] = /i(A) and
f(co(t)) - /(«@)) - /0' LnJ{to(s))ds is a P"-martingale for every f eC2K.
The set (P") is weakly relatively compact because condition i) in Theorem
A.8) is obviously satisfied and condition ii) follows from the boundedness of a
and b and the Burkholder-Davis-Gundy inequalities applied on the space Q.
Let P be a limit point of (P") and (P") be a subsequence converging to
P. We leave as an exercise to the reader the task of showing that, since for
fixed / the functions f0 L" f{co(s))ds are equi-continuous on Wrf and converge to
f^Lsf(o(s))ds, then
Ep\(f(co(t))-j Lsf(oj(s))ds^] =
\( V • \ 1
lim EP,,' I f(w{t)) — I L" f(co(s))ds ) ф
«'-~ LV Jo ) J
for every continuous bounded function ф. If t\ < /2 and 0 is .^"-measurable it
follows that
EP \(f(a>{h)) - f(co(tO) - j 2 Lsf(co(s))ds\ J = 0
since the corresponding equality holds for P" and L"s . By the monotone class
theorem, this equality still holds if ф is merely bounded and .^"-measurable; as
a result, /(&>(?)) — f(o)@)) — f0 Lsf(co(s))ds is a P-martingale and the proof is
complete. ?
Remarks. With respect to the results in Sect. 2 Chap. IX, we see that we have
dropped the Lipschitz conditions. In fact, the hypothesis may be further weakened
by assuming only the continuity in со of a and b for each fixed t. On the other
hand, the existence result we just proved is not of much use without a uniqueness
result which is a much deeper theorem.
A.11) Exercise. Г) If (X") converges in distribution to X, prove that (X")*
converges in distribution to X* where, as usual, X* = supf<, \XS\.
2°) Prove the reflection principle for BM (Sect. 3 Chap. Ill) by means of the
analogous reflection principle for random walks. The latter is easily proved in

§1. Convergence in Distribution 521
the case of the simple random walk, namely with the notation of Theorem A.9),
* A.12) Exercise. Prove that a family (Рк) of probability measures on С ((R+)k, R)
is weakly relatively compact if there exist constants a, p, y, p > 0 such that
supA Ex[ |Xo|^] < oo, and for every pair (s, t) of points in (R+)k
supEk[\Xs-Xt\a]<P\s-t\k+r
x
where X is the coordinate process.
* A.13) Exercise. Let /3", s e [0, 1] and y", t e [0, 1] be two independent se-
sequences of independent standard BM's. Prove that the sequence of doubly indexed
processes
n
V"! — И~1/2 Vfl'V
1
converges in distribution to the Brownian sheet. This is obviously an infinite-
dimensional central-limit theorem.
A.14) Exercise. In the setting of Donsker's theorem, prove that the processes
(ay/H)~l (Slnt] + (nt - [nt])Slnt]+i -tSH), 0 < t < 1,
converge in distribution to the Brownian Bridge.
A.15) Exercise. Let (M") be a sequence of (super) martingales defined on the
same filtered space and such that
i) the sequence (M") converges in distribution to a process M;
ii) for each t, the sequence (M") is uniformly integrable.
Prove that M is a (super) martingale for its natural filtration.
* A.16) Exercise. Let (M") be a sequence of continuous local martingales vanish-
vanishing at 0 and such that ((M", M")) converges in distribution to a deterministic
function a. Let Pn be the law of M".
1°) Prove that the set (Р„) is weakly relatively compact.
[Hint: One can use Lemma D.6) Chap. IV.]
2°) If, in addition, the M"'s are defined on the same filtered space and if, for
each t, there is a constant <x(t) such that (M", M")t < a(t) for each n, show that
(Pn) converges weakly to the law Wu of the gaussian martingale with increasing
process a(t) (see Exercise A.14) Chap. V).
[Hint: Use the preceding exercise and the ideas of Proposition A.23) Chap.
IV.]
3°) Let (M") = (M", i = 1,... ,k) be a sequence of multidimensional local
martingales such that (M") satisfies for each i all the above hypotheses and, in
addition, for i ф j, the processes (M", M") converge to zero in distribution. Prove
that the laws of M" converge weakly to Wa, <8>... <8> Wat.
[Hint: One may consider the linear combinations ?и,-Л/".]

522 Chapter XIII. Limit Theorems in Distribution
The two following exercises may be solved by using only elementary properties
of BM.
* A.17) Exercise (Scaling and asymptotic independence). 1°) Using the notation
of the following section, prove that if yS is a BM, the processes /3 and /3(c) are
asymptotically independent as с goes to 0.
[Hint: For every A > 0, (Д.2,, t < A) and (f)c2A+u — f)c2A, и > 0) are
independent.]
2°) Deduce from 1°) that the same property holds as с goes to infinity. (See
also Exercise B.9).)
Prove that for с ф 1, the transformation x —> X(c) which preserves the Wiener
measure, is ergodic. This ergodic property is the key point in the proof of Exercise
C.20), 1°), Chap. X.
3°) Prove that if (yt, t < 1) is a process whose law Py on C([0, 1], R) satisfies
P^ < W\.%- for every t < 1,
then the two-dimensional process V,(t> = ((y,(c), yt), t < 1) converges in law as с
goes to 0 towards ((/J,, yt), t < 1), where f$ is a BM which is independent of y.
[Hint: Use Lemma E.7) Chap. 0.]
4°) Prove that the law of y(c) converges in total variation to the law of fi i.e. the
Wiener measure. Can the convergence in 3°) be strengthened into a convergence
in total variation?
5°) Prove that V(c) converges in law as с goes to 0 whenever у is a BB, a
Bessel bridge or the Brownian meander and identify the limit in each case.
* A.18) Exercise. (A Bessel process looks eventually like a BM). Let Л be a
BES^(r) with S > 1 and r > 0. Prove that as t goes to infinity, the process
(R,+s — R,, s > 0) converges in law to a BM1.
[Hint: Use the canonical decomposition of Л as a semimartingale. It may be
necessary to write separate proofs for different dimensions.]
§2. Asymptotic Behavior of Additive Functionals
of Brownian Motion
This section is devoted to the proof of a limit theorem for stochastic integrals with
respect to BM. As a corollary, we will get (roughly speaking) the growth rate of
occupation times of BM.
In what follows, В is a standard linear BM and L" the family of its local
times. As usual, we write L for L°. The Lebesgue measure is denoted by m.
B.1) Proposition. If f is integrable,
f(nBs)ds =m(f)L a.s.,
lim n I
"->¦<» Jo

§2. Asymptotic Behavior of Additive Functionals of Brownian Motion 523
and, for each t, the convergence ofn /„' f(nBs)ds to m(f)L, holds in Lp for every
p > 1. Both convergences are uniform in t on compact intervals.
Proof. By the occupation times formula
f(nBs)ds = / f(a)L?"da.
J-oc
For fixed t, the map a —>¦ L° is a.s. continuous and has compact support; thus, the
r.v. supa L" is a.s. finite and by the continuity of L; and the dominated convergence
theorem,
limn / f(nB,)ds = m(f)L, a.s.
" Jo
Hence, this is true simultaneously for every rational t; moreover, it is enough to
prove the result for / > 0 in which case all the processes involved are increasing
and the proof of the first assertion is easily completed.
For the second assertion, we observe that
f f{a)La,lnda < ||/Hi (supz/
and, since by Theorem B.4) in Chap. XI, supa L° is in Lp for every p, the result
follows from the dominated convergence theorem.
The uniformity follows easily from the continuity of L° in both variables. D
The following is a statement about the asymptotic behavior of additive func-
functionals, in particular occupation times. The convergence in distribution involved
is that of processes (see Sect. 1), not merely of individual r.v.'s.
B.2) Proposition. If A is an integrable CAF,
lim —=An =vA(l)L in distribution.
n-+oc ^Jn
Proof. Since (see Exercise B.11) Chap. VI) Lan = y/nL"/^", it follows that
4=AB. = 4= f К vA{da) (=» f La^nvA(da)
and the latter expression converges a.s. to v^(l)L by the same reasoning as in
the previous proposition. о
The above result is satisfactory for vA(\) Ф 0; it says that a positive inte-
integrable additive functional increases roughly like vA(\)*/i. On the contrary, the
case v,i(l) = 0 must be further investigated and will lead to a central-limit type
theorem with interesting consequences. Moreover, measures with zero integral are
important when one wants to associate a potential theory with linear BM.
If we refer to Corollary B.12) in Chap. X, we see that we might as well work
with stochastic integrals and that is what we are going to do. To this end, we

524 Chapter XIII. Limit Theorems in Distribution
need a result which will be equally very useful in the following section. It is an
asymptotic version of Knight's theorem (see Sect. 1 Chap. V).
In what follows, (Af", 1 < j < k) will be a sequence of ^-tuples of continuous
local martingales vanishing at 0 and such that {M", M")x = oo for every n and
j. We call T"(t) the time-change associated with {M",M") and $n. the DDS
Brownian motion of M".
B.3) Theorem. If, for every t, and every pair (/, j) with i ф j
lim (Mf, M">r-(f) = lim <M,", M/W) = 0
in probability, then the k-dimensionaI process /3" — 1/5", 1 < j < kj converges in
distribution to a BM*.
Proof. The laws of the processes $" are all equal to the one-dimensional Wiener
measure. Therefore, the sequence ф"} is weakly relatively compact and we must
prove that, for any limit process, the components, which are obviously linear BM's,
are independent.
It is no more difficult to prove the results in the general case than in the case
к = 2 for which we introduce the following handier notation. We consider two
sequences of continuous local martingales (ЛР) and (N"). We call /u"(f) and rf(t)
the time-changes associated with (Mn, M") and (N", N") respectively and /3" and
y" the corresponding DDS Brownian motions.
If 0 = ?o < h < • • • < tp = t and if we are given scalars /0,..., /p_i and
go gp-u we set
g = E sj¦ i ]r,,r,+,i. у" ш =
j
Let us first observe that if we set
U;= [' f((M",M")u)dMnu, Vs"= f g((N",N")u)dN"u,
Jo Jo
then P(f) = U^ and y"(g) = V^. Therefore writing E {Z (i (U" + V"))TO] = 1
yields
?[exp(i(/5"(/) + yn{g))) ¦ H"] = exp (-^ J {f + g2) (t)dt^j
where
Я" =
= exp^
V i.j
«W"> ^Я>мЧ<,+1)л,-(<>+,) - (Л/", ^">M»ft>v,-@>) ) ¦
I

§2. Asymptotic Behavior of Additive Functionals of Brownian Motion 525
The hypothesis entails plainly that H" converges to 1 in probability; thus the
proof will be finished if we can apply the dominated convergence theorem. But
Kunita-Watanabe's inequality (Proposition A.15) Chap. IV) and the time-change
formulas yield
and we are done. D
We will make a great use of a corollary to the foregoing result which we now
describe. For any process X and for a fixed real number h > 0, we define the
scaled process Х(Л) by
The importance of the scaling operation has already been seen in the case of BM.
If M is a continuous local martingale and /3 its DDS Brownian motion, then /3(/l)
is the DDS Brownian motion of A M as is stated in Exercise A.17) of Chap. V.
We now consider a family M,, i = 1, 2,... ,k of continuous local martingales
such that {M^ M,)^ — oo for every i and call Д their DDS Brownian motions.
We set M" = Mi/^/n and call Д" the DDS Brownian motion of M". As observed
above, #(О =
B.4) Corollary. The k-dimensional process /J" — (Д", i — 1,..., к) converges in
distribution to a BM* as soon as
almost surely for every i, j < к with i ф j.
Proof. If r,(f) (resp. r"(f)) is the time-change associated with (Л/,-, Af;)
(resp. (Mf,M(")), then r?(t) = T,(nt) and consequently (Aff, MJ)T»U) =
n~l (M/, Mj)Ti(nt). The hypothesis entails that f~'(Af(-, Afj)Ti@ converges a.s. to
0 as / goes to +00, so that the result follows from Theorem B.3). о
The foregoing corollary has a variant which often comes in handy.
B.5) Corollary. If there is a positive continuous strictly increasing function ф on
R+ such that
i) фA)~1 (Mj, Mi), —> Ui, i = 1, 2, ..., Jk where Uj is a strictly positive r.v.,
ii) 0(f)~' supJ<; |(M/, Afy).v| —> 0 in probability for every i, j < к with i ф j,
1 t-*oo
then the conclusion of Corollary B.4) holds.
Proof. Again it is enough to prove that, for / ф j, t"' (M,-, Mj)TiU) converges to 0
in probability and in fact we shall show that t~lX, converges to 0 in probability
where X, — sup {|(Л/,-, Mj)s\; s < r,(f)}.
Hypothesis i) implies that f~'</>(ri@) converges to U~l in distribution. For
к > 0 and x > 0, we have, using the fact that X is increasing,

526 Chapter XIII. Limit Theorems in Distribution
P[XrAn>Xt] < Я[0(т,(О) >tx] + P[XTi@ >Af;r,(f) < 0
< P [ф(т; @) > tx] + P [Хф-, [lx) > Xt] .
Pick e > 0; since (/, is strictly positive we may choose x sufficiently large and
T > 0, such that, for every t > T,
Р[ф(т,Ц))>1х]<е.
Hypothesis ii) implies that there exists T' > T such that for every t > T',
P [Хф^(и) > Xt] < e.
It follows that for t > T,
P [XZiU) > Xt] < 2e,
which is the desired result. a
We now return to the problem raised after Proposition B.2). We consider Borel
functions fj, i = 1, 2,..., к in Ll(m) П L2(m) which we assume to be pairwise
orthogonal in L2, i.e. / fifjdm = 0 for i ф j. We set
Щ = JH f fi(nB5)dBs.
B.6) Theorem (Papanicolaou-Stroock-Varadhan). The (k+\)-dimensionalpro-
(k+\)-dimensionalprocess (B, M" , Ml) converges in distribution to (?, И/НЬк/,' = 1,2 к)
where (^, /',..., /*) is a BM*+I and I is the local time of ft at zero.
Proof. We have
(M;",M"), =n f (fifj
Jo
so that by Proposition B.1), we have a.s.
lim (M,«, Ml), = \\М\Ц- for i ф j, lim (M,", M"), = lim (Mf, В), = О
uniformly in t on compact intervals. Thus it is not difficult to see that the hypothe-
hypotheses of Theorem B.3) obtain; as a result, if we call B" the DDS Brownian motion
of M", the process (В, В",..., B?) converges in distribution to (/J, /',..., yk).
Now (B, Ml,..., M"k) is equal to (В, В," ((Л/f, A/f)), i = 1,... ,k) and it is
plain that (B, (Mf, M,"), / = 1 ,k,B",i=\ k) converges in distribution
to (p, \\fi\\\l,i = 1 Jfc, у'\/ = 1 /t). The result follows. D
Remark. Instead of n, we could use any sequence (an) converging to +oo or, for
that matter, consider the real-indexed family
fi(XBs)dBs
o
and let X tend to +oo. The proof would go through just the same.

§2. Asymptotic Behavior of Additive Functionals of Brownian Motion 527
We will now draw the consequences we have announced for an additive func-
functional A which is the difference of two integrable positive continuous additive
functionals (see Exercise B.22) Chap. X) and such that vA(l) — 0. In order to be
able to apply the representation result given in Theorem B.9) of Chap. X, we will
have to make the additional assumption that
\x\\vA\(dx) < oo.
As in Sect. 3 of the Appendix, we set
F(x) = j \x-y\vA{dy).
The function F is the difference of two convex functions and its second derivative
in the sense of distributions is 2vA. Let F'_ be its left derivative which is equal to
vA(] — oo, ¦[). We have the
B.7) Lemma. The function F is bounded and F'_ is in.(/5l(m)r\%2(m). Moreover
where I(vA) = -A/2)// \x - y\vA(dx)vA(dy) is called the energy ofvA.
Proof. Since
f\x- y\\vA\(dy) < И|М| + j \y\\vA\(dy),
the integral I(vA) is finite and we may apply Fubini's theorem to the effect that
l(vA) - - // (x - y)vA(dx)vA(dy)
J J
= - /77 vA(dx)vA(dy)dz
JJJx>z>y
The set of z's such that vA({z}) ф 0 is countable and for the other z's, it follows
from the hypothesis vA A) = 0 that
Thus the proof of the equality in the statement is complete.
By the same token
ЛОО /»00 |»00
/ \F'_{x)\dx = / dx\vA(-\x,oo[)\ < / \vA\(-\x,oo[)dx
Jo Jo Jo
x\vA\(dx) < oo,

528 Chapter XIII. Limit Theorems in Distribution
and likewise
/ \F'_{x)\dx < f \x\\vA\{dx) < oo.
— oo J~ oo
Consequently, F'_ is in 5t' and it follows that F is bounded. D
We may now prove that additive functionals satisfying the above set of hy-
hypotheses are, roughly speaking, of the order of f'/4 as t goes to infinity.
B.8) Proposition. IfvA(\) = 0 and j \x\\vA\{dx) < oo, the 2-dimensional pro-
process (п~х12В„., и~'/4Л„.) converges in distribution to (/6, I(vA)^2yi), where (/3, y)
is a BM2 and I the local time of fi at 0.
Proof. By the representation result in Theorem B.9) of Chap. X and Tanaka's
formula,
F(B,,) - F@)] - n~'/4 f F'_(B,)dBs.
Jo
Since F is bounded, the first term on the right goes to zero as n goes to infinity
and, therefore, it is enough to study the stochastic integral part.
Setting s = nu, we see that we might as well study the limit of
F'_(Bnu)dB
nuY
and since В„. — y/nBy this process has the same law as
Because F'_ is in %x(m) П J/^2(m), it remains to apply the remark following
Theorem B.6). ?
Remark. Propositions B.2) and B.8) are statements about the speed at which
additive functionals of linear BM tend to infinity. In dimension d > 2, there is no
such question as integrable additive functionals are finite at infinity but, for the
planar BM, the same question arises and it was shown in Sect. 4 Chap. X that
integrable additive functionals are of the order of logf. However, as the limiting
process is not continuous, one has to use other notions of convergence.
* B.9) Exercise. 1°) In the situation of Theorem B.3), if there is a sequence of
positive random variables Ln such that
i) \\.mn(M", M")Ln — +oo in probability for each /;
ii) limn sups<Ln {M", M")s = 0 in probability for each pair i, j with / / j,
prove that the conclusion of the Theorem holds.
2°) Assume now that there are only two indexes and write M" for M" and N"
for Л/". Prove that if there is a sequence (Ln) of positive random variables such
that

§2. Asymptotic Behavior of Additive Functionals of Brownian Motion 529
i') limn{M", Mn)Ln = oo in probability,
ii) limnsupsiLii\(Mn,Nn)s\=O,
then the conclusion of the Theorem holds.
3°) Deduce from the previous question that if Д is a BM, and if с converges
to +oo, then ft and /S(c) are asymptotically independent.
Remark however that the criterion given in Corollary B.4) does not apply in
the particular case of a pair (M, ^M) as с -> oo. Give a more direct proof of the
asymptotic independence of/6 and /}(c).
* B.10) Exercise. For / in L2 П L1, prove that for fixed t, the random variables
¦s/n fQ f{nBs)dBs converge weakly to zero in L1 as n goes to infinity. As a
result, the convergence in Theorem B.6) cannot be improved to convergence in
probability.
* B.11) Exercise. Let 0 < a\ < ... < a^ < oo be a finite sequence of real numbers.
Prove that the (k + 1)-dimensional process
.^(L«'"-tf-''"),/ = 1,2,...,*)
converges in distribution to
where (/', i = 1, 2,... ,k) is a ^-dimensional BM independent of /6 and / is the
local time of fi at 0.
* B.12) Exercise. 1°) Let
X(t,a)= f l[0M](Bs)dBs.
Jo
Prove that for p > 2, there exists a constant Cp such that for 0 < s < t < 1 and
0 <a <b <l,
E [ \X(t, b) - X(s, a)\P] < Cp ((f - sY11 + (b- a)p/2).
2°) Prove that the family of the laws /\ of the doubly indexed processes
is weakly relatively compact.
[Hint. Use Exercise A.12).]
3°) Prove that, as к goes to infinity, the doubly-indexed processes
converge in distribution to (В,,Ш (L^\ a)), where В is a Brownian sheet indepen-
independent of B.
[Hint: Use the preceding Exercise B.11).]

530 Chapter XIII. Limit Theorems in Distribution
4°) For v > 0, prove that
«-C/2+u) (Ц -L°t)da^2 e-("+i/2)-B (Ll e") du.
E^-° Jo
5°) Let xx = inf {u : L°u > jc}; the processes Xх'1 (ьат{х - х\ /2 converge in
distribution, as A. tends to +oo, to the process -Jx~ya where ya is a standard BM.
This may be derived from 3°) but may also be proved as a consequence of the
second Ray-Knight theorem (Sect. 2 Chap. XI).
* B.13) Exercise. With the notation of Theorem A.10) in Chap. VI, prove that
lim 4= (W) - \l ) = Yi
in the sense of finite distributions, where as usual, / is the local time at 0 of a BM
independent of у.
[Hint: If Mf = J= /„' eesdBs and />E is the law of (B,, L,, Mf), prove that the
set (Pe, s > 0) is relatively compact.]
* B.14) Exercise. In the notation of this section, if (jc,-), i — 1,. • •, к is a sequence
of real numbers, prove that E, ?~1/2 (Z/i+E — Z/'), i = 1,... ,k) converges in
distribution as e -> 0, to (B, 2^[.t,, i = 1 k), where (Я, /31,..., j8*) is a
BM*+1.
** B.15) Exercise. Prove, in the notation of this section, that for any x € K,
?~1/2 [e /0 l[x,x+E]Ej)di - Lx] converges in distribution to B/V3) j8/.t, as ?
tends to 0. The reader will notice that this is the "central-limit" theorem associated
with the a.s. result of Corollary A.9) in Chap. VI.
[Hint: Extend the result of the preceeding exercise to (L*l+ez — Z/1) and get
a doubly indexed limiting process.]
* B.16) Exercise (A limit theorem for the Brownian motion on the unit sphere).
Let Z be a BMd(a) with а ф 0 and d > 2; set p — \Z\. Let V be the process
with values in the unit sphere of Kd defined by
where C, = /0' p~2ds. This is the skew-product decomposition of BMd.
1°) Prove that there is a BMd, say B, independent of p and such that
/•' d- 1 V
a(V,)dB, — / Vsds
Jo 2 Jo
where a is the field of matrices given by

§3. Asymptotic Properties of Planar Brownian Motion 531
2°) If X, = /0' a{Vs)dBs, prove that (X\ X'), = (X1, B*),.
[Hint: Observe that tx(x)x = 0, <x(x)y = у if (x, y) = 0, hence <x2(x) = cr(x).]
3°) Show that
lim t~l{X1, B-'), = Sij(\ — d~l) a.s.
t-*oo
4°) Prove that the 2rf-dimensional process (c~lBc2,, Bc) /0' ' Vsdsj con-
converges in distribution, as с tends to oo, to the process
where E, 5') is a BMM.
§3. Asymptotic Properties of Planar Brownian Motion
In this section, we take up the study of some asymptotic properties of complex BM
which was initiated in Sect. 4 of Chap. X. We will use the asymptotic version of
Knight's theorem (see the preceding section) which gives a sufficient condition for
the DDS Brownian motions of two sequences of local martingales to be asymp-
asymptotically independent. We will also have to envisage below the opposite situation
in which these BM's are asymptotically equal. Thus, we start this section with a
sufficient condition to this effect.
C.1) Theorem. Let (M"), i — 1, 2, be two sequences of continuous local martin-
martingales and fi" their associated DDS Brownian motions. If Rn(t) is a sequence of
processes of time-changes such that the following limits exist in probability
i) Umn(Ml Mn,)Rn{l) = Нт„(М2", M)RM = t,
ii) Нт„(М^ - M\, Mnx - M)RAn = 0,
then, Нт„ sups<, \fi"(s) - /62"(s)| = 0 in probability.
Proof. If T" is the time-change associated with {M", M"),
+ \M2'(Rn(t))-M(T2"(t))
By Exercise D.14) Chap. IV, for fixed t, the left-hand side converges in probability
to zero if each of the terms
(mi Mi)j;;;, (мг-м^л^-м^,,,, (mim^
converges in probability to zero. Since (M[\ M")T»{t) = t, this follows readily
from the hypothesis.
As a result, \fi" — Р2\ —* 0. On the other hand, Kolmogorov's criterion A.8)
entails that the set of laws of the processes /6" — ^ is weakly relatively compact;
thus, PI - ft -^ 0. This implies (Exercise A.11)) that sups<, \^(s) - ^(s)\
converges in distribution, hence in probability, to zero. ?

532 Chapter XIII. Limit Theorems in Distribution
The following results are to be compared with Corollary B.4). We now look
for conditions under which the DDS Brownian motions are asymptotically equal.
C.2) Corollary. If Mh i = 1,2, are continuous local martingales and R is a
process of time-changes such that the following limits exist in probability
i) lim -{Mi, Mi)RM — \ for i = 1, 2,
u-voo И
ii) lim -(M, - M2, M, - M2)R(U) = 0,
u-*co и
then -ул (fii(u-) — f52(u-)) converges in distribution to the zero process as и tends
to infinity.
Proof. By the remarks in Sect. 5 Chap. 0, it is equivalent to show that the con-
convergence holds in probability uniformly on every bounded interval. Moreover, by
Exercise A.17) Chap. V (see the remarks before Corollary B.4)), "tjA(m) is the
DDS Brownian motion of -j^M-,. Thus, we need only apply Theorem C.1) to
The above corollary will be useful later on. The most likely candidates for R
are mixtures of the time-changes /xj associated with {M',M') and actually the
following result shows that д,1 V fij will do.
C.3) Proposition. The following two assertions are equivalent:
(i) lim -(Mi - Mi, M\ — M2)uivu2 = 0 in probability;
I—>oo t
(ii) lim -{M\, M\)a2 — lim -{Mi, M2)ui = 1 in probability,
f->oo t l-t-oo t
and lim -{M\ — M2, M\ - M2>/tiA/u2 = 0 in probability.
Under these conditions, the convergence stated in Corollary C.2) holds.
Proof. From the "Minkowski" inequality of Exercise A.47) in Chap. IV, we con-
conclude that
|(r-1<Af1. АГ!>М?I/2 — 11 <{Г{{М{ -M2,M,-M2>M?)'/2.
By means of this inequality, the proof that i) implies ii) is easily completed.
To prove the converse, notice that
(M, - M2, M, - MifJ^
- M2, Mi - М2>ц,1 - {Mi - M2, M, - М2)Д2|
Л*1>м? - (M2, M2)rf + 2 ((Mi, M2)„I - (Mu

§3. Asymptotic Properties of Planar Brownian Motion 533
Since by Kunita-Watanabe inequality
|(м,, м2)д, - <мьм2)Д/2| < \t - (Mi, м,)д?|1/21/ - (M2, м2)^\у\
the equivalence of ii) and i) follows easily. ?
The foregoing proposition will be used under the following guise.
C.4) Corollary. If(Mu M|)«, = (M2, M2)oo = oo and
lim {M\ — M2, M\ — М2),/(Л/,-, M/), = 0 almost-surely,
r-»oo
for i = 1,2, then the conclusion of Proposition C.3) holds.
Proof. The hypothesis implies that fj.'t is finite and increases to +00 as t goes to
infinity. Moreover
(M, - M2, Mi - M2)J(Mh Mi),,, =-<«,- M2, M\ - M2)M;,
so that condition i) in the Proposition is easily seen to be satisfied. ?
From now on, we consider a complex BM Z such that Zo = zq a.s. and pick
points zi, ¦. ¦, zp in С which differ from zo- For each j, we set
X/ = / ¦= =
Jo Ls — Zj
Z, - Zj
- Zj
+ i0/.
/7 \
where в/ is the continuous determination of arg I ' _ J I which vanishes for
Y ZO Zj J
t — 0. The process Bл)~]в/ is the "winding number" of Z around Zj up to time ?;
we want to further the results of Sect. 4 in Chap. X by studying the simultaneous
asymptotic properties of the в/, j = 1,..., p.
Let us set
C{ = / \ZS-Zj\~2ds
Jo
and denote by 7"/ the time-change process which is the inverse of C'. As was
shown in Sect. 2 Chap. V, for each j, there is a complex BM t,' — ft +iyJ such
that
We observe that up to a time-change, fiJ is < 0 when Z is inside the disk D/ =
D (zj, \zo - Zj\) and > 0 when Z is outside Dj.
We now recall the notation introduced in Chap. VI before Theorem B.7). Let
{5 be a standard linear BM and / its local time at 0. We put
o

534 Chapter XIII. Limit Theorems in Distribution
and call a* the time-changes associated with <Л/±, Л/*). Let /3+ and f}~ be the
positive and negative parts of fi and put pf = f}*±. By the results in Chapter VI,
(S+, <Г) = (м++, M"_) is a planar BM such that
The process S± is the DDS Brownian motion of M±. Moreover, p± are reflecting
BM's, [p±, \laf) have the same law as {\p\, /) and
-/„+ = sup (-8+), -/ - = sup E;).
*¦ s<t *¦ s<l
The processes la± are the local times at 0 of p± (Exercise B.14) Chap. VI).
C.5) Proposition. The processes p+ and p~ are independent. Moreover, there are
measurable functions f and gfrotn WxWroW such that
fi = f{p+,p-)=g(8+,&-).
Proof. The first part follows from the independence of <5+ and 8~. To prove the
second part, we observe that 0, = p+ ((M+, M+),) + p~ ((Af~, M~)t); thus, it is
enough to prove that (M±, Af*) are measurable functions of p±. Calling L* the
local time of p± at zero, we have
/ = L+ ((M+, M+)) = L
Moreover, as (M+, M+), + (M~, M~)t = t, one can guess that
(M+, M+), = \nf\s : L+ > L; s]
which is readily checked. Since p+ and p~ are functions of S+ and S~, the proof
is complete.
Remark. To some extent, this is another proof of the fact that Brownian motion
may be recovered from its excursion process (Proposition B.5) Chap. XII), as
p+ and p~ may be seen as accounting respectively for the positive and negative
excursions.
In the sequel, we are going to use simultaneously the above ± notational pattern
for several BM's which will be distinguished by superscripts; the superscripts will
be added to the ±. For instance, if fiJ is the real part of the process t,' defined
above
pi = g (Sj+, 8>-).
The following remark will be important.
C.6) Lemma. The process 8J+ is the DDS Brownian motion of the local martin-
martingale

§3. Asymptotic Properties of Planar Brownian Motion 535
JO
^ - Zj
The same result holds for &'~ with Dj instead ofDj and, naturally, the correspond-
corresponding local martingale will be called Nj~.
Proof. It is easily seen that N/+ = M'* and since Sj+ is the DDS Brownian
motion of MJ+, by Exercise A.17) in Chap. V, it is also the DDS Brownian
motion of Nj+. a
We now introduce some more notation pertaining to the imaginary part y1 of
?¦/. We call y'+ and y,J~ the DDS Brownian motions of the local martingales
V
' ~ Jo D) *' '
As in the previous proof, it is seen that
= f
and, by the same reasoning, y,J+ is also the DDS Brownian motion of /0' \aJ>Q)dy/,
namely
Y,J+
= I lu>!>0)dr*J-
j о
The same result holds for y' with the obvious changes.
Moreover, it is plain that y> = yJ+((Mj+, Mj+)) + yj-((Mj~, Mj~)) so that,
by Proposition C.5), the knowledge of the four processes (pi+, p'~, y'+, yJ~) is
equivalent to the knowledge of {p1, y').
Our next result will make essential use of the scaling operation. Let us insist
that for h > 0,
In particular, we denote by ?;'(A) the Brownian motion
We must observe that the family (/6y±, Mj±, Sj±, pj±) of processes associated
with the planar BM ?-/(A) by the above scheme is actually equal to
Indeed, it is obvious for fi* and we have

536 Chapter XIII. Limit Theorems in Distribution
As &i+ is the DDS Brownian motion of MJ+, Exercise A.17) in Chap. V tells us
that Si+ih) is the DDS Brownian motion of M/+(A) which entails our claim in the
case of 8±. Finally, the claim is also true for p± since it is a function of S±.
We may now state
C.7) Theorem. The 2p-dimensional process (?1(/l),..., t,p(/l)) converges in dis-
distribution as h tends to infinity to a process (floc, ..., ^p<x>), the law of which is
characterized by the following three properties:
i) each t;J0C is a complex BM;
ii) if we keep the same notational device as above with the obvious changes,
then the processes pi+°° + iy>+ca are all identical;
Hi) if we call p+o° + iy+o° the common value of the processes py+0° + iyj+cc,
then the processes p+x + iy+o°, p'~°° + iyl~°°,..., pp^°° + iyp~°°, are inde-
independent.
Proof. By Corollary A.6), the set of laws under consideration is weakly relatively
compact. Therefore, all we have to prove is that every limit law satisfies i) through
iii) of the statement.
We first observe that property i) is obvious. Next, to prove that the pj+x are
identical, we may as well prove that the 5y+0° are identical. Now, by Corollary
E.8) Chap. 0, the processes 5J+0° are the limits in distribution of the processes
SJ+(h). Furthermore, by Lemma C.6), the processes Sj+(h) are the scaled DDS
Brownian motions of the local martingales Л^+. Thus, it remains to prove that we
can apply Corollary C.4) to the local martingales NJ+. But by Sect. 2 Chap. V
(Ni+,Ni+), = f f4Z,)ds
Jo
with fj(z)= \z — Zj\ 1d'(z) and likewise
(Ni+~Nk+,Ni+-Nk+), = f fik(Zs)ds
Jo
with fJk(z) = jzt^d-(z) - ^Ц(г) . As the functions f'k are integrable
with respect to the Lebesgue measure in the plane, whereas the functions f> are
not, the ergodic theorem of Sect. 3 Chap. X (see Exercise C.15) Chap. X) shows
that the hypotheses of Corollary C.4) are satisfied. As a result, the processes pi+co
are identical. The same pattern of proof applies to the processes yJ+0° without
any changes. This proves ii).
We now turn to the proof of iii). By the same reasoning as in the proof of ii),
it is enough to prove that
converges in distribution to a BM2p+2. By Lemma C.6) and Exercise A.17) in
Chap. V, 51+(A) (resp. 5-'-(Л)) is the DDS Brownian motion of ±N]+ (resp. j;Nj~)
and likewise y'+W (resp. yJ-{h)) is the DDS Brownian motion of i<?1+ (resp.
\в>~). Thus, we need only apply Corollary B.4) to the

§3. Asymptotic Properties of Planar Brownian Motion 537
local martingales
Let M be any of these martingales; then, as in the first part of the proof
(M, M), = f f(Zs)ds
Jo
for a function / which is not integrable with respect to the Lebesgue measure. On
the other hand, if M, N are two local martingales of the above list
{M, N), = I f(Zs)ds
Jo
where, this time, / is integrable. For instance, for N}~ and Nk~, we get
f(z) = (z-Zj,z- zk) \z - zj\~2 \z -zk\~2 \DjnDk(z)
which is integrable since
\f(z)\ S \z-zj
~
Zk\~l
the other cases are either trivial or similar. In any case, it is easily deduced from
the ergodic theorem (see Exercise C.15) Chap. X) that the hypotheses of Corollary
B.4) are satisfied. This completes the proof. ?
The foregoing theorem allows to generalize Theorem D.2) of Chap. X to
several points. As in there, A, will be an additive functional and we will assume
that \\А\\=2л.
C.8) Theorem. As t goes to infinity,
2
log t
converges in distribution to ((W+, W>~), j = I,..., p, A) where, for each j, the
triple (W+, Wi~, A) has the law described in Theorem D.2) of Chap. X, and,
conditionally on A, the p+\ variables (W+, W;~, j = 1,..., p) are independent.
Proof. From Theorem D.2) in Chap. X, we know that for each j, ^Ь(#/ + , #/'
A,) converges to (W}+, Wj~, A); thus what we have to prove is the relationship
between these triples when j varies, that is between WJ+ and WJ~, j = 1,..., p,
given A. In the remark after Theorem D.2) Chap. X, we pointed out that

538 Chapter XIII. Limit Theorems in Distribution
where Tj = inf{f : j6/ = a), converges in probability to zero. With each planar
BM Z = X + iY we associate a bidimensional r.v. W(Z) by setting
= / hx,>O)dYs, / \iX,<o)dYs
\Jo Jo /
mz)
\Jo Jo /
where T\ — inf{/ : X, = 1}. Thanks to the scaling properties of the family Tj, it
is not hard to see that
logf
if h = ^logf. By another application of Corollary E.8) Chap. 0, it fol-
follows that j^7(@/+,0/~), j = 1, •.., p) converges in distribution to
As a result, the r.v.'s WJ+ which depend on pj+o° alone are all equal to the
same variable W+. For the same reason, T{°° does not depend on j. Furthermore,
conditionally on Л, each W;~ is independent of W+, hence of Т^°°, and becomes
a function of pj~°° + iyj~°° alone. The independence follows from Theorem
C.7). d
We now record the asymptotic distribution for the windings 6* themselves.
C.9) Corollary. The limiting distribution ofi^-ft1,, j — 1, 2,..., p\ is the law
of(Wj = W+ + Wj~, j = \,..., p) which may be described as follows:
i) Wj~ = HYj, where
ii) the r.v.'s Yj are independent Cauchy variables with parameter 1 which are
also independent of the pair (W+, H);
Hi) the Laplace-Fourier transform of the pair (W+, H) is given by
E [exp (-aW + j'wW+)] = [cosh и + (a/v) sir
Proof. This is a reformulation of Corollary D.4) in Chap. X with H = Л/2. а
In Theorem D.2) of Chap. X, we saw that the result is independent of the
radius of the disk used to distinguish between "small" and "big" windings. In this
section, we have used, for convenience sake, the disks D, of radius |zo — Zj\, but
it is, likewise, inessential. This is implied by the next result which will also be
used in the proof of the last theorem of this section.
C.10) Proposition. If f is locally bounded and square-integrable with respect to
the 2-dimensional Lebesgue measure, then
converges in probability to zero as t goes to infinity.

§3. Asymptotic Properties of Planar Brownian Motion 539
Proof. Since M is conformal,
<ReM, ReM), = {ImM, ImM), = / \f\2{Zs)ds
Jo
and we know that ¦?- J'Q \f\2(Zs)ds converges in distribution to a finite r.v. It
follows that (ReM, ReM),/(log/J converges in probability to zero and by Exer-
Exercise D.14) in Chap. IV, ReM,/(log/) and lmMt/(\ogt) converge in probability
to zero. ?
Remark. The assumption that / is locally bounded is only made to ensure that
/„' \f\2(Zs)ds is P0-a.s. finite for every t > 0.
The foregoing discussion entails further asymptotic results. We keep the same
setting and notation and we write Res(/, a) for the residue of / at a.
C.11) Theorem. Let f be holomorphic in C\ jzi, ..., Zp\ and Г an open, rela-
relatively compact set such that {zi, ¦ ¦ ¦, ZP\ С Г; then
2 Г id) ^
/ f(Zs)\r(Zs)dZs^Y
log / Jo '"°°y~T
If f is moreover holomorphic at infinity with lim^oo f(z) — 0, then
2 [>
— / f(Zs)dZs
log/ Jo
converges in distribution as t —» oo, to
Yl Res(/- zy) I у + i Wj~ \ + Res(/, oo) I - - 1 + i W+ \.
y=i I 2 J I 2 ]
Proof. By the preceding Proposition, we may as well suppose that Г is the union
of disjoint disks /) = D{zj, ey) with ey > 0 and sufficiently small, so that we look
for the limit of Y.j=i Ft w'th F/ - ^ /„' f(Zs)\rj(Zs)dZs. Within Гу we may
write f(z) = hj(z) + gj [ j^-) with Ay holomorphic in a neighborhood of /} and
gj(z) = Res(/, Zj)z + gj{z)z2
for an entire function gj. We set
„/ 2 Г' _ч. / 1 \ dZs
/ \rj(Zs)gj
Jo \Zs-ZjJ
log/ Jo \ZS- Zj J (Z, - zjJ
By Proposition C.10),
F/ - Res(/, zj) ( ^- [ \r, (Z,)-^- 1 - Я/
[log/ J Zs - Zj J

540 Chapter XIII. Limit Theorems in Distribution
converges to zero in probability. We moreover claim that #/ converges to zero
in probability.
Let Gj be the antiderivative of gj vanishing at 0. By Ito's formula for conformal
martingales,
Since Z, converges to infinity in probability, the left hand side converges to
Gj @) = 0 in probability. As a result
1 \ dZ,
log/Л OJ\Zs-Zj) (Zs-ZjJ
converges to 0 in probability. But the real part of the conformal martingale
ft / 1 \ Й7
/'
Jo
has a bracket equal to /J <p(Zs)ds where
-4
Z - Zj
lr;(z)
is integrable. By the same reasoning as in Proposition C.10), our claim is proved.
The first statement is then an easy consequence of Theorem C.8) and of Propo-
Proposition D.6) of Chap. X.
For the second statement, we write f(z) = —Res(/, 00O + ^g{\/z) f°r
\z\ > rj and g holomorphic in a neighborhood of {|г| < 1//?} where rj has been
chosen sufficiently large. We have to add to the previous limit that of
2 f Г' dZ, f ( 1 \ dZ% \
-Res(/,oo)/ 1(Z<>4)—i+/ \{Z gl \ \.
log/ I Jo Zs Jo \ZJ Zj \
This first part converges in distribution to —Res(/, 00) (-? - 1 + iW+) thanks to
Theorem C.8) and to Proposition D.6) of Chap. X and the second part converges
in probability to zero by the same reasoning as for gj above. ?
* C.12) Exercise. Let n be an integer and let z, be the time-change inverse of
Jo
/0
Prove that, with the notation of this section
2 ,„,
log? v r' ' r</
converges in law to n (Wl~ + W+,..., W~ + W+).
[Hint: Use Theorem C.7).]

Notes and Comments 541
** C.13) Exercise (Mutual windings). Let В',..., Bp be p complex BM's on a
filtered probability space {?2, Ж, .Щ, P) which are correlated as follows: for every
к and l, к Ф I, there exists a matrix Akj such that for every и and v in E2(~ C),
(u,Bf)(v,B't)-(u,AkJv)t, t>0,
is a martingale.
1°) Show that, if for every к ф I, the matrix Akj is not an orthogonal matrix
and if B'o Ф 0 a.s. for every i, then
where the C,'s are independent Cauchy r.v.'s with parameter 1.
[Hint: Show that for i ф j, /J \d(9',0J)S\/logt, /J |d(log|fl''|,0'),|/logr
and /J |flf(log|Z?'|, log|-Ву|>л| /log? are bounded in probability as t -> oo.]
2°) Let В be a BM3 and ?>',..., Dp, p different straight lines which intersect
at zero. Assume that Bo is a.s. not in D' for every ('. Define the winding numbers
в], i < p, of В around D', i < p. Show that as a consequence of the previous
question, the same convergence in law as in 1°) holds.
3°) Let B\ ..., B", be n independent planar BM's such that B'o ф BJ0 a.s.
Call e'J the winding number of B' - BJ around 0. Show that
logr
where the C'';'s are independent Cauchy r.v.'s with parameter 1.
Notes and Comments
Sect. 1. For the basic definitions and results, as well as for those in Sect. 5 Chap. 0,
we refer to the books of Billingsley [1] and Parthasarathy [1]. A more recent
exposition is found in the book of Jacod and Shiryaev [1]. Our proof of Donsker's
theorem (Donsker [1]) as well as Exercise A.14) are borrowed from the former.
This theorem is constantly being used as a tool to obtain properties of Brownian
motion which have first been remarked on its random walk skeletons. This method,
which we have refrained from using in this book, is, for instance, found in Pitman
[1] and Le Gall [5]. It is interesting to note that conversely some original limit
laws on random walks can only be understood in terms of Brownian motion as is
seen in the work of Le Gall [7] completing former work of Jain and Pruitt.
Proposition A.10) is taken from Stroock-Varadhan [1]; although it is of
marginal importance in our development it is fundamental in theirs and, as is
more generally the case with Martingale problems, has been extended to many
situations.
For Exercise A.13), see Nualart [1] and Yor [14]. Exercise A.15) is due to
Pages [1] and Exercise A.16) is inspired by Rebolledo ([1] and 2]).

542 Chapter XIII. Limit Theorems in Distribution
Exercises A.17) and A.18) were suggested respectively by J. Pitman and
L. Dubins. Exercise A.17) allows to simplify the proofs of some limit theorems
found in Getoor-Sharpe [4] (see also Jeulin [2] page 128).
Sect. 2. The main result of this section is due to Papanicolaou et al. [1], but
their proof was different. The asymptotic version of Knight's theorem comes from
Pitman-Yor [5]; another proof is found in Le Gall-Yor [2] and Exercise B.9) is a
variation on the same theme.
Kasahara and Kotani [1] have studied the same problem as Papanicolaou et al.
[1] in the case of BM2. We also refer to Kasahara [4] to whom Exercise B.13)
is due. Biane [3] unifies the asymptotic limit theorems for (multiple) additive
functionals of several Brownian motions in M.d.
A number of extensions of Exercises B.11) through B.15) have been obtained
in recent years by Berman, Borodin [1] and in particular Rosen in the case of
stable Levy processes.
The SDE presentation of the Brownian motion on the sphere found in Exercise
B.16), is a very particular case of that given in Lewis [1] and Van den Berg-Lewis
[1]; more generally, see Rogers-Williams [1] and Elworthy [1] for constructions
of Brownian motions on surfaces.
Sect. 3. This section is entirely taken from Pitman and Yor ([4], [5] and [7]).
Exercise C.13) is taken from Yor [22] who answers a question of Mitchell Berger.
The result in question 2°) of this exercise was originally obtained in a different
manner in Le Gall-Yor [3]. More general asymptotic studies for the windings of
BM3 around curves in R3 are obtained in Le Gall-Yor [4]; the computation of the
characteristic functions of the limit laws led the authors to some extension of the
Ray-Knight theorems for Brownian local times, presented in Le Gall-Yor [5]; see
also the Notes and Comments on Sect. 4 of Chap. XII.
Knight [11] and Yamazaki [1] give convergence results in the sense of fdd's
which are closely related to what is called "log-scaling laws", namely limit theo-
theorems such as
в(ехрХи)/и -^i F(X)
u—»oo
found in Pitman-Yor [7].
Another extension of these results is provided by Watanabe [5] who studies
asymptotics of Abelian differentials along Brownian paths on a Riemann surface.
Supplementing these multidimensional limits in law, there are also deep in-
investigations of the pathwise behavior of multiwindings, such as for example of
their speed of transience originating with Lyons-McKean [1] and continuing with
Gruet [1], Gruet-Mountford [1] and Mountford [1].
We also mention that limit theorems for a large class of diffusions, including
the Jacobi processes (see, e.g., Warren-Yor [1]) are developed in Hu-Shi-Yor [1].
These limit theorems are closely related to the asymptotics of diffusion processes
in random environments (Kawazu-Tanaka [1], Tanaka [4]).
Intensive discussions of recent studies on the geometry of the planar Brownian
curve are found in Le Gall [9] and Duplantier et al. [1].

Appendix
§1. Gronwall's Lemma
Theorem. If(p is a positive locally bounded Borel function on Ж+ such that
(p(t) <a + b <l>(s)ds
Jo
for every t and two constants a and b, then <p(t) < aehl. If in particular a = 0
then ф = 0.
Proof. Plainly,
- a+abt+b2f (t-u№(u)du <a + abt+b2t <$>(u)du.
h Jo
Proceeding inductively one gets in this fashion
</>@ < a + abt + ... + ab"-- +
n\ Jo
n\ n\
Since ф is locally bounded, the last term on the right converges to zero as n tends
to infinity and the result follows.
§2. Distributions
Let U be a fixed open set in W1. We denote by C?? the space of infinitely differ-
entiable functions on U which have a compact support contained in U.
B.1) Definition. A sequence (</>„) in C?° is said to converge to an element ф of
C?° if the supports of the ф„ 's are contained in a fixed compact subset ofU and if
the k-th derivatives of фп — ф converge uniformly to zero for every к > 0.
B.2) Definition. A distribution T on U is a linear form on C?? such that Т{ф„)
converges to 0 whenever (</>„) is a sequence in С™ which converges to zero as n
tends to infinity.

544 Appendix
We will also write (T, </>) for the value taken by the distribution T on the
function ф of Cf?. With every Radon measure jx on U, we associate a distribution
Tf, by setting
Likewise, if / is a locally integrable Borel function we write Tf for Tf, where
/x(dx) = f(x)dx; in other words
(Тг,ф)= / 4>(x)f(,x)dx.
B.3) Definition. If T is a distribution and 9"/Эх ... дха/ a partial derivation
operator, we define the corresponding partial derivative ofT by setting for ф € Cj?
where \a\ — oi\ + .. .aj.
This obviously defines another distribution and in the case of Tf above, if /
is |a| times continuously differentiable, then
L—_ = j j
where
daf
g =
ad ¦
Эх?... Эх,
§3. Convex Functions
We recall that a real-valued function / defined on an open interval / of К is
convex if
f(tx + A - t)y) < tf{x) + A - t)f(y),
for every 0 < t < 1 and x,y e /.It follows from this definition that for fixed x, the
ratio (f(y) — f(x))/(y—x) increases with y. This, in turn, entails immediately that
in each point x the function / has a left-hand derivative f'_{x) and a right-hand
derivative f'+(x) and that, for у > x
У
We moreover have the
C.1) Proposition. The functions f'_ and f'+ are increasing, respectively left and
right-continuous and the set [x : f'_{x) ф /+(*)} « at most countable.

§3. Convex Functions 545
Proof. Since f!_(x) < f'+(x), the first property follows at once from the above
inequality. To prove that /_J. is right-continuous, we interchange increasing limits
to the effect that if an \, 0 and bm \, 0,
.. ,, . , „ i- Л- fix+an+bm)-fix + an)\
lim /|(х+а„) = lim j lim — I
И —* OO tl —> OO \ ffl —> CO l?*m /
fix+bm)- fix) _ ,
m^°° bm
Finally, f'_ and /| have only countably many discontinuities and where f'_ is
continuous, we have f'_ — f'+ thanks to the above inequalities. D
We now study the second derivative of /. If / is C2, then /" is positive, as
is easily seen. More generally, we have the
C.2) Proposition. The second derivative f" of f in the sense of distributions is
a positive Radon measure; conversely, for any Radon measure ц on R, there is a
о
convex function f such that f" = //. and for any interval I and x e /,
1 f
fix) = ~ / I* -a\fi(da) +сцх + р,
1 Ji
1 f
/-00 — - / sgn(x — a)jiida) +a/
2 Л
where a/ and /3/ are constants and sgn x — 1 if x > 0 and —1 if x < 0.
Proof. Let ф е C??; the derivative Df of/ in the sense of distributions is given
by
lim Ф{Х +e)? ^) fix)dx
> = - / fix)<t>\x)dx = -f (li
= -lim / ф(х) ( X ~ f(X) )dx= f <l>{x)fL(x)dx.
By the integration by parts formula for Stieltjes integrals, the second derivative is
the measure associated with the increasing function f_. Of course by the above
results, we could have used f'+ instead of f'_ without altering the result.
Conversely, if / С J the integrals
- / \x-a\ii(da) and - / \x-
2 Ji t Jj
a\nida)
are convex on / and differ by an affine function. As a result one can define a
о
convex function / on the whole line such that on /
fix) = - \x- a\nida) + a,x +

546 Appendix
О
An application of Lebesgue's theorem yields, for x e I,
/1С*) — ~ I sgn(* - a)ix{da) + a,
z Ji
о
and if ф is a test function with support in /, then
I f'_{xL>\x)dx = f ix{da) (- I <t>'(x)sgn(x - a)dx\ = - f ф(а)ц{йа)
which proves that the second derivative of / is //..
The convex function determined by //. is of course unique only up to addition
of an affine function. If the measure //. is such that / \x — a\fi(da) is finite for
every x, which will in particular be the case if //. has compact support, then one
can globally state that
-!/„-.
f(x) = -
The constants a and fi can be fixed by specifying special values for / in two
points. If in particular for a < b we demand that f(a) = f(b) = 0 then one can
give for / a more compact expression which we now describe in a slightly more
general setting.
Let s be a continuous, strictly increasing function on / = [a, b]. We will say
that / is s-convex if ft. i < c\ < x < ci < b,
(s(cz) - s(a)) f(x) < (s(C2) - six)) fid) + (six) - j(c,)) /(c2).
Exactly as above one can define the right and left s-derivatives df±/ds by taking
the appropriate limits of the ratios (f(y)-f(x))/is(y)-six)). At the points where
they are equal we say that / has an s-derivative. The functions thus defined are
increasing and determine as above a measure //..
If for x < у we set
G(x, y) = G(y, x) - (s{x) - s(a))(s(b) - s(y))/(s(b) -
then if / is s-convex and if f(a) = f(b) — 0,
= - /
Ja
/(*) = - / G(x,y)/i(dy).
Ja
Indeed, using the integration by parts formula for Stieltjes integrals,
b
/b
G(x, y)ii{dy) = , ; ' I (s(y) - s(a))ix(dy)
s(b) -s(a) Jh,x]
s(x) -s(a)
s(b)-s(a)

§4. Hausdorff Measures and Dimension 547
s(b)-s(x)
s(b)-s(a)
, s(x)~s(a)
s(b) - s(a) L ds" Л ds
= -fix).
Naturally, all we have said is valid for concave functions with the obvious changes.
§4. Hausdorff Measures and Dimension
Let h be a strictly increasing continuous function on E+ such that /z@) = 0
and A(oo) = oo. Let В be a Borel subset of a metric space E. The Hausdorff
h-measure of В is the number
Л\В) = \iminf $
where the infimum is over all coverings [J /,, of В where /„ is a closed set in E
with diameter |/„| < s. Of special interest is the case where h(t) = ta, a > 0,
in which case we will write Aa and speak of a-measure. If E — WLd, Ad is the
ordinary Lebesgue measure.
D.1) Lemma. Ifh(t) — v{t)k{t) with \im,^Qv{t) = 0, then mh{F) > 0 implies
mk{F) — oo.
Proof. Pick e > 0; there is an /j > 0 such that v < ц implies v(v) < e. Let (J /„
be a covering of F with \In\ < v < ц. Then
it follows that
hence
mk(F) > -mh(F),
E
and since ? is arbitrary, the proof is complete.
A consequence of this lemma is that there is a number a0 such that Aa(F) =
+oo if a < ao and Aa(F) = 0 if a > щ (the number Aa°(F) itself may be zero,
non zero and finite or infinite). The number ao is called the Hausdorff dimension
of F. For instance, one can prove that the dimension of the Cantor "middle third"
set is log 2/log 3.

548 Appendix
§5. Ergodic Theory
Let (E, %,m) be a ст-finite measure space. A positive contraction T of Ll(m)
is a linear operator on Ll(m) with norm < 1 and mapping positive (classes of)
functions into positive (classes of) functions. A basic example of such a contraction
is the map / —> / о в where в is a measurable transformation of (E, /6) which
leaves m invariant.
E.1) Theorem. (HopPs decomposition theorem). There is an m-essentially
unique partition С U D of E such that for any f e Ll+(m)
i) 5X0 Tkf = 0 or +oo on C,
H) YT=QTkf < oo on D.
If D — 0, the contraction T is said to be conservative. In that case, the
sums YlT=o Tk f f°r / ? Ll+(m) take on only the values 0 and +oo. The sets
{So° Tk f — oo} where / runs through Ll+(m) form a a -algebra denoted by
К and called the invariant ст-algebra. If all these sets are either 0 or ? (up to
equivalence) or in other words if Y, is /и-а.е. trivial then T is called ergodic.
We now state the basic Chacon-Ornstein theorem.
E.2) Theorem. If T is conservative and g is an element of Ll+{m) such that
m(g) > 0, then for every f e L{(m),
m-a.e.
A™ (E
The conditional expectations on the right are taken with respect to m. If m is
unbounded this means that the quotient is equal to E[(f/h) \ K]/E[(g/h) | Y]
where h is a strictly positive element in Ll{m) and the conditional expectations
are taken with respect to the bounded measure h ¦ m; it can be shown that the
result does not depend on h. If T is ergodic the quotient on the right is simply
m(f)/m{g).
The reader is referred to Revuz [3] for the proof of these results.
§6. Probabilities on Function Spaces
Let ? be a Polish space and set ?2 — C(R+, E). Let us call X the canonical
process and set .V, =a (Xs, s < t) and ,%o = a (Xs,s > 0).
F.1) Theorem. If for every t > 0, there exists a probability measure P' on .Ytsuch
that for every s < t, P' coincides with Ps on .3^, then there exists a probability
measure P on .W^ which for every t coincides with P' on.V{.
For the proof of this result the reader can refer to the book of Stroock and
Varadhan [1] p. 34; see also Azema-Jeulin [1].

§7. Bessel Functions 549
§7. Bessel Functions
The modified Bessel function /„ is defined for v > — 1 and x > 0, by
oo
2*у + к + 1).
Observe that for v = -1 and к — 0 the term F(v + к + 1) is infinite, and
therefore the first term in the above series vanishes. By using the relationship
r(z + 1) = zF(z), one thus sees that /| = 1_\. For some details about these
functions we refer the reader to Lebedev [1], pages 108-111.
This family of functions occurs in many computations of probability laws. Call
for instance d$.v) the density of a random variable with conditional law
where к is random with a Poisson law of parameter x > 0 and v > — 1. Then
/t=0
Replacing x and v by x/2t and y/2t we find that, for v > — 1,
qjv\x, у) = A/2/)ехр(-(дг + y)/2t)(y/x)v/2Iv
where t > 0, x > 0, у > 0, is also a probability density, in fact the density of
BESQ(i;) as found in section 1, Chapter IX. At that point we needed to know the
Laplace transform of this density which is easily found from the above. Indeed, the
Laplace transform of y* is equal to (A + 1)"* and therefore the Laplace transform
of d^v) is equal to
= (X + l)-(u+1>exp(-Ax/(A + 1)).
k=0
From this, using the same change of variables as before, one gets that the Laplace
transform of qjv\x, ¦) is equal to
Bkt + 1 Г("+1) exp(-A*/BA/ + 1)).
Another formula involving Bessel functions and which was of interest in Sect.3,
Chap. VIII, is the following. If x e Rd we call ?(*) the angle of Ox with a fixed
axis, and if /id is the uniform probability measure on the unit sphere Sd~l, then
Js*-<
Js*-<
where v = (d/2) — 1. This can be proved directly from the definition of Iv by
writing the exponential as a series and computing /s,,_, cosi;(x)ppd(dx); to this
end, it is helpful to use the duplication formula
ГBг) = Bл-Г1/222*-A/2)Г(г)Г(г + A/2)).

550 Appendix
§8. Sturm-Liouville Equation
Let д be a positive Radon measure on R. Then there exists a unique, positive,
decreasing function фц such that
(*) Фц@)=1, Ф^=ФЦ11,
where ф"^ is the second derivative in the sense of distributions (Appendix 2).
Observe from (*) that since ф^ is positive, it is convex, and ф" is equal to йф'
where ф'^ is the right derivative of ф^ (Appendix 3).
To prove this existence and uniqueness result we transform (*) into the Riccati
equation
(+) g(x) = 1 + iiQa, x]) - f g\y)dy,
Ja
where agl. We claim that this equation has a unique solution g on [a, oo[ which
satisfies the inequality g(x) > 1/A + x — a), for x > a.
Indeed, since the function x -> x2 is locally Lipschitz, there is a unique
maximal solution to (+) on an interval [a, a[ with a > a. Obviously g is of finite
variation on every bounded interval. We will first prove that g is > 0 on [a, a[.
Indeed, suppose that g(x—) < 0 for some x e [a, a[ and set у = infjx e [a, or[:
g(x-) < 0). For a < x < y, we have
(I) g(x) = g(x-) + ii({x}) > 0.
On the other hand, by Proposition D.6) Chap. 0, we may write
\/g(x) = I- [ (g(y)g(y-))-1 dg(y)
Jja.x]
= 1 + (x - a) - I (g(y)g(y-))-1 dn(y) <\+{x-a).
J]a.x\
As a result, g(y) > 1/A + у - a) + ii({y)) > 0, and since g is right-continuous
this contradicts the definition of y. That g is > 0 then follows from (|).
Now, since g is > 0 on [a, or[, if a is finite, rewriting (+) as
/
Ja
g2(y)dy = 1 +ii(]a,x]),
we see that g is bounded on ]a. a[ and by letting x increase to a we get
g(or-) + / g2(y)dy = 1 + /x(]a, or[).
Ja
If we set g(a) = g(a—) + /u({a}) and solve the equation (+) for x > a, we see
that a cannot be maximal. As a result a = oo and we have proved our claim.
Next, if g is the solution to (+), we set, for x > a,
= ехрП g(y)dy\.

§8. Sturm-Liouville Equation 551
One sees rapidly that xj/(x) > ] +x —a on [a. oo[ and that xj/" = xj/ji. We further
set
/D
The function ф is > 0 and is another solution to the equation ф" — фц.. Moreover,
because xj/' is increasing, we have
ф'(х) = xj/'(x) I xj/(yJdy -
С00
Л
(\/xj/(x))
which shows that ф is decreasing.
The space of solutions to the equation ф" — фц is the space of functions
uxj/ + уф with м, v e R. Since i/r increases to +oo at infinity, the only positive
bounded solutions are of the form ьф with v > 0. If for a < 0 we put ф)х = ф/ф@)
we get the unique solution to (*) that we were looking for.

Bibliography
Airault, H. and Follmer, H.
[1] Relative densities of semimartingales. Invent. Math. 27 A974) 299-327.
Albeverio, S., Fenstad, J.E., Hoegh-Krohn, R., and Lindstrom, T.
[1] Non standard methods in stochastic analysis and mathematical physics. Academic
Press, New York 1986.
Aldous, D.
[1] The continuous random tree II: an overview. In: M.T. Barlow and N.H. Bingham
(eds.) Stochastic analysis. Camb. Univ. Press 1991.
Alili, L.
[1] Fonctionnelles exponentielles et certaines valeurs principales des temps locaux brown-
iens. These de doctorat de 1'universite de Paris VI. 1995
Alili, L, Dufresne, D., and Yor, M.
[1] Sur l'identite de Bougerol pour les fonctionnelles exponentielles du mouvement
brownien avec drift. In: Biblioteca de la Revista Matematica Iberoamericana, 1997,
pp. 3-14.
Aronszajn, N.
[1] Theory of reproducing kernels. Trans. Amer. Math. Soc. 68 A950) 337^04.
Atsuji, A.
[1] Some inequalities for some increasing additive functionals of planar Brownian motion
and an application to Nevanlinna theory. J.F. Sci. Univ. Tokyo Sect. I-A, 37 A990)
171-187.
[2] Nevanlinna theory via stochastic calculus. J. Funct. Anal. 132, 2 A995) 437-510.
[3] On the growth of meromorphic functions on the unit disc and conformal martingales.
J. Math. Sci. Univ. Tokyo 3 A996) 45-56.
Attal, S., Burdzy, K., Emery, M., and Hu, Y.
[1] Sur quelques nitrations et transformations browniennes. Sem. Prob. XXIX. Lecture
Notes in Mathematics, vol. 1613. Springer, Berlin Heidelberg New York 1995, 56-69.
Auerhan, J., and Lepingle, D.
[I] Les filtrations de certaines martingales du mouvement brownien dans R" (II). Sem.
Prob. XV. Lecture Notes in Mathematics, vol. 850. Springer, Berlin Heidelberg New
York 1981, pp. 643-668.
Azema, J.
[1] Quelques applications de la theorie generale des processus I. Invent. Math. 18 A972)
293-336.
[2] Representation multiplicative d'une surmartingale bornee. Z.W. 45 A978) 191-212.
[3] Sur les fermes aleatoires. Sem. Prob. XIV. Lecture Notes in Mathematics, vol. 1123.
Springer, Berlin Heidelberg New York 1985, pp. 397-495.

554 Bibliography
Azema, J., Duflo, M., and Revuz, D.
[1] Mesure invariante sur les classes recurrentes des processus de Markov. Z.W. 8 A967)
157-181.
Azema, J., Gundy, R.F., and Yor, M.
[1] Sur l'integrabilite uniforme des martingales continues. Sem. Prob. XIV. Lecture Notes
in Mathematics, vol. 784. Springer, Berlin Heidelberg New York 1980, pp. 53-61.
Azema, J., and Jeulin, T.
[1] Precisions sur la mesure de Follmer. Ann. I.H.P. 22 C) A976) 257-283.
Azema, J., and Rainer, С
[1] Sur Г equation de structure d[X, X], = dt - X+.dX,. Sem. Prob. XXVIII, Lecture
Notes in Mathematics, vol. 1583. Springer, Berlin Heidelberg New York 1994, 236-
255.
Azema, J., and Yor, M.
[1] En guise d'introduction. Asterisque 52-53, Temps Locaux A978) 3-16
[2] Une solution simple au probleme de Skorokhod. Sem. Prob. XIII. Lecture Notes in
Mathematics, vol. 721. Springer, Berlin Heidelberg New York 1979, pp. 90-115 and
625-633
[3] Etude d'une martingale remarquable. Sem. Prob. XXIII. Lecture Notes in Mathemat-
Mathematics, vol. 1372. Springer, Berlin Heidelberg New York 1989, pp. 88-130.
[4] Sur les zeros des martingales continues. Sem. Prob. XXVI. Lecture Notes in Mathe-
Mathematics, vol. 1526. Springer, Berlin Heidelberg New York 1992, pp. 248-306.
Azencott, R.
[1] Grandes deviations et applications. Ecole d'Ete de Probabilites de Saint-Flour VIII.
Lecture Notes in Mathematics, vol. 774. Springer, Berlin Heidelberg New York 1980,
pp. 1-176.
Barlow, M.T.
[1] L(B,,t) is not a semi-martingale. Sem. Prob. XVI. Lecture Notes in Mathematics,
vol. 920. Springer, Berlin Heidelberg New York 1982, pp. 209-211.
[2] One-dimensional stochastic differential equation with no strong solution. J. London
Math. Soc. 26 A982) 335-345.
[3] Continuity of local times for Levy processes. Z.W. 69 A985) 23-35.
[4] Skew Brownian motion and a one dimensional stochastic differential equation.
Stochastics25A988) 1-2.
[5] Necessary and sufficient conditions for the continuity of local times of Levy processes.
Ann. Prob. 16 A988) 1389-1427.
Barlow, M.T., Emery, M., Knight, F.B., Song, S., Yor, M.
[1] Autour d'un theoreme de Tsirel'son sur des filtrations browniennes et non-brown-
iennes. Sem. Probab. XXXII, Lect. Notes in Mathematics, vol. 1686. Springer, Berlin
Heidelberg New York 1998, pp. 264-305.
Barlow, M.T., Jacka, S.D., and Yor, M.
[1] Inequalities for a pair of processes stopped at a random time. Proc. London Math.
Soc. 52 A986) 142-172.
Barlow, M.T., and Perkins, E.
[1] One-dimensional stochastic differential equations involving a singular increasing pro-
process. Stochastics 12 A984) 229-249.
[2] Strong existence and non-uniqueness in an equation involving local time. Sem. Prob.
XVII. Lecture Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York
1986, pp. 32-66.

Bibliography 555
Barlow, M.T., Pitman, J.W., and Yor, M.
[1] Une extension multidimensionnelle de la loi de l'arc sinus. Sem. Prob. XXIII. Lecture
Notes in Mathematics, vol. 1372. Springer, Berlin Heidelberg New York 1989, pp
294-314.
[2] On Walsh's Brownian motions. Sem. Prob. XXIII. Lecture Notes in Mathematics, vol.
1372. Springer, Berlin Heidelberg New York 1989, pp. 275-293.
Barlow, M.T., and Yor, M.
[1] (Semi-)martingale inequalities and local times. Z.W. 55 A981) 237-254.
[2] Semi-martingale inequalities via the Garsia-Rodemich-Rumsey lemma and applica-
applications to local times. J. Funct. Anal. 49 198-229.
Bass, R.F.
[1] Joint continuity and representation of additive functionals of (/-dimensional Brownian
motion. Stoch. Proc. Appl. 17 A984) 211-228.
[2] Lp-inequalities for functionals of Brownian motion. Sem. Prob. XXI. Lecture Notes
in Mathematics, vol. 1247. Springer, Berlin Heidelberg New York 1987, pp. 206-217.
[3] Probabilistic techniques in Analysis. Prob. and its App. Springer, Berlin Heidelberg
New York 1995.
[4] Diffusions and Elliptic Operators. Springer, Berlin Heidelberg New York 1997.
Bass, R.F., and Burdzy, K.
[1] Stochastic Bifurcation Models. Preprint 1998
Bass, R.F., and Griffin, P.S.
[1] The most visited site of Brownian motion and simple random walk. Z.W. 70 A985)
417^C6.
Benes, V.
[1] Non existence of strong non-anticipating solutions to SDE's; Implications for func-
functional DE's, filtering and control. Stoch. Proc. Appl. 5 A977) 243-263.
[2] Realizing a weak solution on a Probability space. Stoch. Proc. Appl. 7 A978) 205-225.
Bertoin, J.
[1] Sur une integrate pour les processus a or-variation bornee. Ann. Prob. 17 A989) 1521
1535.
[2] Applications de la theorie spectrale des cordes vibrantes aux fonctionnelles additives
principales d'un brownien reflechi. Ann. I.H.P. 25, 3 A989) 307-323.
[3] Complements on the Hilbert transform and the fractional derivatives of Brownian
local times. J. Math. Kyoto Univ. 30 D) A990) 651-670.
[4] On the Hilbert transform of the local times of a Levy process. Bull. Sci. Math. 119
B) A995) 147-156.
[5] An extension of Pitman's theorem for spectrally positive Levy processes. Ann. Prob.
20C) A993) 1464-1483.
[6] Excursions of a BES0№ and its drift term @ < d < 1). Prob. Th. Rel. F. 84 A990)
231-250.
[7] Levy processes. Cambridge Univ. Press 1996.
[8] Subordinators: examples and applications. XXVIIе Ecole d'Ete de St. Flour: Summer
1997. Lecture Notes in Mathematics. Springer, Berlin Heidelberg New York 1998.
[9] Cauchy's principal value of local times of Levy processes with no negative jumps via
continuous branching processes. Elec. J. of Prob. 2 1997, Paper 6.
Bertoin, J., and Le Jan, Y.
[1] Representation of measures by balayage from a regular recurrent point. Ann. Prob.
20A992M38-548.

556 Bibliography
Bertoin, J., and Pitman, J.
[1] Path transformations connecting Brownian bridge, excursion and meander. Bull. Sci.
Math. 118A994) 147-166.
Bertoin, J., and Werner, W.
[1] Comportement asymptotique du nombre de tours effectues par la trajectoire browni-
enne plane. Sem. Prob. XXVIII Lecture Notes in Mathematics, vol. 1583. Springer,
Berlin Heidelberg New York 1994, pp. 164-171.
[2] Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck
process. Sem. Prob. XXVIII Lecture Notes in Mathematics, vol. 1583. Springer, Berlin
Heidelberg New York 1994, pp. 138-152.
[3] Stable windings. Ann. Prob. 24, 3, July 1996, pp. 1269-1279.
Bernard, A., and Maisonneuve, B.
[1] Decomposition atomique de martingales delaclasse Hl. Sem. Prob. XI. Lecture Notes
in Mathematics, vol. 581. Springer, Berlin Heidelberg New York 1977, pp. 303-323.
Besicovitch, A.S., and Taylor, S.J.
[1] On the complementary intervals of a linear closed set of zero Lebesgue measure.
J. London Math. Soc. A954) 449-459.
Biane, P.
[ 1 ] Relations entre pont brownien et excursion normalisee du mouvement brownien. Ann.
I.H.P. 22, 1 A986) 1-7.
[2] Sur un calcul de F. Knight. Sem. Prob. XXII. Lecture Notes in Mathematics, vol.
1321. Springer, Berlin Heidelberg New York 1988, pp. 190-196.
[3] Comportement asymptotique de certaines fonctionnelles additives de plusieurs mou-
vements browniens. Sem. Prob. XXIII. Lecture Notes in Mathematics, vol. 1372.
Springer, Berlin Heidelberg New York 1989, pp. 198-233.
[4] Decomposition of Brownian trajectories and some applications. In: A. Badrikian, P.A.
Meyer, J.A. Yan (eds.) Prob. and Statistics; rencontres franco-chinoises en Probabilites
et Statistiques. Proceedings of WuHan meeting, pp. 51-76. World Scientific, 1993.
Biane, P., Le Gall, J.F., and Yor, M.
[1] Un processus qui ressemble au pont brownien. Sem. Prob. XXI. Lecture Notes in
Mathematics, vol. 1247. Springer, Berlin Heidelberg New York 1987, pp. 270-275.
Biane, P., Pitman, J., and Yor, M.
[1] Representations of the Jacobi theta and Riemann zeta functions in terms of Brownian
excursions. In preparation A998).
Biane, P., and Yor, M.
[1] Valeurs principales associees aux temps locaux browniens. Bull. Sci. Math. Ill A987)
23-101.
[2] Variations sur une formule de P. Levy. Ann. I.H.P. 23 A987) 359-377.
[3] Quelques precisions sur le meandre brownien. Bull. Sci. Math. 112 A988) 101-109.
[4] Sur la loi des temps locaux browniens pris en un temps exponentiel. Sem. Prob. XXII,
Lecture Notes in Mathematics, vol. 1321. Springer, Berlin Heidelberg New York 1988,
pp. 454-466.
Billingsley, P.
[1] Convergence of probability measures. Wiley and Sons, New York 1979.
Bingham, N.H.
[I] Limit theorem for occupation times of Markov processes. Z.W. 17 A971) 1-22.
[2] The strong arc sine law in higher dimensions. In: Bergelson, March, Rosenblatt (eds.)
Convergence in Ergodic Theory and Probability, de Gruyter 1996, pp. 111-116.

Bibliography 557
Bismut, J.M.
[1] Martingales, the Malliavin calculus and Hypoellipticity under general Hormander's
conditions. Z.W. 56 A981) 469-506.
[2] On the set of zeros of certain semi-martingales. Proc. London Math. Soc. C) 49 A984)
73-86.
[3] Last exit decomposition and regularity at the boundary of transition probabilities. Z.W.
69 A985) 65-98.
[4] The Atiyah-Singer theorems. J. Funct. Anal. 57 A984) 56-99 and 329 348.
[5] Formules de localisation et formules de Paul Levy. Asterisque 157-158, Colloque
Paul Levy sur les processus stochastiques A988) 37—58.
Blumenthal, R.M.
[1] Excursions of Markov processes. Probability and its applications. Birkhauser 1992.
Blumenthal, R.M., and Getoor, R.K.
[1] Markov processes and potential theory. Academic Press, New York 1968.
Borodin, A.N.
[1] On the character of convergence to Brownian local time I and II. Z.W. 72 A986)
231-250 and 251-277.
Borodin, A.N., and Salminen, P.
[1] Handbook of Brownian motion - Facts and formulae. Birkhauser 1996.
Bougerol, P.
[1] Exemples de theoremes locaux sur les groupes resolubles. Ann. 1HP 29 D) A983)
369-391.
Bouleau, N.
[1] Sur la variation quadratique de certaines mesures vectorielles. Zeit. fur Wahr. 61
A982J61-270.
Bouleau, N., and Yor, M.
[1] Sur la variation quadratique des temps locaux de certaines semimartingales. C.R.
Acad. Sci. Paris, Serie I 292 A981) 491^94.
Boylan, E.S.
[1] Local times of a class of Markov processes. 111. J. Math. 8 A964) 19-39.
Brassesco, S.
[1] A note on planar Brownian motion. Ann. Prob. 20 C) A992) 1498-1503.
Breiman, L.
[1] Probability. Addison-Wesley Publ. Co., Reading, Mass. 1968.
Brosamler, G.
[1] Quadratic variation of potentials and harmonic functions. Trans. Amer. Math. Soc.
149A970J43-257.
Brossard, 1., and Chevalier, L.
[1] Classe L logL et temps local. C.R. Acad. Sci. Paris, Ser. A Math. 305 A987) 135-137.
Brown, T.C., and Nair, M.G.
[1] A simple proof of the multivariate random time change theorem for point processes.
J. Appl. Prob. 25 A988) 210-214.
Burdzy, K.
[1] Brownian paths and cones. Ann. Prob. 13 A985) 1006-1010.
[2] Cut points and Brownian paths. Ann. Prob. 17 A989) 1012-1036.

558 Bibliography
[3] Geometric properties of two-dimensional Brownian paths. Prob. Th. Rel. F. 81 A989)
485-505.
Burkholder, D.L.
[1] Martingale transforms. Ann. Math. Stat. 37 A966) 1494-1504.
[2] Distribution function inequalities for martingales. Ann. Prob. 1 A973) 19-42.
[3] Exit times of Brownian motion, harmonic majorization and Hardy spaces. Adv. Math.
26A977) 182-205.
Burkholder, D.L., and Gundy, R.F.
[1] Extrapolation and interpolation of quasi-linear operators on martingales. Acta. Math.
124 A970) 249-304.
Calais, J.Y., and Genin, M.
[1] Sur les martingales locales continues indexees par ]0, oo[. Sem. Proba. XVII, Lecture
Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1988, pp.
454-466.
Cameron, R.H., and Martin, W.T.
[ 1 ] Transformation of Wiener integrals under translations. Ann. Math. 45 A944) 386-396.
[2] Evaluations of various Wiener integrals by use of certain Sturm-Liouville differential
equations. Bull. Amer. Math. Soc. 51 A945) 73-90.
[3] Transformation of Wiener integrals under a general class of linear transformations.
Trans. Amer. Math. Soc. 18 A945) 184-219.
Carlen, E.
[1] The pathwise description of quantum scattering in stochastic mechanics. In: S. Albev-
erio et al. (eds.) Stochastic processes in quantum physics, Lecture Notes in Physics,
vol. 262. Springer, Berlin Heidelberg New York 1986, pp. 139-147.
Carlen, E., and Elworthy, D.
[1] Stochastic and quantum mechanical scattering on hyperbolic spaces. In: D. Elworthy
and N. Ikeda (eds.) Asymptotic problems in probability theory: stochastic models and
diffusions on fractals. Pitman Research Notes A993) vol. 283.
Carlen, E., and Kree, P.
[1] Sharp /.''-estimates on multiple stochastic integrals. Ann. Prob. 19 A) A991) 354-368.
Carmona, P., Petit, F., and Yor, M.
[1] Some extensions of the Arcsine law as partial consequences of the scaling property
of Brownian motion. Prob. Th. Rel. Fields 100 A994) 1-29.
[2] Beta variables as times spent in [0, oo) by certain perturbed Brownian motions. J. Lon-
London Math. Soc. A998).
[3] Further relations between perturbed reflecting Brownian motions and Bessel processes.
Preprint A998)
Carne, Т.К.
[1] The algebra of bounded holomorphic functions. J. Funct. Anal. 45 A982) 95-108.
[2] Brownian motion and Nevanlinna Theory. Proc. London Math. Soc. C) 52 A986)
349-368.
[3] Brownian motion and stereographic projection. Ann. I.H.P. 21 A985) 187-196.
Centsov, N.V.
[1] Limit theorems for some classes of random functions. Selected Translations in Math-
Mathematics. Statistics and Probability 9 A971) 37^42.

Bibliography 559
Chacon, R., and Walsh, J.B.
[1] One-dimensional potential embedding. Sem. Prob. X. Lecture Notes in Mathematics,
vol. 511. Springer, Berlin Heidelberg New York 1976, pp. 19-23.
Chaleyat-Maurel, M., and Yor, M.
[1] Les filrrations de X et X+, lorsque X est une semimartingale continue. Asterisque
52-53 A978) 193-196.
Chan, J., Dean, D.S., Jansons, K.M., and Rogers, L.C.G.
[1] On polymer conformations in elongational flows. Comm. Math. Phys. 160 B) A994)
239-257.
Chen, L.H.Y.
[1] Poincare-type inequalities via stochastic integrals. Z.W. 69 A985) 251-277.
Chevalier, L.
[1] Un nouveau type d'inegalites pour les martingales discretes. Z.W. 49 A979) 249-256.
Chitashvili, R., and Mania, M.
[1] On functions transforming a Wiener process into a semimartingale. Prob. Th. Rel.
Fields 109 A997) 57-76.
Chitashvili, R.J., and Toronjadze, T.A.
[1] On one dimensional stochastic differential equations with unit diffusion coefficient;
structure of solutions. Stochastics 4 A981) 281-315.
Chou, S.
[1] Sur certaines generalisations de l'inegalite de Fefferman. Sem. Prob. XVIII, Lecture
Notes in Mathematics 1059. Springer, Berlin Heidelberg New York 1984, pp. 219-
222.
Chover, J.
[1] On Strassen's version of the log log law. Z.W. 8 A967) 83-90.
Chung, K.L.
[1] Excursions in Brownian motion. Ark. for Math. 14 A976) 155-177.
[2] Lectures from Markov processes to Brownian motion. Springer, Berlin Heidelberg
New York 1982.
[3] Green, Brown and Probability. World Scientific, Singapore A995).
Chung, K.L., and Durrett, R.
[1] Downcrossings and local time. Z.W. 35 A976) 147-149.
Chung, K.L., and Williams, R.T.
[1] Introduction to stochastic integration. Second edition. Birkhauser, Boston 1989.
Chung, K.L., and Zhao, Z.
[1] From Schrodinger's equation to Brownian motion. Grundlehren 312. Springer, Berlin
Heidelberg New York 1995.
Cinlar, E., Jacod, J., Protter, P., and Sharpe, M.J.
[1] Semi-martingales and Markov processes. Z.W. 54 A980) 161-219.
Clark, J.M.C.
[1] The representation of functionals of Brownian motion by stochastic integrals. Ann.
Math. Stat. 41 A970) 1282-1295; 42 A971) 1778.

560 Bibliography
Cocozza, С, and Yor, M.
[1] Demonstration d'un theoreme de Knight a l'aide de martingales exponentielles. Sem.
Prob. XIV. Lecture Notes in Mathematics, vol. 784. Springer, Berlin Heidelberg New
York 1980, pp. 496^199.
Csaki, E.
[1] On some distributions concerning maximum and minimum of a Wiener process. In
B. Gyires, editor, Analytic Function Methods in Probability Theory, number 21 in
Colloquia Mathematica Societatis Janos Bolyai, p. 43-52. North-Holland, 1980 A977,
Debrecen, Hungary).
Dambis, K.E.
[1] On the decomposition of continuous martingales. Theor. Prob. Appl. 10 A965) 401-
410.
Darling, D.A., and Kac, M.
[1] Occupation times for Markov processes. Trans. Amer. Math. Soc. 84 A957) 444-458.
Davis, B.
[1] Picard's theorem and Brownian motion. Trans. Amer. Math. Soc. 213 A975) 353-362.
[2] On Kolmogorov's inequality \\f\\p < cp||/||i, 0 < p < 1. Trans. Amer. Math. Soc.
222 A976) 179-192.
[3] Brownian motion and analytic functions. Ann. Prob. 7 A979) 913-932.
[4] On Brownian slow points. Z.W. 64 A983) 359-367.
[5] On the Barlow-Yor inequalities for local time. Sem. Prob. XXI. Lecture Notes in
Mathematics, vol. 1247. Springer, Berlin Heidelberg New York 1987, pp. 218-220.
[6] Weak limits of perturbed random walks and the equation Y, = B, + a sups</ Ys +
P infi?, Ys. Ann. Prob. 24 A996) 2007-2017.
[7] Perturbed Brownian motion. Preprint A998)
Davis, M.H.A., and Varaiya, P.
[1] The multiplicity of an increasing family of ст-fields. Ann. Prob. 2 A974) 958-963.
Dean, D.S., and Jansons, K.M.
[1] A note on the integral of the Brownian Bridge. Proc. R. Soc. London A 437 A992)
792-730.
De Blassie, R.D.
[1] Exit times from cones in K" of Brownian motion. Prob. Th. Rel. F. 74 A989) 1-29.
[2] Remark on: Exit times from cones in K" of Brownian motion. Prob. Th. Rel. F. 79
A989)95-97.
Deheuvels, P.
[1] Invariance of Wiener processes and Brownian Bridges by integral transforms and
applications. Stoch. Proc. Appl. 13, 3, A982) 311-318.
Delbaen, F., and Schachermayer, W.
[1] Arbitrage possibilities in Bessel processes and their relations to local martingales.
Prob. Th. Rel. F. 102 C) A995) 357-366.
Dellacherie, С
[1] Integrates stochastiques par rapport aux processus de Wiener et de Poisson. Sem.
Prob. VIII. Lecture Notes in Mathematics, vol. 381. Springer, Berlin Heidelberg New
York 1974, pp. 25-26.
Dellacherie, C, Maisonneuve, В., and Meyer, P.A.
[1] Probabilites et Potentiel. Hermann, Paris, vol. V. Processus de Markov (fin). Comple-
Complements de Calcul stochastique. 1992.

Bibliography 561
Dellacherie, С, and Meyer, P.A.
[1] Probabilites et potentiel. Hermann, Paris, vol. I 1976, vol. II 1980, vol. Ill 1983, vol.
IV 1987.
Dellacherie, C, Meyer, P.A., and Yor, M.
[1] Sur certaines proprietes des espaces H1 et BMO. Sem. Prob. XII. Lecture Notes in
Mathematics, vol. 649. Springer, Berlin Heidelberg New York 1978, pp. 98-113.
De Meyer, B.
[1] Une simplification de l'argument de Tsirel'son sur le caractere non-brownien des
processes de Walsh. Sem. Prob. XXXIII. Lecture Notes in Mathematics. Springer,
Berlin Heidelberg New York A999).
Denisov, I.V.
[1] A random walk and a Wiener process near a maximum. Theor. Prob. Appl. 28 A984)
821-824.
Deuschel, J.D., and Stroock, D.W.
[1] An introduction to the theory of large deviations. Academic Press, New York 1988.
Doleans-Dade, С
[1] Quelques applications de la formule du changement de variables pour les semi-
martingales. Z.W. 16 A970) 181-194.
[2] On the existence and unicity of solutions of stochastic integral equations. Z.W. 36
A976)93-101.
Doleans-Dade, C, and Meyer, P.A.
[1] Integrates stochastiques par rapport aux martingales locales. Sem. Prob. IV. Lecture
Notes in Mathematics, vol. 124. Springer, Berlin Heidelberg New York 1970, pp.
77-107.
Donati-Martin, C.
[1] Transformation de Fourier et temps d'occupation browniens. C.R. Acad. Sci. Paris,
Serie 1 308A989M15-517.
Donati-Martin, C, and Yor, M.
[1] Mouvement brownien et inegalite de Hardy dans L2. Sem. Prob. XXIII. Lecture Notes
in Mathematics, vol. 1372. Springer, Berlin Heidelberg New York 1989, pp. 315-323.
[2] Fubini's theorem for double Wiener integrals and the variance of the Brownian path.
Ann. I.H.P. 27A991) 181-200.
[3] Some Brownian Functionals and their laws. Ann. Prob. 25, 3 A997) 1011-1058.
Donsker, M.D.
[1] An invariance principle for certain probability limit theorems. Mem. Amer. Math.
Soc. 6A951) 1-12.
Donsker, M.D., and Varadhan, S.R.S.
[1] Asymptotic evaluation of certain Markov processes expectations for large time I, II
and III. Comm. Pure Appl. Math. 28 A975) 1-47; 29 A976) 279-301 and 389^61.
Doob, J.L.
[1] Stochastic processes. Wiley, New York 1953.
[2] Classical potential Theory and its probabilistic counterpart. Springer, New York Berlin
1984.
Doss, H.
[1] Liens entre equations differentielles stochastiques et ordinaires. C.R. Acad. Sci. Paris,
Ser. A 233 A976) 939-942 et Ann. I.H.P. 13 A977) 99-125.

562 Bibliography
Doss, H., and Lenglart, E.
[1] Sur 1'existence, l'unicite et le comportement asymptotique des solutions d'equations
differentielles stochastiques. Ann. I.H.P. 14 A978) 189-214.
Douglas, R.
[1] On extremal measures and subspace density. Michigan Math. J. 11 A964) 644-652.
Dubins, L.
[1] Rises and upcrossings of non-negative martingales, III. J. Math. 6 A962) 226-241.
[2] A note on upcrossings of martingales. Ann. Math. Stat. 37 A966) 728.
[3] On a theorem of Skorokhod. Ann. Math. Stat. 39 A968) 2094-2097.
Dubins, L., Emery, M., and Yor, M.
[1] On the Levy transformation of Brownian motions and continuous martingales. Sem.
Prob. XXVII. Lecture Notes in Mathematics, vol. 1577. Springer, Berlin Heidelberg
New York 1993, pp. 122-132.
Dubins, L., Feldman, J., Smorodinsky, M., and TsireFson, B.
[1] Decreasing sequences of <r-fields and a measure change for Brownian motion. Ann.
Prob. 24A996)882-904.
Dubins, L., and Gilat, D.
[1] On the distribution of maxima of martingales. Proc. Amer. Math. Soc. 68 A978)
337-338.
Dubins, L., and Schwarz, G.
[1] On continuous martingales. Proc. Nat. Acad. Sci. USA 53 A965) 913-916.
[2] On extremal martingales distributions. Proc. Fifth Berkeley Symp. 2A) A967) 295-
297.
Dubins, L., and Smorodinsky, M.
[1] The modified discrete Levy transformation is Bernoulli. Sem. Prob. XXVI. Lecture
Notes in Mathematics, vol. 1528. Springer, Berlin Heidelberg New York 1992, pp.
157-161.
Dudley, R.M.
[1] Wiener functionals as Ito integrals. Ann. Prob. 5 A) A977) 140-141.
Duplantier, B.
[I] Areas of planar Brownian curves. J. Phys. A. Math. Gen. 22 A3) A989) 3033-3048.
Duplantier, В., Lawler, G.F., LeGall, J.F., and Lyons, T.J.
[1] The geometry of the Brownian curve. Bull. Sci. Math. 117 A993) 91-106.
Durbin, J.
[1] The first passage density of the Brownian motion process to a curved boundary (with
an appendix by D. Williams). J. Appl. Prob. 29 B) A992) 291-403.
[2] A reconciliation of two different expressions for the first passage density of Brownian
motion to a curved boundary. J. Appl. Prob. 25 A988) 829-832.
Durrett, R.
[1] A new proof of Spitzer's result on the winding of 2-dimensional Brownian motion.
Ann. Prob. 10 A982) 244-246.
[2] Brownian motion and martingales in analysis. Wadsworth, Belmont, Calif. 1984.
[3] Probability: Theory and examples. Duxbury Press, Wadsworth, Pacific Grove, Cal.
2nd ed. 1996.

Bibliography 563
Durrett, R., and Iglehart, D.L.
[1] Functionals of Brownian meander and Brownian excursion. Ann. Prob. 5 A977) 130-
135.
Durrett, R.T., Iglehart, D.L., and Miller, D.R.
[ 1 ] Weak convergence to Brownian meander and Brownian excursion. Ann. Prob. 5 A977)
117-129.
Dvoretsky, A., Erdos, P., and Kakutani, S.
[1] Non increase everywhere of the Brownian motion process. Proc. Fourth Berkeley
Symposium 2 A961) 103-116.
Dwass, H.
[1] On extremal processes I and II. Ann. Math. Stat. 35 A964) 1718-1725 and 111. J. Math.
10A966) 381-391.
Dynkin, E.B.
[1] Markov processes. Springer, Berlin Heidelberg New York 1965.
[2] Self-intersection gauge for random walks and Brownian motion. Ann. Prob. 16 A988)
1-57.
[3] Local times and random fields. Seminar on Stoch. Proc. 1983. Birkhauser 1984, pp.
69-84.
El Karoui, N.
[1] Sur les montees des semi-martingales. Asterisque 52-53, Temps Locaux A978) 63-88.
El Karoui, N., and Chaleyat-Maurel, M.
[1] Un probleme de reflexion et ses applications au temps local et aux equations
differentielles stochastiques sur K-cas continu. Asterisque 52-53, Temps Locaux
A978) 117-144.
El Karoui, N., and Weidenfeld, G.
[1] Theorie generale et changement de temps. Sem. Prob. XI. Lecture Notes in Mathe-
Mathematics, vol. 581. Springer, Berlin Heidelberg New York 1977, pp. 79-108.
El worthy, K.D.
[1] Stochastic differential equations on manifolds. London Math. Soc. Lecture Notes Se-
Series 70. Cambridge University Press A982).
Elworthy, K.D., Li, X.M., and Yor, M.
[1] On the tails of the supremum and the quadratic variation of strictly local martingales.
Sem. Prob. XXXI, Lect. Notes in Mathematics 1655. Springer, Berlin Heidelberg New
York 1997, pp. 113-125.
Elworthy, K.D., and Truman, A.
[1] The diffusion equation: an elementary formula. In: S. Albeverio et al. (eds.) Stochastic
processes in quantum physics. Lecture Notes in Physics, vol. 173. Springer, Berlin
Heidelberg New York 1982, pp. 136-146.
Emery, M.
[1] Une definition faible de BMO. Ann. I.H.P. 21 @A985M9-71.
[2] On the Azema martingales. Sem. Prob. XXIII, XXIV. Lect. Notes in Mathematics
1372, 1426. Springer, Berlin Heidelberg New York A989), A990), pp. 66-87; pp.
442^47.
[3] Quelques cas de representation chaotique. Sem. Prob. XXV. Lect. Notes in Mathe-
Mathematics 1485. Springer, Berlin Heidelberg New York A991), pp. 10-23.
[4] On certain probabilities equivalent to coin-tossing, d'apres Schachermayer. Sem. Prob.
XXXIII, Lect. Notes in Mathematics, Springer, Berlin Heidelberg New York A999).

564 Bibliography
Emery, M., and Perkins, E.
[1] La filtration de В + L. Z.W. 59 A982) 383-390.
Emery, M, and Schachermayer, W.
[1] Brownian filtrations are not stable under equivalent time changes. Sem. Prob. XXXIII,
Lect. Notes in Mathematics, Springer Berlin Heidelberg New York A999).
[2] A remark on Tsirel'son's stochastic differential equation. Sem. Prob. XXXIII, Lect.
Notes in Mathematics, Springer Berlin Heidelberg New York A999).
Emery, M., Strieker, C, and Yan, J.A.
[1] Valeurs prises par les martingales locales continues a un instant donne. Ann. Prob. 11
A983N35-641.
Emery, M., and Yor, M.
[1] Sur un theoreme de Tsirel'son relatif a des mouvements browniens correles et a la
nullite de certains temps locaux. Sem. Prob. XXXII, Lect. Notes in Mathematics 1686.
Springer, Berlin Heidelberg New York 1998, pp. 306-312.
Engelbert, H.J., and Hess, J.
[1] Integral representation with respect to stopped continuous local martingales. Stochas-
tics4A980) 121-142.
Engelbert, H.J., and Schmidt, W.
[1] On solutions of stochastic differential equations without drift. Z.W. 68 A985) 287-
317.
Ethier, S.N., and Kurtz, T.G.
[1] Markov processes. Characterization and convergence. Wiley and Sons, New York
1986.
Evans, S.
[1] On the Hausdorff dimension of Brownian cone points. Math. Proc. Camb. Philos. Soc.
98 A985) 343-353.
Ezawa, H., Klauder, J.R., and Shepp, L.A.
[1] Vestigial effects of singular potentials in diffusion theory and quantum mechanics.
J. Math. Phys. 16, 4 A975) 783-799.
[2] A path-space picture for Feynman-Kac averages. Ann. Phys. 88, 2 A974) 588-620.
[3] On the divergence of certain integrals of the Wiener process. Ann. Inst. Fourier, XXIV,
2A974) 189-193
Feldman, J., and Smorodinsky, M.
[1] Simple examples of non-generating Girsanov processes. Sem. Prob. XXXI, Lect. Notes
in Mathematics 1655. Springer, Berlin Heidelberg New York 1997, pp. 247-251.
Feldman, J., and Tsirel'son, B.
[1] Decreasing sequences of a -fields and a measure change for Brownian motion, II. Ann.
Prob. 24A996)905-911.
Feller, W.
[1] Diffusion process in one dimension. Trans. Amer. Math. Soc. 77 A954) 1-31.
[2] On second order differential operators. Ann. Math. 61 A955) 90-105.
[3] Differential operators with the positive maximum property. 111. J. Math. 3 A959)
182-186.
Fisk, D.L.
[1] Sample quadratic variation of sample continuous, second order martingales. Z.W. 6
A966) 273-278.

Bibliography 565
Fitzsimmons, PJ.
[1] A converse to a theorem of P. Levy. Ann. Proh. 15 A987) 1515-1523.
Fitzsimmons, P.J., and Getoor, R.K.
[1] Limit theorems and variation properties for fractional derivatives of the local time of
a stable process. Ann. I.H.P. 28 A992) 311-333.
[2] On the distribution of the Hilbert transform of the local time of a symmetric Levy
process. Ann. Prob. 20 A992) 1484-1497.
Fitzsimmons, P.J., Pitman, J.W., and Yor, M.
[1] Markovian bridges: Construction, Palm interpretation, and splicing. Seminar on
Stochastic Processes 1992. Birkhauser, pp. 101-134.
Follmer, H.
[1] Stochastic holomorphy. Math. Ann. 207 A974) 245-265.
[2] Calcul d'lto sans probabilites. Sem. Prob. XV. Lecture Notes in Mathematics, vol.
850. Springer, Berlin Heidelberg New York 1981, pp. 143-150.
[3] Von der Brownsche Bewegung zum Brownschen Blatt: einige neuere Richtungen in
der Theorie der stochastischen Prozesse. In: W. Jager, J. Moser, R. Remmert (eds.)
Perspectives in Mathematics, Anniversary of Oberwolfach, 1984. Birkhauser, Basel
1984.
[4] The exit measure of a supermartingale. Z.W. 21 A972) 154-166.
[5] On the representation of semimartingales. Ann. Prob. 1 D) A973) 580-589.
Follmer, H., and Imkeller, P.
[1] Anticipation cancelled by a Girsanov transformation: a paradox on Wiener space. Ann.
I.H.P. 29 D) A993) 569-586.
Follmer, H., and Protter, P.
[1] On Ito's formula for d!-dimensional Brownian motion. Preprint A998).
Follmer, H., Protter, P., and Shiryaev, A.N.
[1] Quadratic covariation and an extension of Ito's formula. Bernoulli 1 A/2) A995)
149-169.
Follmer, H., Wu, СТ., and Yor, M.
[1] Canonical decomposition of linear transformations of two independent Brownian mo-
motions. Preprint 1998
Freedman, D.
[1] Brownian motion and diffusion. Holden-Day, San Francisco 1971.
Freidlin, M.I.
[1] Action functionals for a class of stochastic processes. Theor. Prob. Appl. 17 A972)
511-515.
Friedman, A.
[1] Stochastic differential equations and applications 1 and II. Acadamic Press, New York
1975 and 1976.
Fujisaki, M., Kallianpur, G., and Kunita, H.
[1] Stochastic differential equations for the non-linear filtering problem. Osaka J. Math.
9A972) 19-40.
Fukushima, M., and Takeda, M.
[1 ] A transformation of symmetric Markov processes and the Donsker-Varadhan theory.
Osaka J. Math. 21 A984) 311-326.

566 Bibliography
Galmarino, A.R.
[1] Representation of an isotropic diffusion as a skew product. Z.W. 1 A963) 359-378.
Galtchouk, L., and Novikov, A.A.
[ 1 ] On Wald's equation. Discrete time case. Sem. Prob. XXXI, Lect. Notes in Mathematics
1655. Springer, Berlin Heidelberg New York 1997, pp. 126-135.
Garsia, A., Rodemich, E., and Rumsey, J., Jr.
[1] A real variable lemma and the continuity of paths of some Gaussian processes. Indiana
Univ. Math. J. 20 A970/1971) 565-578.
Gaveau, B.
[1] Principe de moindre action, propagation de la chaleur et estimees sous-elliptiques sur
certains groupes nilpotents. Acta. Math. 139 A977) 96-153.
Geiger, J., and Kersting, G.
[1] Winding numbers for 2-dimensional positive recurrent diffusions. Potential Analysis
3, 2A994) 189-201.
Geman, D., and Horowitz, J.
[1] Occupation densities. Ann. Prob. 8 A980) 1-67.
Geman, D., Horowitz, J., and Rosen, J.
[1] A local time analysis of intersections of Brownian paths in the plane. Ann. Prob. 12
A985)86-107.
Getoor, R.K..
[1] Another limit theorem for local time. Z.W. 34 A976) 1-10.
[2] The Brownian escape process. Ann. Prob. 7 A979) 864-867.
[3] Infinitely divisible probabilities on the hyperbolic plane. Pacific J. Math. 11 A961)
1287-1308.
Getoor, R.K., and Sharpe, M.J.
[1] Conformal martingales. Invent. Math. 16 A972) 271-308.
[2] Last exit times and additive functional. Ann. Prob. 1 A973) 550-569.
[3] Last exit decompositions and distributions. Indiana Univ. Math. J. 23 A973) 377-404.
[4] Excursions of Brownian motion and Bessel processes. Z.W. 47 A979) 83-106.
[5] Excursions of dual processes. Adv. Math. 45 A982) 259-309.
[6] Naturality, standardness and weak duality for Markov processes. Z.W. 67 A984) 1-62.
Gikhman, 1.1., and Skorokhod, A.V.
[1] Theory of stochastic processes I, II and III. Springer, Berlin Heidelberg New York
1974, 1975, and 1979.
Girsanov, I.V.
[1] On transforming a certain class of stochastic processes by absolutely continuous sub-
substitution of measures. Theor. Prob. Appl. 5 A960) 285-301.
f2] An example of non-uniqueness of the solution to the stochastic differential equation
of K. Ito. Theor. Prob. Appl. 7 A962) 325-331.
Going, A.
[1] Some generalizations of Bessel processes. Ph. D. Thesis, ETH Zurich (January 1998).
Goswami, A., and Rao, B.V.
[1] Conditional expectation of odd chaos given even. Stochastics and Stochastics Reports
35A991J13-224.

Bibliography 567
Graversen, S.E.
[1] "Polar"-fiinctions for Brownian motion. Z.W. 61 A981) 261-270.
Greenwood, P., and Perkins, E.
[1] A conditional limit theorem for random walks and Brownian local times on square
root boundaries. Ann. Prob. 11 A983) 227-261.
Gruet, J.C.
[1] Sur la transience et la recurrence de certaines martingales locales continues a valeurs
dans le plan. Stochastics 28 A989) 189-207.
[2] Semi-groupe du mouvement brownien hyperbolique. Stochastics and Stochastic Re-
Reports 56 A996) 53-61.
[3] Windings of hyperbolic Brownian motion. In: M. Yor (ed.) Exponential Functionals
and Principal Values related to Brownian Motion. Bib. Rev. lbero-Amer. A997)
[4] On the length of the homotopic Brownian word in the thrice punctured sphere. Prob.
Th. Rel. F. A998).
Gruet, J.C, and Mountford, T.S.
[1] The rate of escape for pairs of windings on the Riemann sphere. J. London Math.
Soc. 48 B) A993) 552-564.
Gundy, R.F.
[1] Some topics in probability and analysis. Conf. Board of the Math. Sciences. Regional
Conf. Series in Maths. 70 A989)
Hardin, CD.
[1] A spurious Brownian motion. Proc. Amer. Math. Soc. 93 A985) 350.
Harrison, J.M., and Shepp, L.A.
[1] On skew Brownian motion. Ann. Prob. 9 A981) 309-313.
Hartman, P., and Wintner, A.
[1] On the law of the iterated logarithm. Amer. J. Math. 63 A941) 169-176.
Haussmann, U.G.
[1] Functionals of Ito processes as stochastic integrals. Siam J. Contr. Opt. 16, 2 A978)
252-269.
Hawkes, J., and Barlow, M.T.
[1] Application de l'entropie metrique a la continuite des temps locaux des processus de
Levy. C.R. Acad. Sci. Paris 301, Serie I, 5 A985) 237-239.
Hawkes, J., and Truman, A.
[1] Statistics of local time and excursions for the Ornstein-Uhlenbeck process. In: M.T.
Barlow, N. Bingham (eds.) Stochastic Analysis. London Math. Soc. Lect. Notes, vol.
167, 1991.
Hobson, D.
[1] The maximum maximum of a martingale. Sem. Prob. XXXII. Lecture Notes in Math-
Mathematics, vol. 1686. Springer, Berlin Heidelberg New York 1998, pp. 250-263.
Hu, Y., Shi, Z., and Yor, M.
[1] Rates of convergence of diffusions with drifted Brownian potentials. Trans. Amer.
Math. Soc. A999).
Hunt, G.A.
[1] Markov processes and potentials I, II and III. III. J. of Math. 1 A9657) 44-93 and
316-369; 111. J. Math. 2 A958) 151-213.

568 Bibliography
Ibragimov, I.
[1] Sur la regularite des trajectoires des fonctions aleatoires. C.R. Acad. Sci. Paris, Serie
A 289 A979) 545-547.
Idrissi Khamlichi, A.
[1] Problemes de polarite pour des semimartingales bidimensiormelles continues. These
de 3° Cycle, Universite de Paris VII, 1986.
Ikeda, N., and Manabe, S.
[1] Integral of differential forms along the paths of diffusion processes. Publ. R.I.M.S.,
Kyoto Univ. 15 A978) 827-852.
Ikeda, N., and Watanabe, S.
[ 1 ] A comparison theorem for solutions of stochastic differential equations and its appli-
applications. Osaka J. Math. 14 A977) 619-633.
[2] Stochastic differential equations and diffusion processes. Second edition North-
Holland Publ. Co., Amsterdam Oxford New York; Kodansha Ltd., Tokyo 1989.
Imhof, J.P.
[1] Density factorizations for Brownian motion and the three-dimensional Bessel Pro-
Processes and applications. J. Appl. Prob. 21 A984) 500-510.
[2] On Brownian bridge and excursion. Studia Scient. Math. Hungaria 20 A985) 1-10.
[3] A simple proof of Pitman's 2M - X theorem. Adv. Appl. Prob. 24 A992) 499-501.
Isaacson, D.
[1] Stochastic integrals and derivatives. Ann. Math. Sci. 40 A969) 1610-1616.
Ito, K.
[1] Stochastic integral. Proc. Imp. Acad. Tokyo 20 A944) 519-524.
[2] Stochastic differential equations. Memoirs A.M.S. 4, 1951.
[3] Multiple Wiener integrals. J. Math. Soc. Japan 3 A951) 157-169.
[4] Lectures on stochastic processes. Tata Institute, Bombay 1961.
[5] Poisson point processes attached to Markov processes. Proc. Sixth Berkeley Symp.
Math. Stat. Prob., vol. 3. University of California, Berkeley, 1970, pp. 225-239.
[6] Extension of stochastic integrals. Proc. lnternat. Symp. SDE Kyoto 1976, pp. 95-109.
[Most of K. Ito's papers are reprinted in: D. Stroock, S. Varadhan (eds.) Kiyosi ltd Selected
Papers. Springer A987)]
ltd, K., and McKean, H.P.
[1] Diffusion processes and their sample paths. Springer, Berlin Heidelberg New York
1965.
ltd, K., and Watanabe, S.
[1] Transformation of Markov processes by multiplicative functional. Ann. Inst. Fourier
15A965) 15-30.
Jacod, J.
[1] A general theorem of representation for martingales. Proc. AMS Prob. Symp. Urbana
1976, 37-53.
[2] Calcul stochastique et problemes de martingales. Lecture Notes in Mathematics, vol.
714. Springer, Berlin Heidelberg New York 1979.
[3] Equations differentielles lineaires: la methode de variation des constantes. Sem. Prob.
XVI. Lecture Notes in Mathematics, vol. 920. Springer. Berlin Heidelberg New York
1982, pp. 442-446.

Bibliography 569
Jacod, J., and Memin, J.
[1] Caracteristiques locales et conditions de continuity absolue pour les semimartingales.
Z.W. 35 A976) 1-37.
[2] Sur l'integrabilite uniforme des martingales exponentielles. Z.W. 42 A978) 175-204.
Jacod, J., and Shiryaev, A.N.
[1] Limit theorems for stochastic processes. Springer, Berlin Heidelberg New York 1987.
Jacod, J., and Yor, M.
[1] Etude des solutions extremales et representation integral e des solutions pour certains
problemes de martingales. Z.W. 38 A977) 83-125.
Janson, S.
[1] On complex hypercontractivity. J. Funct. Anal. 151 A997) 270-280.
[2] Gaussian Hilbert spaces. Cambridge Univ. Press A997)
Jeulin, T.
[1] Un theoreme de J. Pitman. Sem. Prob. XIII. Lecture Notes in Mathematics, vol. 721.
Springer, Berlin Heidelberg New York 1979, pp. 332-359.
[2] Semi-martingales et grossissement d'une filtration. Lecture Notes in Mathematics, vol.
833. Springer, Berlin Heidelberg New York 1980.
[3] Application de la theorie du grossissement a l'etude des temps locaux browniens. In:
Grossissements de filtrations: Exemples et applications. Lecture Notes in Mathematics,
vol. 1118. Springer, Berlin Heidelberg New York 1985, pp. 197-304.
[4] Sur la convergence absolue de certaines integrates. Sem. Prob. XVI. Lecture Notes in
Mathematics, vol. 920. Springer, Berlin Heidelberg New York 1982, pp. 248-256.
[5] Ray-Knight's theorems on Brownian local times and Tanaka formula. Seminar on
Stoch. Proc. 1983, Birkhauser 1984, pp. 131-142.
Jeulin, Т., and Yor, M.
[1] Autour d'un theoreme de Ray. Asterisque 52-53, Temps locaux A978) 145-158.
[2] Inegalite de Hardy, semimartingales et faux-amis. Sem. Prob. XIII. Lecture Notes in
Mathematics, vol. 721. Springer, Berlin Heidelberg New York 1979, pp. 332-359.
[3] Sur les distributions de certaines fonctionnelles du mouvement brownien. Sem. Prob.
XV. Lecture Notes in Mathematics, vol. 850. Springer, Berlin Heidelberg New York
1981, pp. 210-226.
[4] Filtration des ponts browniens et equations differentielles lineaires. Sem. Proba. XXIV,
Lecture Notes in Mathematics, vol., 1426. Springer, Berlin Heidelberg New York 1990,
pp. 227-265.
[5] Une decomposition non canonique du drap brownien. Sem. Prob. XXVI. Lecture
Notes in Mathematics, vol. 1528. Springer, Berlin Heidelberg New York 1992, pp.
322-347.
[6] Moyennes mobiles et semimartingales. Sem. Prob. XXVII, Lect. Notes in Mathemat-
Mathematics, vol. 1557. Springer, Berlin Heidelberg New York 1993, pp. 53-77.
Johnson, G., and Helms, L.L.
[1] Class (D) supermartingales. Bull. Amer. Math. Soc. 69 A963) 59-62.
Kac, M.
[1] On distribution of certain Wiener functionals. Trans. Amer. Math. Soc. 65 A949)
1-13
[2] On some connections between probability theory and differential and integral equa-
equations. Proc. Second Berkeley Symp. Math. Stat. Prob. Univ. California Press 1951,
pp. 189-251.
[3] Integration in Function spaces and some of its applications. Lezioni Fermiane. Ace.
Nat. dei Lincei. Scuola Normale Sup. Pisa A980).

570 Bibliography
Kahane, J.P.
[1] Some random series of functions. Second edition, Cambridge studies in Advanced
Mathematics, 5, 1985.
[2] Brownian motion and classical analysis. Bull. London Math. Soc. 7 A976) 145-155.
[3] Le mouvement brownien. In: Materiaux pour 1'histoire des Mathematiques au XXе
siecle. Soc. Math. France (Dec. 1997).
Kallenberg, O.
[1] Some time change representation of stable integrals, via predictable transformations
of local martingales. Stoch. Proc. and their App. 40 A992) 199-223.
[2] On an independence criterion for multiple stochastic integrals. Ann. Prob. 19, 2 A991)
483^85.
[3] On the existence of universal Functional Solutions to classical SDE's. Ann. Prob. 24,
1 A996) 196-205.
[4] Foundations of Modern Probability. Springer 1997.
Kallianpur, G.
[1] Stochastic filtering theory. Springer, Berlin Heidelberg New York 1980.
[2] Some recent developments in nonlinear filtering theory. In: N. Ikeda, S. Watanabe,
M. Fukushima, H. Kunita (eds.) Ito's Stochastic Calculus and Probability Theory.
Springer, Berlin Heidelberg New York 1996, pp. 157-170.
Kallianpur, G., and Robbins, H.
[1] Ergodic property of Brownian motion process. Proc. Nat. Acad. Sci. USA 39 A953)
525-533.
Kallssen, J.
[1] An up to indistinguishability unique solution to a stochastic differential equation that
is not strong. Sem. Prob. XXXIII, Lect. Notes in Mathematics. Springer, Berlin Hei-
Heidelberg New York 1999.
Karandikar, R.L.
[1] A.s. approximation results for multiplicative stochastic integrals. Sem. Prob. XVI.
Lecture Notes in Mathematics, vol. 920. Springer, Berlin Heidelberg New York 1982,
pp. 384-391.
Karatzas, I., and Shreve, S.E.
[1] Brownian motion and stochastic calculus. Springer, Berlin Heidelberg New York 1988.
Karlin, S., and Taylor, H.M.
[1] A second course in stochastic processes. Academic Press 1981.
Kasahara, Y.
[1] Limit theorems of occupation times for Markov processes. R.I.M.S. 3 A977) 801-818.
[2] Limit theorems for occupation times of Markov processes. Japan J. Math. 7 A981)
291-300.
[3] A limit theorem for slowly increasing occupation times. Ann. Prob. 10 A982) 728-
736.
[4] On Levy's downcrossing theorem. Proc. Japan Acad. 56 A A0) A980) 455-458.
Kasahara, Y., and Kotani, S.
[1] On limit processes for a class of additive functional of recurrent diffusion processes.
Z.W. 49A979) 133-143.
Kawabata, S., and Yamada, T.
[1] On some limit theorems for solutions of stochastic differential equations. Sem. Prob.
XVI. Lecture Notes in Mathematics, vol. 920. Springer, Berlin Heidelberg New York
1982, pp. 412-441.

Bibliography 571
Kawazu, К., and Tanaka, H.
[1] A diffusion process in a Brownian environment with drift. J. Math. Soc. Japan 49
A997) 189-211.
Kazamaki, N.
[1] Change of time, stochastic integrals and weak martingales. Z.W. 22 A972) 25-32.
[2] On a problem of Girsanov. Tohoku Math. J. 29 A977) 597-600.
[3] A remark on a problem of Girsanov. Sem. Prob. XII. Lecture Notes in Mathematics,
vol. 649. Springer, Berlin Heidelberg New York 1978, pp. 47-50.
[4] Continuous Exponential martingales and BMO. Lect. Notes in Mathematics, vol. 1579.
Springer, Berlin Heidelberg New York 1994.
Kazamaki, N., and Sekiguchi, T.
[1] On the transformation of some classes of martingales by a change of law. Tohoku
Math. J. 31 A979J61-279.
[2] Uniform integrability of continuous exponential martingales. Tohoku Math. J. 35
A983J89-301.
Kennedy, D.
[1] Some martingales related to cumulative sum tests and single-server queues. Stoch.
Proc. Appl. 4 A976) 261-269.
Kent, J.
[1] The infinite divisibility of the Von Mises-Fischer distribution for all values of the
parameter in all dimensions. Proc. London Math. Soc. 35 A977) 359-384.
[2] Some probabilistic properties of Bessel functions. Ann. Prob. 6 A978) 760-770.
Kertz, R.P., and Rosier, U.
[1] Martingales with given maxima and terminal distributions. Israel J. of Maths. 69, 2
A990) 173-192.
Khintchine, A.Y.
[1] Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. (Ergebnisse der Mathema-
tik, Bd. 2). Springer, Berlin Heidelberg 1933, pp. 72-75.
Knight, F.B.
[1] Brownian local time and taboo processes. Trans. Amer. Math. Soc. 143 A969) 173-
185.
[2] Random walks and a sojourn density process of Brownian motion. Trans. Amer. Math.
Soc. 107A963M6-86
[3] A reduction of continuous square-integrable martingales to Brownian motion. Lecture
Notes in Mathematics, vol. 190. Springer, Berlin Heidelberg New York 1970, pp.
19-31.
[4] An infinitesimal decomposition for a class of Markov processes. Ann. Math. Stat. 41
A970) 1510-1529.
[5] Essentials of Brownian motion and diffusion. Math. Surv. 18 Amer. Math. Soc,
Providence, Rhode-Island 1981.
[6] On the duration of the longest excursion. Sem. Stoch. Prob. 1985. Birkhauser, Basel
1986, pp. 117-147.
[7] On invertibility of Martingale time changes. Sem. Stoch. Proc. 1987. Birkhauser, Basel
1988, pp. 193-221.
[8] Inverse local times, positive sojourns, and maxima for Brownian motion. Asterisque
157-158 A988) 233-247.
[9] Calculating the compensator: methods and examples. Sem. Stoch. Proc. 1990. Birk-
Birkhauser, Basel 1991, pp. 241-252.
[10] Foundations of the prediction process. Clarendon Press, Oxford 1992.

572 Bibliography
[11] Some remarks on mutual windings. Sem. Prob. XXVII. Lecture Notes in Mathematics,
vol. 1557. Springer, Berlin Heidelberg New York 1993, pp. 36-43.
[12] A remark about Walsh's Brownian motion. Colloque in honor of J.P. Kahane (Orsay,
June 1993). In: The Journal of Fourier Analysis and Applications, Special Issue, 1995,
pp. 1600-1606.
[13] On the upcrossing chains of stopped Brownian motion. Sem. Prob. XXXII, Lect. Notes
in Mathematics 1686. Springer, Berlin Heidelberg New York 1998, pp. 343-375.
Kolmogorov, A.N.
[1] Ober das Gesetz des itierten Logarithmus. Math. Ann. 101 A929) 126-135.
Koval'chik, I.M.
[1] The Wiener integral. Russ. Math. Surv. 18 A963) 97-135.
Kono, N.
[I] Demonstration probabiliste du theoreme de d'Alembert. Sem. Prob. XIX. Lecture
Notes in Mathematics, vol. 1123. Springer, Berlin Heidelberg New York 1985, pp.
207-208.
Khoshnevisan, D.
[1] Exact rates of convergence to Brownian local time. Ann. Prob. 22,3 A994) 1295-1330.
Krengel, U.
[1] Ergodic theorems, de Gruyter, Berlin New York 1985.
Kiichler, U.
[1] On sojourn times, excursions and spectral measures connected with quasi-diffusions.
Jour. Math. Kyoto U. 26, 3 A986) 403-421.
Kunita, H.
[1] Absolute continuity of Markov processes and generators. Nagoya Math. J. 36 A969)
1-26.
[2] Absolute continuity of Markov processes. Sem. Prob. X. Lecture Notes in Mathemat-
Mathematics, vol. 511. Springer, Berlin Heidelberg New York 1976, pp. 44-77.
[3] On backward stochastic differential equations. Stochastics 6 A982) 293-313.
[4] Stochastic differential equations and stochastic flows of diffeomorphisms. Ecole d'Ete
de Probabilites de Saint-Flour XII. Lecture Notes in Mathematics, vol. 1097. Springer,
Berlin Heidelberg New York 1984, pp. 143-303.
[5] Some extensions of Ito's formula. Sem. Prob. XV. Lecture Notes in Mathematics, vol.
850. Springer, Berlin Heidelberg New York 1981, pp. 118-141.
[6] Stochastic flows and stochastic differential equations. Cambridge University Press
1990.
Kunita, H., and Watanabe, S.
[1] On square-integrable martingales. Nagoya J. Math. 30 A967) 209-245.
Kurtz, T.G.
[1] Representations of Markov processes as multiparameter time changes. Ann. Prob. 8
A980N82-715.
Lamperti, J.
[1] Continuous state branching processes. Bull. A.M.S. 73 A967) 382-386.
[2] Semi-stable Markov processes I. Z.W. 22 A972) 205-255.
Lane, D.A.
[1] On the fields of some Brownian martingales. Ann. Prob. 6 A978) 499-508.

Bibliography 573
Lebedev, N.N.
[1] Special functions and their applications. Dover Publications, 1972.
Ledoux, M.
[I] Inegalites isoperimetriques et calcul stochastique. Sem. Prob. XXII, Lecture Notes in
Mathematics, vol. 1321. Springer, Berlin Heidelberg New York 1988, pp. 249-259.
Ledoux, M., and Talagrand, M.
[1] Probability on Banach spaces. Springer, Berlin Heidelberg New York 1991.
Le Gall, J.F.
[I] Applications du temps local aux equations differentielles stochastiques unidimension-
nelles. Sem. Prob. XVII. Lecture Notes in Mathematics, vol. 986. Springer, Berlin
Heidelberg New York 1983, pp. 15-31.
[2] Sur la saucisse de Wiener et les points multiples du mouvement Brownien. Ann. Prob.
14A986) 1219-1244.
[3] Sur la mesure de Hausdorff de la courbe brownienne. Sem. Prob. XIX. Lecture Notes
in Mathematics, vol. 1123. Springer, Berlin Heidelberg New York 1985, pp. 297-313.
[4] Sur le temps local d'intersection du mouvement brownien plan et la methode de
renormalisation de Varadhan. Sem. Prob. XIX. Lecture Notes in Mathematics, vol.
1123. Springer, Berlin Heidelberg New York 1985, pp. 314-331.
[5] Une approche elementaire des theoremes de decomposition de Williams. Sem. Prob.
XX. Lecture Notes in Mathematics, vol. 1204. Springer, Berlin Heidelberg New York
1986, pp. 447^64.
[6] Sur les fonctions polaires pour le mouvement brownien. Sem. Prob. XXII. Lecture
Notes in Mathematics, vol. 1321. Springer, Berlin Heidelberg New York 1988, pp.
186-189.
[7] Proprietes d'intersection des marches aleatoires, I. Convergence vers le temps local
d'intersection, II. Etude des cas critiques. Comm. Math. Phys. 104 A986) 471-507
and 509-528.
[8] Introduction au mouvement brownien. Gazette des Mathematiciens 40, Soc. Math.
France A989) 43-64.
[9] Some properties of planar Brownian motion. Ecole d'ete de Saint-Flour XX, 1990.
Lecture Notes in Mathematics, vol. 1527. Springer, Berlin Heidelberg New York 1992,
pp. 112-234.
[10] The uniform random tree in a Brownian excursion. Prob. Th. Rel. Fields 96 A993)
369-383.
[II] Mouvement brownien et calcul stochastique. Cours de troisieme cycle. Universite
Paris VI, 1994.
Le Gall, J.F., and Meyre, T.
[1] Points cones du mouvement brownien plan, le cas critique. Prob. Th. Rel. Fields 93
A992J31-247.
Le Gall. J.F., and Yor, M.
[1] Sur l'equation stochastique de Tsirel'son. Sem. Prob. XVII. Lecture Notes in Mathe-
Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1983, pp. 81-88.
[2] Etude asymptotique de certains mouvements browniens complexes avec drift. Z.W.
71 A986) 183-229.
[3] Etude asymptotique des enlacements du mouvement brownien autour des droites de
l'espace. Prob. Th. Rel. Fields 74 A987) 617-635.
[4] Enlacements du mouvement brownien autour des courbes de l'espace. Trans. Amer.
Math. Soc. 317 A990) 687-722.
[5] Excursions browniennes et carres de processus de Bessel. C.R. Acad. Sci. Paris, Serie
I 303 A986) 73-76.

574 Bibliography
Lehoczky, J.
[1] Formulas for stopped diffusion processes with stopping times based on the maximum.
Ann. Prob. 5 A977) 601-608.
Le Jan, Y.
[1] Martingales et changements de temps. Sem. Prob. XIII. Lecture Notes in Mathematics,
vol. 721. Springer, Berlin Heidelberg New York 1979, pp. 385-389.
Lenglart, E.
[1] Sur la convergence p.s. des martingales locales. C.R. Acad. Sci. Paris 284 A977)
1085-1088.
[2] Relation de domination entre deux processus. Ann. I.H.P. 13 A977) 171-179.
[3] Transformation de martingales locales par changement absolument continu de proba-
bilites. Z.W. 39 A977) 65-70.
Lenglart, E., Lepingle, D., and Pratelli, M.
[I] Une presentation unifiee des inegalites en theorie des martingales. Sem. Prob. XIV.
Lecture Notes in Mathematics, vol. 784. Springer, Berlin Heidelberg New York 1980,
pp. 26^8.
Lepingle, D.
[1] Sur le comportement asymptotique des martingales locales. Sem. Prob. XII. Lecture
Notes in Mathematics, vol. 649. Springer, Berlin Heidelberg New York 1978, pp.
148-161.
[2] Une remarque sur les lois de certains temps d'atteinte. Sem. Prob. XV. Lecture Notes
in Mathematics, vol. 850. Springer, Berlin Heidelberg New York 1981, pp. 669-670.
Lepingle, D., and Memin, J.
[1] Sur l'integrabilite des martingales exponentielles. Z.W. 42 A978) 175-203.
Leuridan, C.
[1] Une demonstration elementaire d'une identite de Biane et Yor. Sem. Prob. XXX,
Lecture Notes in Mathematics, vol. 1626. Springer, Berlin Heidelberg New York
1996, pp. 255-260.
[2] Le theoreme de Ray-Knight a temps fixe. Sem. Prob. XXXII, Lecture Notes in Math-
Mathematics, vol. 1686. Springer, Berlin Heidelberg New York 1998, pp. 376-406.
[3] Les theoremes de Ray-Knight et la mesure d'lto pour le mouvement brownien sur le
tore R/Z. Stochastics and Stochastic Reports 55 A995) 109-128.
Levy, P.
[1] Le mouvement brownien plan. Amer. J. Math. 62 A940) 487-550.
[2] Processus stochastiques et mouvement brownien. Gauthier-Villars, Paris 1948.
[3] Wiener random functions and other Laplacian random functions. Proc. Second Berke-
Berkeley Symp., vol. II, 1950, pp. 171-186.
[4] Sur un probleme de M. Marcinkiewicz. C.R. Acad. Sci. Paris 208 A939) 318-321.
Errata p. 776.
[5] Sur certains processus stochastiques homogenes. Compositio Math. 7 A939) 283-339.
Lewis, J.T.
[1] Brownian motion on a submanifold of Euclidean space. Bull. London Math. Soc.
A986) 616-620.
Liptser, R.S., and Shiryaev, A.N.
[1] Statistics of random processus, I and II. Springer Verlag, Berlin, 1977 and 1978.

Bibliography 575
Lorang, G., and Roynette, B.
[1] Etude d'une fonctionnelle Нее аи pont de Bessel. Ann. Inst. H. Poincare 32 A996)
107-133.
Lyons, T.J., and McKean, H.P., Jr.
[1] Windings of the plane Brownian motion. Adv. Math. 51 A984) 212-225.
Lyons, T.J., and Zhang, T.S.
[1] Decomposition of Dirichlet processes and its application. Ann. Prob. 22 A994) 494-
524.
McGill, P.
[1] A direct proof of the Ray-Knight theorem. Sem. Prob. XV. Lecture Notes in Mathe-
Mathematics, vol. 850. Springer, Berlin Heidelberg New York 1981, pp. 206-209.
[2] Markov properties of diffusion local time: a martingale approach. Adv. Appl. Prob.
14A982O89-810.
[3] Integral representation of martingales in the Brownian excursion filtration. Sem. Prob.
XX. Lecture Notes in Mathematics, vol. 1204. Springer, Berlin Heidelberg New York
1986, pp. 465-502.
McGill, P., Rajeev, В., and Rao, B.V.
[1J Extending Levy's characterization of Brownian motion. Sem. Prob. XXII, Lecture
Notes in Mathematics, vol. 1321. Springer, Berlin Heidelberg New York 1988, pp.
163-165.
McKean, H.P., Jr.
[1] The Bessel motion and a singular integral equation. Mem. Coll. Sci. Univ. Kyoto,
Ser. A, Math. 33 A960) 317-322.
[2] Stochastic integrals. Academic Press, New York 1969.
[3] Brownian local time. Adv. Math. 15 A975) 91-111.
Maisonneuve, B.
[1] Systemes regeneratifs. Asterisque 15. Societe Mathematique de France 1974.
[2] Exit systems. Ann. Prob. 3 A975) 399^11.
[3] Une mise au point sur les martingales locales continues definies sur un intervalle
stochastique. Sem. Prob. XI. Lecture Notes in Mathematics, vol. 528. Springer, Berlin
heidelberg New York 1977, pp. 435-445.
[4] On Levy's downcrossing theorem and various extensions. Sem. Prob. XV. Lecture
Notes in Mathematics, vol. 850. Springer, Berlin Heidelberg New York 1981, pp.
191-205.
[5] Ensembles regeneratifs de la droite. Z.W. 63 A983) 501-510.
[6] On the structure of certain excursions of a Markov process. Z.W. 47 A979) 61-67.
[7] Excursions chevauchant un temps aleatoire quelconque. In: Hommage a P.A. Meyer
et J. Neveu. Asterisque 236 A996) 215-226.
Malliavin, P.
[1] Stochastic Analysis. Springer, Berlin Heidelberg New York 1997.
Malric, M.
[1] Filtrations browniennes et balayage. Ann. I.H.P. 26 A990) 507-540.
Mandelbrot, B.B., and Van Ness, J.W.
[1] Fractional Brownian motions, fractional noises and applications. S1AM Rev. 10 A968)
422-437.

576 Bibliography
Mandl, P.
[1] Analytical treatment of one-dimensional Markov process. Springer, Berlin Heidelberg
New York 1968.
March, P., and Sznitman, A.S.
[1] Some connections between excursion theory and the discrete random Schrodinger
equation with random potentials. Prob. Th. Rel. Fields 67 A987) 11-54.
Marcus, M.B., and Rosen, J.
[1] Sample path properties of the local times of strongly symmetric Markov processes
via Gaussian processes. Ann. Prob. 20, 4 A992) 1603-1684.
[2] Gaussian chaos and sample path properties of additive functionals of symmetric
Markov processes. Ann. Prob. 24, 3 A996) 1130-1177.
Maruyama, G.
[1] On the transition probability functions of Markov processes. Nat. Sci. Rep. Ochano-
mizu Univ. 5A954) 10-20.
[2] Continuous Markov processes and stochastic equations. Rend. Circ. Palermo 10 A955)
48-90.
Maruyama, G., and Tanaka, H.
[1] Ergodic property of я-dimensional Markov processes. Mem. Fac. Sci. Kyushu Univ.
13A959) 157-172.
Meilijson, I.
[1] There exists no ultimate solution to Skorokhod's problem. Sem. Prob. XVI. Lecture
Notes in Mathematics, vol. 920. Springer, Berlin Heidelberg New York 1982, pp.
392-399.
[2] On the Azema-Yor stopping time. Sem. Prob. XVII. Lecture Notes in Mathematics,
vol. 986. Springer, Berlin Heidelberg New York 1983, pp. 225-226.
Meleard, S.
[1] Application du calcul stochastique a Г etude des processus de Markov reguliers sur
[0, 1]. Stochastics 19 A986) 41-82.
Messulam, P., and Yor, M.
[1] On D. Williams' "pinching method" and some applications. J. London Math. Soc. 26
A982) 348-364.
Metivier, M.
[1] Semimartingales: a course on stochastic processes, de Gruyter, Berlin New York 1982.
Meyer, P.A.
[1] Processus de Markov. Lecture Notes in Mathematics, vol. 26. Springer, Berlin Hei-
Heidelberg New York 1967.
[2] Processus de Markov: La frontiere de Martin. Lecture Notes in Mathematics, vol. 77.
Springer, Berlin Heidelberg New York 1979.
[3] Demonstration simplified d'un theoreme de Knight. Sem. Prob. V. Lecture Notes in
Mathematics, vol. 191. Springer, Berlin New York 1971, pp. 191-195.
[4] Processus de Poisson ponctuels, d'apres K. Ito. Sem. Prob. V. Lecture Notes in Math-
Mathematics, vol. 191. Springer, Berlin Heidelberg New York 1971, pp. 177-190.
[5] Un cours sur les integrates stochastiques. Sem. Prob. X. Lecture Notes in Mathematics,
vol. 511. Springer, Berlin Heidelberg New York 1976, pp. 245-400.
[6] Demonstration probabiliste de certaines inegalites de Littlewood-Paley. Sem. Prob.
XII. Lecture Notes in Mathematics, vol. 511. Springer, Berlin Heidelberg New York
1976, pp. 125-183.

Bibliography 577
[7] La formule d'lto pour le mouvement brownien d'apres Brosamler. Sem. Prob. XII.
Lecture Notes in Mathematics, vol. 649. Springer, Berlin Heidelberg New York 1978,
pp. 763-769.
[8] Flot d'une equation differentielle stochastique. Sem. Prob. XV. Lecture Notes in Math-
Mathematics, vol. 850. Springer, Berlin Heidelberg New York 1981, pp. 103-117.
[9] Sur 1'existence de Poperateur carre du champ. Sem. Prob. XX. Lecture Notes in
Mathematics, vol. 1204, Springer, Berlin Heidelberg New York 1986, pp. 30-33.
[10] Formule d'lto generalisee pour le mouvement brownien lineaire, d'apres Follmer,
Protter, Shiryaev. Sem. Prob. XXXI, Lect. Notes in Mathematics 1655. Springer,
Berlin Heidelberg New York 1997, pp. 252-255.
Meyer, P.A., Smythe, R.T., and Walsh, J.B.
[1] Birth and death of Markov processes. Proc. Sixth Berkeley Symposium III. University
of California Press, 1971, pp. 295-305.
Millar, P.W.
[1] Martingale integrals. Trans. Amer. Math. Soc. 133 A968) 145-166.
[2] Stochastic integrals and processes with stationary independent increments. Proc. Sixth
Berkley Symp. 3 A972) 307-332.
Mokobodzki, G.
[1] Operateur carre du champ: un contre-exemple. Sem. Prob. XXIII. Lecture Notes in
Mathematics, vol. 1372. Springer, Berlin Heidelberg New York 1989, pp. 324-325.
Molchanov, S.
[1] Martin boundaries for invariant Markov processes on a solvable group. Theor. Prob.
Appl. 12A967K10-314.
Molchanov, S.A., and Ostrovski, E.
[1] Symmetric stable processes as traces of degenerate diffusion processes. Theor. Prob.
Appl. 14 A) A969) 128-131.
Monroe, I.
[1] Processes that can be imbedded in Brownian motion. Ann. Prob. 6, 1 A978) 42-56.
Mortimer, T.M., and Williams, D.
[1] Change of measure up to a random time: theory. J. Appl. Prob. 28 A991) 914-918.
Mountford, T.S.
[1] Transience of a pair of local martingales. Proc. A.M.S. 103 C) A988) 933-938.
[2] Limiting behavior of the occupation of wedges by complex Brownian motion. Prob.
Th. Rel. F. 84 A990) 55-65.
[3] The asymptotic distribution of the number of crossings between tangential circles by
planar Brownian motion. J. London Math. Soc. 44 A991) 184—192.
Mueller, С
[1] A unification of Strassen's law and Levy's modulus of continuity. Z.W. 56 A981)
163-179.
Nagasawa, M.
[1] Time reversions of Markov processes. Nagoya Math. J. 24 A964) 177-204.
[2] Transformations of diffusions and Schrodinger processes. Prob. Th. Rel. F. 82 A989)
106-136.
[3] Schrodinger equations and diffusion theory. Birkhauser 1993.
Nakao, S.
[1] On the pathwise uniqueness of solutions of one-dimensional stochastic differential
equations. Osaka J. Math. 9 A972) 513-518.

578 Bibliography
[2] Stochastic calculus for continuous additive functionals of zero energy. Zeit. fur Wahr.
68A985M57-578.
Neveu, J.
[1] Processus aleatoires gaussiens. Les Presses de l'Univ. de Montreal, 1968.
[2] Integrates stochastiques et application. Cours 3° Cycle. Laboratoire de Probabilites,
Universite de Paris VI, 1972.
[3] Sur l'esperance conditionnelle par rapport a un mouvement Brownien. Ann. I.H.P.
12,2A976) 105-110.
[4] Bases mathematiques du calcul des probabilites, 2eme edition. Masson, Paris 1971.
Ndumu, N.M.
[1] The heat kernel formula in a geodesic chart and some applications to the eigenvalue
problem of the 3-sphere. Prob. Th. Rel. F. 88 C) A991) 343-361.
Norris, J.R., Rogers, L.C.G., and Williams, D.
[1] Self-avoiding random walk: A Brownian model with local time drift. Prob. Th. Rel.
F. 74 A987) 271-287.
Novikov, A.A.
[1] On moment inequalities for stochastic integrals. Theor. Prob. Appl. 16 A971) 538-
541.
[2] On an identity for stochastic integrals. Theor. Prob. Appl. 17 A972) 717-720.
Nualart, D.
[1] Weak convergence to the law of two-parameter continuous processes. Z.W. 55 A981)
255-269.
[2] The Malliavin calculus and related topics. Springer, Berlin Heidelberg New York
1995.
Ocone, D.L.
[1] A symmetry characterization of conditionally independent increment martingales. In:
D. Nualart and M. Sanz (eds.) Proceedings of the San Felice workshop on Stochastic
Analysis A991). Birkhauser 1993, 147-167.
[2] Malliavin's calculus and stochastic integral representations of functionals of diffusion
processes. Stochastics and Stochastic Reports 12 A984) 161-185.
Orey, S.
[1] Two strong laws for shrinking Brownian tubes. Z.W. 63 A983) 281-288.
Orey, S., and Pruitt, W.
[1] Sample functions of the /V-parameter Wiener process. Ann. Prob. 1 A973) 138-163.
Orey, S., and Taylor, S.J.
[1] How often on a Brownian path does the law of the iterated logarithm fail? Proc.
London Math. Soc. 28 A974) 174-192.
Oshima, Y., and Takeda, M.
[1] On a transformation of symmetric Markov processes and recurrence property. Lecture
Notes in Mathematics, vol. 1250, Proceedings Bielefeld. Springer, Berlin Heidelberg
New York 1987.
Ouknine, Y., and Rutkowski, M.
[1] Local times of functions of continuous semimartingales. Stochastic Anal, and Appli-
Applications 13A995J11-232.
Pages, H.
[1] Personal communication.

Bibliography 579
Paley, R., Wiener, N., and Zygmund, A.
[1] Note on random functions. Math. Z. 37 A933) 647-668.
Papanicolaou, G., Stroock, D.W., and Varadhan, S.R.S.
[1] Martingale approach to some limit theorems. Proc. 1976. Duke Conf. On Turbulence.
Duke Univ. Math. Series III, 1977.
Parthasarathy, K.R.
[1] Probability measures on metric spaces. Academic Press, New York 1967.
Pazy, A.
[1] Semi-groups of linear operators and applications to partial differential equations. (Ap-
(Applied Mathematical Sciences, vol. 44). Springer, Berlin Heidelberg New York 1983.
Perkins, E.
[1] A global intrinsic characterization of Brownian local time. Ann. Prob. 9 A981) 800-
817.
[2] The exact Hausdorff measure of the level sets of Brownian motion. Z.W. 58 A981)
373-388.
[3] Local time is a semimartingale. Z.W. 60 A982) 79-117.
[4] Weak invariance principles for local time. Z.W. 60 A982) 437--451.
[5] Local time and pathwise uniqueness for stochastic differential equations. Sem. Prob.
XVI. Lecture Notes in Mathematics, vol. 920. Springer, Berlin Heidelberg New York
1982, pp. 201-208.
[6] On the Hausdorff dimension of Brownian slow points. Z.W. 64 A983) 369-399.
[7] The Cereteli-Davis solution to the H' -embedding problem and an optimal embedding
in Brownian motion. Seminar on stochastic processes. Birkhauser, Basel 1985, pp.
172-223.
Petit, F.
[1] Quelques extensions de la loi de l'arcsinus. C.R. Acad. Sci. Paris 315 Serie I A992)
855-858.
Pierre, M.
[1] Le probleme de Skorokhod: une remarque sur la demonstration d'Azema-Yor. Sem.
Prob. XIV. Lecture Notes in Mathematics, vol. 784. Springer, Berlin Heidelberg New
York 1980, pp. 392-396.
Pitman, J.W.
[1] One-dimensional Brownian motion and the three-dimensional Bassel process. Adv.
Appl. Prob. 7 A975) 511-526.
[2] A note on L2 maximal inequalities. Sem. Prob. XV. Lecture Notes in Mathematics,
vol. 850. Springer, Berlin Heidelberg New York 1981, pp. 251-258.
[3] Stationary excursions. Sem. Prob. XVI. Lecture Notes in Mathematics, vol. 1247.
Springer, Berlin Heidelberg New York 1987, pp. 289-302.
[4] Levy systems and path decompositions. Seminar on stochastic processes. Birkhauser,
Basel 1981, pp. 79-110.
[5] Cyclically stationary Brownian local time processes. Prob. Th. Ret. Fields 106 A996)
299-329.
[6] The SDE solved by local times of a Brownian excursion or bridge derived from the
height profile of a random tree or forest. To appear in Ann. Prob. A999).
[7] The distribution of local times of a Brownian bridge. Sem. Prob. XXXIII, Lect. Notes
in Mathematics. Springer, Berlin Heidelberg New York 1999.
Pitman, J.W., and Rogers, L.C.G.
[1] Markov functions. Ann. Prob. 9 A981) 573-582.

580 Bibliography
Pitman, J.W., and Yor, M.
[I] Bessel processes and infinitely divisible laws. In: D. Williams (ed.) Stochastic inte-
integrals. Lecture Notes in Mathematics, vol. 851. Springer, Berlin Heidelberg New York
1981.
[2] A decomposition of Bessel bridges. Z.W. 59 A982) 425^457.
[3] Sur une decomposition des ponts de Bessel. In: Fractional Analysis in Markov pro-
processes. Lecture Notes in Mathematics, vol. 923. Springer, Berlin Heidelberg New York
1982, pp. 276-285.
[4] The asymptotic joint distribution of windings of planar Brownian motion. Bull. Amer.
Math. Soc. 10 A984) 109-111.
[5] Asymptotic laws of planar Brownian motion. Ann. Prob. 14 A986) 733-779.
[6] Some divergent integrals of Brownian motion. In: D. Kendall (ed.) Analysis and
Geometric Stochastics, Supplement to Adv. Appl. Prob. A986) 109-116.
[7] Further asymptotic laws of planar Brownian motion. Ann. Prob. 17 C) A989) 965-
1011.
[8] Arcsine laws and interval partitions derived from a stable subordinator. Proc. London
Math. Soc. 65 C) A992) 326-356.
[9] Dilatations d'espace-temps, rearrangements des trajectoires browniennes, et quelques
extensions d'une identite de Knight. C.R. Acad. Sci. Paris, t 316, Serie I A993)
723-726.
[10] Decomposition at the maximum for excursions and bridges of one dimensional dif-
diffusions. In: N. Ikeda, S. Watanabe, M. Fukushima, H. Kunita (eds.) Ito's stochastic
calculus and Probability Theory. Springer, Berlin Heidelberg New York 1996, pp.
293-310.
[II] Quelques identites en loi pour les processus de Bessel. Hommage a P.A. Meyer and
J. Neveu. Asterisque 236, Soc. Math. France 1996.
[12] Ranked Functionals of Brownian excursions. Comptes Rendus Acad. Sci. Paris, t 326,
Serie I, January 1998, 93-97.
[ 13] Path decompositions of a Brownian bridge related to the ratio of its maximum and
amplitude. Preprint A999).
Port, S.C., and Stone, C.J.
[1] Brownian motion and classical potential theory. Academic Press, New York 1978.
Pratelli, M.
[1] Le support exact du temps local d'une martingale continue. Sem. Prob. XIII. Lecture
Notes in Mathematics, vol. 721. Springer, Berlin Heidelberg New York 1979, pp.
126-131.
Priouret, P.
[1] Processus de diffusion et equations differentielles stochastiques. Ecole d'Ete de Prob-
abilites de Saint-Flour III. Lecture Notes in Mathematics, vol. 390. Springer, Berlin
Heidelberg New York 1974, pp. 38-113.
Priouret, P., and Yor, M.
[1] Processus de diffusion a valeurs dans R et mesures quasi-invariantes sur C(R, R).
Asterisque 22-23 A975) 247-290.
Rauscher, B.
[1] Some remarks on Pitman's theorem. Sem. Prob. XXXI, Lect. Notes in Mathematics
1655. Springer, Berlin Heidelberg New York 1997, pp. 266-271.
Ray, D.B.
[1] Sojourn times of a diffusion process. 111. J. Math. 7 A963) 615-630.

Bibliography 581
Rebolledo, R.
[1] La methode des martingales appliquee a l'etude de la convergence en loi des processus
Memoire de la S.M.F. 62 A979)
[2] Central limit theorems for local martingales. Z.W. 51 A980) 269-286.
Resnick, S.I.
[1] Inverses of extremal processes. J. Appl. Prob. A974) 392-405.
Revuz, D.
[1] Mesures associees aux fonctionnelles additives de Markov I. Trans. Amer. Math. Soc.
148A970M01-531.
[2] Lois du tout ou rien et comportement asymptotique pour les probabilites de transition
des processus de Markov. Ann. I.H.P. 19 A983) 9-24.
[3] Markov chains. North-Holland, Amsterdam New York 1984.
[4] Une propriete ergodique des chaines de Harris et du mouvement Brownien lineaire
(unpublished).
Robbins, H., and Siegmund, D.
[1] Boundary crossing probabilities for the Wiener process and sample sums. Ann. Math.
Stat 41 A970) 1410-1429.
Rogers, L.C.G.
[1] Williams characterization of the Brownian excursion law: proof and applications. Sem.
Prob. XV. Lecture Notes in Mathematics, vol. 850. Springer, Berlin Heidelberg New
York 1981, pp. 227-250.
[2] Characterizing all diffusions with the 2M - X property. Ann. Prob. 9A981M61-572.
[3] Ito excursions via resolvents. Z.W. 63 A983) 237-255.
[4] A guided tour through excursions. Bull. London Math. Soc. 21 A989) 305-341.
[5] The joint law of the maximum and terminal value of a martingale. Prob. Th. Rel. F.
95 D) A993) 451^*66.
[6] Continuity of martingales in the Brownian excursion filtration. Prob. Th. Rel. F. 76
A987) 291-298.
Rogers, L.C.G., and Walsh, J.B.
[1] The intrinsic local time sheet of Brownian motion. Prob. Th. Rel. F. 88 A991) 363-
379.
[2] The exact |-variation of a process arising from Brownian motion. Stochastics and
Stoch. Reports 51 A994) 267-291.
[3] A(t, B,) is not a semi-martingale. Seminar on Stoch. processes 1990. Prog, in Prob.
24 A991) 275-283. Birkhauser.
[4] Local time and stochastic area integrals. Ann. Prob. 19 A991) 457-482.
Rogers, L.C.G., and Williams, D.
[1] Diffusions, Markov processes and Martingales, vol. 2: Ito calculus. Wiley and Sons,
New York 1987.
[2] Diffusions, Markov processes and Martingales, vol. 1: Foundations. Wiley and sons,
New York 1994.
Root, D.H.
[1] The existence of certain stopping times on Brownian motion. Ann. Math. Stat. 40
A969) 715-718.
Rosen, J.
[1] A local time approach to the self-intersection of Brownian paths in space. Comm.
Math. Phys. 88 A983) 327-338.

582 Bibliography
Rosen, J., and Yor, M.
[1] Tanaka formulae and renormalization for triple intersections of Brownian motion in
the plane. Ann. Prob. 19 A991) 142-159.
Roth, J.P.
[1] Operateurs dissipatifs et semi-groupes dans les espaces de fonctions continues. Ann.
Inst. Fourier 26 A976) 1-97.
Ruiz de Chavez, J.
[1] Le theoreme de Paul Levy pour des mesures signees. Sem. Prob. XVIII. Lecture Notes
in Mathematics, vol. 1059. Springer, Berlin Heidelberg New York 1984, pp. 245-255.
Saisho, Y., and Tanemura, H.
[1] Pitman type theorem for one-dimensional diffusion processes. Tokyo J. Math. 18 B)
A990) 429-440.
Schilder, M.
[1] Asymptotic formulas for Wiener integrals. Trans. Amer. Math. Soc. 125 A966N3-85.
Schwartz, L.
[1] Le mouvement brownien sur R", en tant que semi-martingale dans SN. Ann. I.H.P.
21, 1 A985) 15-25.
Seshadri, V.
[1] Exponential models, Brownian motion and independence. Can. J. of Stat. 16 A988)
209-221.
[2] The inverse Gaussian distribution. Clarendon Press, Oxford 1993.
Sharpe, M.J.
[1 ] Local times and singularities of continuous local martingales. Sem. Prob. XIV. Lecture
Notes in Mathematics, vol. 784. Springer, Berlin Heidelberg New York 1980, pp. 76-
101.
[2] Some transformation of diffusions by time reversal. Ann. Prob. 8 A980) 6, 1157-1162.
[3] General theory of Markov processes. Academic Press, New York 1989.
Shepp, L.A.
[1] Radon-Nikodym derivatives of Gaussian measures. Ann. Math. Stat. 37 A966) 321-
354.
[2] On the integral of the absolute value of the pinned Wiener process. Ann. Prob. 10
A982J34-239.
[3] The joint density of the maximum and its location for a Wiener process with drift.
J. Appl. Prob. 16 A979) 423-427.
Sheppard, P.
[1] On the Ray-Knight Markov properties of local times. J. London Math. Soc. 31 A985)
377-384.
Shi, Z.
[1] Lim inf behaviours of the windings and Levy's stochastic areas of planar Brownian
motion. Sem. Prob. XXVIII, Lecture Notes in Mathematics, vol. 1583. Springer, Berlin
Heidelberg New York 1994, pp. 122-137.
Shiga, Т., and Watanabe, S.
[1] Bessel diffusion as a one-parameter family of diffusion processes. Z.W. 27 A973)
37-46.

Bibliography 583
Skorokhod, A.
[1] Stochastic equation for diffusion processes in a bounded region, I and II. Theor. Prob.
Appl. 6 A961) 264-274; 7 A962) 3-23.
[2] Studies in the theory of random processes. Addison-Wesley, Reading, Mass. 1965.
[3] On a generalization of a stochastic integral. Theor. Prob. and Appl. 20 A975) 219-233.
Simon, B.
[1] Functional integration and quantum physics. Academic Press, New York 1979.
Smith, L., and Diaconis, P.
[1] Honest Bernoulli excursions. J. Appl. Prob. 25 C) A988) 464-^77.
Song, S.Q., and Yor, M.
[1] Inegalites pour les processus self-similaires arretes a un temps quelconque. Sem. Prob.
XXI. Lecture Notes in Mathematics, vol. 1247. Springer, Berlin Heidelberg New York
1987, pp. 230-245.
Spitzer, F.
[1] Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc.
87A958) 187-197.
[2] Recurrent random walk and logarithmic potential. Proc. Fourth Berkeley Symp. on
Math. Stat. and Prob. II. University of California 1961, pp. 515-534.
[3] Electrostatic capacity in heat flow and Brownian motion. Z.W. 3 A964) 110-121.
[These papers are reprinted in: Durrett, R. and Kesten, H. (eds.), Random walks,
Brownian motion and interacting particle systems. Birkhauser 1992.]
Stall, A.
[1] Self-repellent random walks and polymer measures in two dimensions. Doctoral dis-
dissertation. Bochum, 1985.
Strassen, V.
[1] An invariance principle for the law of the iterated logarithm. Z.W. 3 A964) 211-226.
Strieker, С
[1] Quasi-martingales, martingales locales, semi-martingales et filtrations. Z.W. 39 A977)
55-63.
[2] Sur un theoreme de H.J. Engelbert et J. Hess. Stochastics 6 A981) 73-77.
Strieker, C, and Yor, M.
[1] Calcul stochastique dependant d'un parametre. Z.W. 45 A978) 109-133.
Stroock, D.W.
[1] On the growth of stochastic integrals. Z.W. 18 A971) 340-344.
[2] Topics in stochastic differential equations. Tata Institute, Bombay 1982.
[3] Some applications of stochastic calculus to partial differential equations. Ecole d'Ete
de Probabilites de Saint-Flour XI, Lecture Notes in Mathematics, vol. 976. Springer,
Berlin Heidelberg New York 1983, pp. 268-382.
[4] An introduction to the theory of large deviations. (Universitext). Springer, Berlin
Heidelberg New York 1984.
[5] Probability Theory. An analytic view. Cambridge Univ. Press 1994.
Stroock, D.W., and Varadhan, S.R.S.
[1] Multidimensional diffusion processes. Springer, Berlin Heidelberg New York 1979.
Stroock, D.W., and Yor, M.
[1] On extremal solutions of martingale problems. Ann. Sci. Ecole Norm. Sup. 13 A980)
95-164.

584 Bibliography
[2] Some remarkable martingales. Sem. Prob. XV. Lecture Notes in Mathematics, vol.
850. Springer, Berlin Heidelberg New York 1981, pp. 590-603.
Sussman, H.J.
[1] An interpretation of stochastic differential equations as ordinary differential equations
which depend on the sample point. Bull. Amer. Math. Soc. 83 A977) 296-298.
[2] On the gap between deterministic and stochastic ordinary differential equations. Ann.
Prob. 6A978) 19-41.
Sznitman, A.S., and Varadhan, S.R.S.
[1] A multidimensional process involving local time. Proc. Th. Rel. F. 71 A986) 553-579.
Takaoka, K.
[1] On the martingales obtained by an extension due to Saisho, Tanemura and Yor of
Pitman's theorem. Sem. Prob. XXXI, Lect. Notes in Mathematics 1655. Springer,
Berlin Heidelberg New York 1997, pp. 256-265.
Tanaka, H.
[1] Note on continuous additive functional of the 1-dimensional Brownian path. Z.W. 1
A963J51-257.
[2] Time reversal of random walks in one dimension. Tokyo J. Math. 12 A989) 159-174.
[3] Time reversal of random walks in W. Tokyo J. Math. 13 A990) 375-389.
[4] Diffusion processes in random environments. Proc. 1CM (S.D. Chatterji, ed.). Birk-
hauser, Basel 1995, pp. 1047-1054.
Taylor, S.J.
[1] The a-dimensional measure of the graph and the set of zeros of a Brownian path.
Proc. Cambridge Phil. Soc. 51 A955) 265-274.
Toby, E., and Werner, W.
[1] On windings of multidimensional reflected Brownian motion. Stoch. and Stoch. Re-
Reports, vol. 55 A995), pp. 315-327.
Trotter, H.F.
[1] A property of Brownian motion paths. 111. J. Math. 2 A958) 425-433.
Truman, A.
[1] Classical mechanics, the diffusion heat equation and Schrodinger equation. J. Math.
Phys. 18A977J308-2315.
Truman, A., and Williams, D.
[1] A generalized Arcsine law and Nelson's mechanics of one-dimensional time-homo-
time-homogeneous diffusions. In: M. Pinsky (ed.) Diffusion processes and related problems in
Analysis I. Birkhauser, Boston 1990, pp. 117-135.
Tsirel'son, B.
[1] An example of a stochastic differential equation having no strong solution. Theor.
Prob. Appl. 20 A975) 427-430.
[2] Triple points: from non-brownian filtrations to harmonic measures. Geom. Funct. Anal.
(GAFA), vol. 7 A997) 1071-1118.
Uppman, A.
[1] Sur le flot d'une equation differentielle stochastique. Sem. Prob. XVI. Lecture Notes
in Mathematics, vol. 920. Springer, Berlin Heidelberg New York 1982, pp. 268-284.

Bibliography 585
Vallois, P.
[1] Le probleme de Skorokhod sur R: une approche avec le temps local. Sem. Prob. XVII.
Lecture Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1983,
pp. 227-239.
[2] Sur la loi du maximum et du temps local d'une martingale continue uniformement
integrable. Proc. London Math. Soc. 3 F9) A994) 399-427.
[3] Sur la loi conjointe du maximum et de l'inverse du temps local du mouvement brown-
ien: application a un theoreme de Knight. Stochastics and Stochastics Reports 35
A991) 175-186.
[4] Une extension des theoremes de Ray-Knight sur les temps locaux browniens. Prob.
Th. Rel. Fields 88 A991) 443-482.
[5] Diffusion arretee au premier instant oil l'amplitude atteint un niveau donne. Stoch.
and Stoch. Reports 43 A993) 93-115.
[6] Decomposing the Brownian path via the range process. Stoch. Proc. and their Appl.
55A995J11-226.
Van den Berg, M., and Lewis, J.T.
[1] Brownian motion on a hypersurface. Bull. London Math. Soc. 17 A985) 144-150.
Van Schuppen, J.H., and Wong, E.
[1] Transformations of local martingales under a change of law. Ann. Prob. 2 A974)
879-888.
Ventsel, A.D.
[1] Rough limit theorems on large deviations for Markov processes 1 and 2. Theory Prob.
Appl. 21 A976) 227-242 and 499-512.
Ventsel, A.D., and Freidlin, M.I.
[1] On small random pertubations of dynamical systems. Russ. Math. Surv. 25 A970)
1-55.
[2] Some problems concerning stability under small random perturbations. Theor. Prob.
Appl. 17 A972) 269-283.
Vershik, A.
[1] Decreasing sequence of measurable partitions and their applications. Soviet Math.
Dokl. 11,4A970) 1007-1011.
[2] The theory of decreasing sequences of measurable partitions. Saint-Petersburg Math. J.
6A994O05-761.
Vervaat, W.
[ 1 ] A relation between Brownian bridge and Brownian excursion. Ann. Prob. 7 A) A979)
141-149.
Volkonski, V.A.
[1] Random time changes in strong Markov processes. Theor. Prob. Appl. 3 A958) 310—
326.
von Weizsacker, H.
[1] Exchanging the order of taking suprema and countable intersection of ст-algebras.
Ann. I.H.P. 19A983)91-100.
Vuolle-Apiala, J., and Graversen, S.E.
[1] Duality theory for self-similar processes. Ann. I.H.P. 23, 4 A989) 323-332.
Walsh, J.B.
[1] A property of conformal martingales. Sem. Prob. XI. Lecture Notes in Mathematics,
vol. 581. Springer, Berlin Heidelberg New York 1977, pp. 490-492.
[2] Excursions and local time. Asterisque 52-53, Temps locaux A978) 159-192.

586 Bibliography
[3] A diffusion with a discontinuous local time. Asterisque 52-53, Temps locaux A978)
37-45.
[4] The local time of the Brownian sheet. Asterisque 52-53, Temps locaux A978) 47-62.
[5] Downcrossings and the Markov property of local time. Asterisque 52-53, Temps
locaux A978) 89-116.
[6] Propagation of singularities in the Brownian sheet. Ann. Prob. 10 A982) 279-288.
[7] Stochastic integration with respect to local time. Seminar on Stochastic Processes
1989. Birkhauser 1983, pp. 237-302.
[8] Some remarks on A(t, B,). Sem. Prob. XXVII, Lect. Notes in Mathematics 1557.
Springer, Berlin Heidelberg New York 1993, pp. 173-176.
Wang, A.T.
[1] Quadratic variation of functionals of Brownian motion. Ann. Prob. 5 A977) 756-769.
[2] Generalized Ito's formula and additive functionals of the Brownian path. Z.W. 41
A977) 153-159.
Warren, J.
[ 1 ] Branching processes, the Ray-Knight theorem and sticky Brownian motion. Sem. Prob.
XXXI, Lect. Notes in Mathematics 1655. Springer, Berlin Heidelberg New York 1997,
pp. 1-15.
[2] On the joining of sticky Brownian motion. Sem. Prob. XXXIII, Lect. Notes in Math-
Mathematics. Springer, Berlin Heidelberg New York 1999.
Warren, J., and Yor, M.
[1] Skew-products involving Bessel and Jacobi processes. Sem. Prob. XXXIV, Lect. Notes
in Mathematics. Springer, Berlin Heidelberg New York 2000.
[2] Identifying the Brownian burglar. Sem. Prob. XXXIV, Lect. Notes in Mathematics.
Springer, Berlin Heidelberg New York 2000.
[3] The Brownian burglar: conditioning Brownian motion by its local time process. Sem.
.Prob. XXXII, Lect. Notes in Mathematics 1686. Springer, Berlin Heidelberg New
York 1998, pp. 328-342.
Watanabe, S.
[1] On time-inversion of one-dimensional diffusion processes. Z.W. 31 A975) 115-124.
[2] A limit theorem for sums of i.i.d. random variables with slowly varying tail probability.
P.R. Krishnaia (ed.) Multivariate analysis, vol. 5. North-Holland, Amsterdam 1980,
pp. 249-261.
[3] Generalized Arcsine laws for one-dimensional diffusion processes and random walks.
In: M. Cranston and M. Pinsky (eds.) Proceedings of Symposia in pure mathematics,
vol. 57. Amer. Math. Soc, Providence, Rhode Island A995) 157-172.
[4] Bilateral Bessel diffusion processes with drift and time inversion. To appear A998).
[5] Asymptotic windings of Brownian motion paths on Riemann surfaces. Preprint A998).
[6] The existence of a multiple spider martingale in the natural filtration of a certain
diffusion in the plane. Sem. Prob. XXXIII, Lect. Notes in Mathematics. Springer,
Berlin Heidelberg New York 1999.
Weber, M.
[1] Analyse infinitesimale de fonctions aleatoires. Ecole d'ete de Saint-Flour XI-1981.
Lecture Notes in Mathematics, vol. 976. Springer, Berlin Heidelberg New York 1983,
pp. 383-465.
Weinryb, S.
[1] Etude d'une equation differentielle stochastique avec temps local. Sem. Prob. XVII.
Lecture Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1983,
pp. 72-77.

Bibliography 587
Weinryb, S., and Yor, M.
[1] Le mouvement brownien de Levy indexe par M3 comme limite centrale de temps
locaux d'intersection. Sem. Prob. XXII. Lecture Notes in Mathematics, vol. 1321.
Springer, Berlin Heidelberg New York 1988, pp. 225-248.
Wendel, J.W.
[1] Hitting spheres with Brownian motion. Ann. Prob. 8 A980) 164-169.
[2] An independence property of Brownian motion with drift. Ann. Prob. 8 A980) 600-
601.
Widder, D.V.
[1] The Heat Equation. Academic Press 1975.
Wiener, N.
[1] Differential space. J. Math. Phys. 2 A923) 132-174.
[2] The homogeneous chaos. Amer. J. Math. 60 A930) 897-936.
Williams, D.
[1] Markov properties of Brownian local time. Bull. Amer. Math. Soc. 76 A969) 1035—
1036.
[2] Decomposing the Brownian path. Bull. Amer. Math. Soc. 76 A970) 871-873.
[3] Path decomposition and continuity of local time for one dimensional diffusions I.
Proc. London Math. Soc. C) 28 A974) 738-768.
[4] A simple geometric proof of Spitzer's winding number formula for 2-dimensional
Brownian motion. University College, Swansea A974).
[5] On a stopped Brownian motion formula of H.M. Taylor. Sem. Prob. X. Lecture Notes
in Mathematics, vol. 511. Springer, Berlin Heidelberg New York 1976, pp. 235-239.
[6] On Levy's downcrossing theorem. Z.W. 40 A977) 157-158.
[7] Diffusions, Markov processes and Martingales, vol. 1: Foundations. Wiley and Sons,
New York 1979.
[8] Conditional excursion theory. Sem. Prob. XIII. Lecture Notes in Mathematics, vol.
721. Springer, Berlin Heidelberg New York 1979, pp. 490-494.
[9] Brownian motion and the Riemann zeta function. In: D. Welsh and G. Grimmett
(eds.) Disorder in Physical Systems. Festschrift for J. Hammersley. Oxford 1990, pp.
361-372.
[10] Probability with Martingales. Cambridge Univ. Press 1990.
Wong, E.
[1] The construction of a class of stationary Markoff processes. In: Stoch. Processes in
Math. Physics and Engineering. Proc. of Symposia in Applied Maths., vol. XVI. Am.
Math. Soc. 1964, pp. 264-276.
Wong, E., and Zakai, M.
[ 1 ] The oscillation of stochastic integrals. Z.W. 4 A965) 103-112.
Yamada, T.
[1] On a comparison theorem for solutions of stochastic differential equations. J. Math.
Tokyo Univ. 13 A973) 497-512.
[2] On some representations concerning the stochastic integrals. Proba. and Math. Statis-
Statistics 4, 2 A984) 153-166.
[3] On the fractional derivative of Brownian local times. J. Math. Kyoto Univ. 21, 1
A985) 49-58.
[4] On some limit theorems for occupation times of one-dimensional Brownian motion
and its continuous additive functionals locally of zero energy. J. Math. Kyoto Univ.
26, 2 A986) 309-222.

588 Bibliography
[5] Representations of continuous additive functionals of zero energy via convolution
type transforms of Brownian local time and the Radon transform. Stochastics and
Stochastics Reports 46, 1 A994) 1-15.
[6] Principal values of Brownian local times and their related topics. In: N. Ikeda, S.
Watanabe, M. Fukushima, H. Kunita (eds.) Ito's stochastic calculus and Probability
theory. Springer, Berlin Heidelberg New York 1996, pp. 413^422.
Yamada, Т., and Ogura, Y.
[ 1 ] On the strong comparison theorem for the solutions of stochastic differential equation
Z.W. 56 A981) 3-19.
Yamada, Т., and Watanabe, S.
[1] On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto
Univ. 11 A971) 155-167.
Yamazaki, Y.
[1] On limit theorems related to a class of "winding-type" additive functionals of complex
Brownian motion. J. Math. Kyoto Univ. 32 D) A992) 809-841.
Yan, J.A.
[1] A propos de l'integrabilite uniforme des martingales exponentielles. Sem. Prob. XVI.
Lecture Notes m Mathematics, vol. 920. Springer, Berlin Heidelberg New York 1982,
pp. 338-347.
[2] Sur un theoreme de Kazamaki-Sekiguchi. Sem. Prob. XVII. Lecture Notes in Mathe-
Mathematics vol. 986. Springer, Berlin Heidelberg New York 1983, pp. 121-122.
Yan, J.A., and Yoeurp, C.
[1] Representation des martingales comme integrates stochastiques de processus option-
nels. Sem. Prob. X. Lecture Notes in Mathematics, vol. 511. Springer, Berlin Heidel-
Heidelberg New York 1976, pp. 422-431.
Yoeurp, С
[1] Complements sur les temps locaux et les quasi-martingales. Asterisque 52-53, Temps
locaux A978) 197-218.
[2] Sur la derivation des integrates stochastiques. Sem. Prob. XIV, Lecture Notes in
Mathematics, vol. 784. Springer, Berlin Heidelberg New York 1980, pp. 249-253.
[3] Contribution au calcul stochastique. These de doctorat d'etat, Universite de Paris VI,
1982.
[4] Theoreme de Girsanov generalise et grossissement de filtrations. In: Grossissement
de filtrations: exemples et applications. Lecture Notes in Mathematics, vol. 1118.
Springer, Berlin Heidelberg New York 1985, pp. 172-196.
Yor, M. (ed.)
[1] Exponential functionals and principal values related to Brownian motion. Bib. Revista
Mat. Ibero-Americana, 1997.
Yor, M.
[1] Sur les integrates stochastiques optionnelles et une suite remarquable de formules
exponentielles. Sem. Prob. X. Lecture Notes in Mathematics, vol. 511. Springer, Berlin
Heidelberg New York 1976, pp. 481-500.
[2] Sur quelques approximations d'integrales stochastiques. Sem. Prob. XI. Lecture Notes
in Mathematics, vol. 528. Springer, Berlin Heidelberg New York 1977, pp. 518-528.
[3] Formule de Cauchy relative a certains lacets browniens. Bull. Soc. Math. France 105
A977K-31.
[4] Sur la continuite des temps locaux associes a certaines semi-martingales. Asterisque
52-53, Temps locaux A978) 23-36.

Bibliography 589
[5] Un exemple de processus qui n'est pas une semi-martingale. Asterisque 52—53 A978)
219-221.
[6] Sous-espaces denses dans Ll et Hl et representation des martingales. Sem. Prob. XII.
Lecture Notes in Mathematics, vol. 649. Springer, Berlin Heidelberg New York 1978,
pp. 264-309.
[7] Les filtrations de certaines martingales du mouvement brownien dans R". Sem. Prob.
XIII. Lecture Notes in Mathematics, vol. 721. Springer, Berlin Heidelberg New York
1979, pp. 427-440.
[8] Sur le balayage des semi-martingales continues. Sem. Prob. XIII. Lecture Notes in
Mathematics, vol. 721. Springer, Berlin Heidelberg New York 1979, pp. 453^71.
[9] Sur l'etude des martingales continues extremales. Stochastics 2 A979) 191-196.
[10] Loi de l'indice du lacet brownien et distribution de Hartman-Watson. Z.W. 53 A980)
71-95.
[11] Remarques sur une formule de P. Levy. Sem. Prob. XIV. Lecture Notes in Mathe-
Mathematics, vol. 784. Springer, Berlin heidelberg New York 1980, pp. 343-346.
[12] Sur un processus associe aux temps locaux browniens. Ann. Sci. Univ. Clermont-
Ferrand II, 20 A982) 140-148.
[13] Sur la transformee de Hilbert des temps locaux browniens et une extension de la
formule d'lto. Sem. Prob. XVI. Lecture Notes in Mathematics, vol. 920. Springer,
Berlin Heidelberg New York 1982, pp. 238-247.
[14] Le drap brownien comme limite en loi de temps locaux lineaires. Sem. Prob. XVII.
Lecture Notes in Mathematics, vol. 986. Springer, Berlin Heidelberg New York 1983,
pp. 89-105.
[15] Une inegalite optimale pour le mouvement brownien arrete a un temps quelconque.
C.R. Acad. Sci. Paris, Ser. A 296 A983) 407^09.
[16] A propos de l'inverse du mouvement brownien dans M" (n > 3). Ann. I.H.P. 21, 1
A985) 27-38.
[17] Renormalisation et convergence en loi pour les temps locaux d'intersection du mou-
mouvement brownien dans R3. Sem. Prob. XIX. Lecture Notes in Mathematics, vol. 1123.
Springer, Berlin Heidelberg New York 1985, pp. 350-365.
[18] Sur la representation comme integrate stochastique des temps locaux du mouvement
brownien dans R". Sem. Prob. XX. Lecture Notes in Mathematics, vol. 1204. Springer,
Berlin Heidelberg New York 1986, pp. 543-552.
[19] De nouveaux resultats sur l'equation de Tsirel'son. C.R. Acad. Sci. Paris, Serie I 309,
A989M11-514.
[20] Remarques sur certaines constructions des mouvements browniens fractionnaires.
Sem. Prob. XXII. Lecture Notes in Mathematics, vol. 1321. Springer, Berlin Hei-
Heidelberg New York 1988, pp. 217-224.
[21] On stochastic areas and averages of planar Brownian motion. J. Phys. A. Math. Gen.
22 A989) 3049-3057.
[22] Etude asymptotique des nombres de tours de plusieurs mouvements browniens com-
complexes corretes. In: Durrett, R. and Kesten, H. (eds.), Random walks, Brownian motion
and interacting particle systems. Birkhauser, Basel 1992, pp. 441^455.
[23] Some aspects of Brownian motion. Part 1: Some special functional. Lectures in Math-
Mathematics ETH Zurich. Birkhauser 1992. Part II: Some recent martingale problems. Lect.
Notes in Maths. ETH Zurich. Birkhauser 1997.
[24] The distribution of Brownian quantiles. J. Appl. Prob. 32 A995) 405^16.
[25] Random Brownian scaling and some absolute continuity relationships. In: E. Bolt-
hausen, M. Dozzi, F. Russo (eds.) Progress in probability, vol. 36. Birkhauser A995)
243-252.
[26] Inegalites de martingales continues arretees a un temps quelconque, I, II. In: Grossisse-
ments de filtrations: exemples et applications. Lect. Notes in Mathematics 1118.
Springer, Berlin Heidelberg New York 1985, pp. 110-171.

590 Bibliography
[27J Sur ccrtaincs fonctionncllcs exponentielles du mouvement brownien reel. J. Appl.
Prob. 29 A992) 202-208.
[28] On some exponential functional of Brownian motion. Adv. App. Prob. 24 A992)
509-531.
Zaremba, P.
[1] Skorokhod problem-elementary proof of the Azema-Yor formula. Prob. Math. Stat. 6
A985) 11-17.
Zvonkin, A.K.
[1] A transformation of the phase space of a process that removes the drift. Math. USSR
Sbornik2A974) 129-149.

Index of Notation
a.s., a.e.
BM
BB
BES*, BES*(*), BES11
BESQ*, BESQ"
ВМО, ВМОр
CAF
Cont. semi. mart.
Ск(Е)
Cq(E)
DDS
%*
e(f,g), ex(p,b)
F.V.
Space of finite variation processes 119
Almost sure, almost surely, almost everywhere
Brownian sheet 39
Brownian motion 19
d-dimensional Brownian motion 20
d-dimensional Brownian motion started at x 20
Brownian Bridge 37
Bessel processes of dimension S, of index v 445
Squares of Bessel processes 440
Bessel Bridge 463
Square of Bessel Bridge 463
Space of martingales with bounded mean
oscillation 75
Continuous additive functional 401
Continuous semimartingale 127
Space of continuous functions with compact
support on the space E 289
Space of continuous functions with limit 0
at infinity 88, 281
Space of differentiable functions on a product
space
Dambi s-Dubins-Schwarz 181
ст-field of Borel sets and space of Borel functions
Space of positive Borel functions
Space of bounded Borel functions
Space of universally measurable functions
Exponential martingales 148, 149
Stochastic differential equations 366
17
Finite variation 5, 119
a -field of the stopping time T
44

592 Index of Notation
¦J^"X Right-continuous and complete filtration
generated by X 98
¦^[v Completion of .J^" with respect to v 93
gt, gt(x) Density of centered Gaussian variables
with variance t 17
g,{w), d,{co) Last zero before t, first zero after t 239
H2 Space of L2-bounded martingales 129
H2 Space of L2-bounded continuous martingales 129
Hq Space of L2-bounded continuous martingales
vanishing at 0 129
HP, Я1 59
Д ст-field of invariant events 423
К ¦ M, К ¦ X
/о KsdXs
L2(M), 5t2(M),
L20C(M)
ЩХ)
L\ogL
loc. mart.
log2 = log log
Stochastic integrals 138,
Spaces of processes 137
140
Family of local times of X
Class of martingales 58
Local martingale 123
Iterated logarithm 56
222
Ms Martingale additive functional 149, 284
M*, M* = sups<, |Afj| Bilateral supremum 54
N,Nf,N,M,mN Kernel notation 80
С Optional a -field 172
ODE. Ordinary differential equation 382
OU Ornstein-Uhlenbeck process 37
¦V Predictable a-field 171
PRP Predictable representation property 209
P, Semi-group 80
PT Law of X at time T 104
p^(x,y) Density of Bessel semigroup 446
P^° Law of Bessel bridge 463
P* Law of Bessel process 445
PPP Poisson point process 474
Qsx Law of Square of Bessel process 440
Qsxa Law of Square of Bessel Bridge 463
qf(x, y) Density of Square of Bessel semigroup 441

Index of Notation 593
R Life-time of an excursion 480
r.v. Random variable 2
SDE. Stochastic differential equation 366
S Supremum process of BM: S, — sups</ Bs, 54
t.f. Transition function 80
U,Us Spaces of excursions 480
¦M, Ш& a -fields of spaces of excursions 480
W Wiener measure 35
W, Wd Wiener space 35
J& Space of paths 500
X(c) Scaled process 535
Z Set of zeros of BM 109
в,, вт Shift operators 36
a (X,, t G 7") ст-field generated by the random variables
{X,,teT} 1
A, S Cemetery 84
| Л | Modulus of subdivision Л 4
n{x,a,b) Martingale problem 296
Va Measure associated with the additive
functional A 410
(M, M), (X, X) Brackets 120, 124, 125, 128
(M,N), (X,Y)
= Equality in law 10
—* Convergence in the sense of finite distributions 516
P-lim Convergence in probability 10
1 д Characteristic function of the set A 1
<] Absolute continuity on .Wt 325

Index of Terms
Absorbing point 84, 97
Action functional 342
Adapted process 42
Additive functional 401
continuous - 401
strong - 402
integrable - 410
сг-integrable - 410
signed- 419
Arcsine law
First- 112
Second - 242
Area
Stochastic- 196,396
Associativity of stochastic integrals 139
Asymptotic a -field 99
Atom 76
Augmentation
usual - of a filtration 45
Bachelier's equation 268
Backward
equation 282
integral 144
Bernstein's inequality 153
Bessel
bridges 463
processes 445
squared - - process 440
Bismut description of Ito's measure
502
Blumenthal zero one law 95
BMO-martingales 75
Bougerol's identity 388
Boundary
entrance - 395
natural - 305
Bracket
of two continuous local martingales
125
of two semimartingales 128
Bridge
Brownian- 37, 154, 384
Bessel - 463
Squared Bessel - 463
Brownian - as conditioned Brownian
motion 41
pseudo-Brownian - 248
Brownian
motion 19
J-dimensional - 20
motion, standard linear - 19
Bridge 37, 154, 384
filtrations
motion with drift 73, 352
sheet 39
СЛ")- - motion 97
motion, skew - 87, 292
excursion 480
morion, reflected - 86, 238
motion, killed - 87
meander 493
motion on the sphere 530
Burkholder-Davis-Gundy inequalities 160
Cadlag 34
Cameron-Martin
formula 371, 445
space 339
Canonical
process 34
realization 92
Cauchy
r.v. 13
process 116
Cemetery 84
Chacon-Omstein theorem 548
Chain rule 6
Chaos, Wiener 201,207
Chapman-Kolmogorov equation 80
Characteristic measure of a Poisson point
process 475

596
Index of Terms
Chernov inequality 292
Chung-Jacobi-Riemann identity 509
Clark's formula 341
Comparison theorems 393
Conformal
local martingale 189
invariance of Brownian motion 190
Continuous process 17
Convergence
weak - 10
in distribution 516
in the sense of finite distributions 516
in law 10
Cooptional time 313
Covariance 294
Cramer transform 343
Dambis 181
Debut 46
Deviations of Brownian motion, large -
345
Differentiability of BM, non - 32
Diffusion
process 294
coefficient 294
Discrete
P.P. 471
ст-discrete P.P. 471
Distribution, convergence in - 516
Dominated process 162
Donsker's theorem 518
Doob's inequality 54
Downcrossings 60
Doss-Siissmann method 382
Drift, Brownian motion with - 73, 352
coefficient 294
Dubins-Schwarz 181
Dubins inequality 66
Dynkin's operator 310
Elastic BM 408
Energy 527
Enlargement
of a probability space 182
of a filtration 363
Entrance
boundary 305
law 494
Ergodic theorem 427, 548
for BM 548
Excessive
measure 409
function 423
Excursions
intervals 109
process 480
normalized - 486
Explosions, criterion for - 383
Exponential
formulas 476
local martingales 148
inequality 153
holding point 97
Extension
Kolmogorov - - theorem 34
Extremal
martingales 213
probability measures 210
process 406
Fefferman's inequality 76
Feller
process 90
semi-groups 88
property, strong 423
Feynman-Kac formula 358
Filtered space 41
Filtering 175, 207
Filtration 41
Brownian -
complete - 45
natural - 42
right-continuous - 42
usual augmentation of a - 45
Fine topology 98
Finite dimensional distributions 18
of BM 23
convergence in the sense of - 516
Fokker-Planck equation 282
Forward equation 282
Fractional Brownian motion 38
Fubini's theorem for stochastic integrals
175
Galmarino's test 47
Garsia-Neveu lemma 170
Gaussian
martingales 133, 186
random variable 11
process 36
measure 16
space 12
Markov process 86
Gebelein's inequality 205
Generator
infinitesimal - 281
extended infinitesimal - 285

Index of Terms 597
Girsanov
theorem 327
transformation 329
pair 329
Good X inequality 164
Gronwall's lemma 543
Hardy's inequality 75, 155
Heat process 20
Hermite polynomials 151
Holder properties
of BM 28, 30
ofsemimarts. 187
of local times 237
Holding
point 97
exponential - time 97
Homogeneous
Markov process 81
transition function 80
Hypercontractivity 206
Image of a Markov process 87
Independent, increments 96
Indistinguishable 19
Index
of a stable law 115
of a Bessel process 440
Inequality
Burkholder-Davis-Gundy - 160
exponential - 153
Chernov - 292
Dubins - 66
Bernstein - 153
Fefferman - 76
Gebelein - 205
Doob's - 54
Good X - 164
Hardy's- 155
Kunita-Watanabe - 127
maximal - 53
Infinitely divisible 115
Infinitesimal generator 281
extended - 285
Initial distribution 81
Innovation process 175
Instantaneously reflecting 307
Integral, stochastic - 138, 140, 141
Integration by parts formula 146
Invariant
measure 409
function 423
tr-field 423
events 423
Irregular points 98
Iterated logarithm, law of the 56
Ito
integral 138
formula 147
Tanaka formula 223
processes 298
measure of excursions 482
Kailath-Segal identity 159
Kazamaki's criterion 331
Kernel 79
Killed 100
Brownian motion 87
Knight's
theorem 183
asymptotic version of - 524
identity 504
Kolmogorov's continuity criterion
Kunita-Watanabe inequality 127
19
Lamperti's relation 452
Langevin's equation 378
Large numbers for local martingales
law of- 186
Large deviations of Brownian motion 345
Last exit times 408
Law
of a process 34
convergence in - 10
uniqueness in - 367
Lebesgue theorem for stochastic integrals
142
Levy
characterization theorem 150, 158
measure 115
Khintchine formula 115
process 96
L logL class 58
Limit-quotient theorem 427
Linear continuous Markov processes 300
Linear stochastic equation 377
Localization 123
Local extrema of BM 113
Local martingales 123
exponential - 148
pure - 212
standard - 213
conformal - 189
Local time
of a continuous semimartingale 222
of Brownian motion 238
Locally bounded process 140

598
Index of Terms
Markov
process 81
property 83, 94
property, strong 202
property of BM, strong - 102, 156
processes, linear 300
Markovian 84
Martingale 51
problem 296
Master formula 475
Maximum principle, positive 283
Meander, Brown ian - 493
Measure
associated with an additive functional
410
characteristic - of a Poisson point
process 475
Ito - 482
Levy- 115
Wiener - 35
Measurable process 126
Modification 19
Modulus of continuity, Levy's 30
Monotone class theorem 2
Natural boundary 305
Newtonian potential 100
Normalized excursion 486
Novikov's criterion 332
Occupation
time 20,401
times formula 224
Operateur carre du champ 351
Optional
a -field 172
process 172
projection 173
stopping theorem 69
Ornstein-Uhlenbeck process 37
rf-dimensional - 360
Orthogonal martingales 145
Papanicolaou-Stroock-Varadhan theorem
526
Parameter
of a OU process 37
of a Poisson process 58,471
Path decomposition 255,318
Pathwise uniqueness 367
Pitman's theorem 253
Point process 471
discrete - 471
a -discrete - 471
Points of increase of Brownian motion
461
Poisson
process 471
point process 474
(•^")-point process 474
Polar
functions for BM 24
sets 191
Polarization 124
Potential kernel 100
Newtonian - 100
Predictable 47
a -field 47, 171
process 171
projection 173
stopping 76
stopping time 172
representation property 209
Process
adapted - 42
cadlag - 34
continuous - 17
increasing - 119
finite-variation - 119
locally bounded - 140
measurable - 126
optional - 172
predictable- 171
progressively measurable - 44
Prokhorov' s criterion 10, 516
Pseudo-Brownian bridge 248
Pure martingales 212
Quadratic variation 28
of Brownian motion 29
of local mart. 124
ofsemimarts. 128
Quasi-invariance of Wiener measure 339
Quasi-left continuity 101
Quasi-martingales 134
Ray-Knight theorems 454
Recurrence of Brownian motion 58, 192
Recurrent
Harris - 425
process 424
set 424
Reduced Gaussian random variable 11
Reflecting
instantaneously - 307
slowly - 307
Brownian motion 86, 239

Index of Terms
599
Reflection
principle 105
stochastic differential equation with -
385
Regular
linear Markov process 300
points 98
Reproducing kernel Hilbert space 21, 39,
339
Resolvent equation 89
Scale function 302
of Bessel processes 442, 446
Scaling invariance
of Brownian motion 21
of local time 244
of Bessel process 446
Section theorem 172
Semimartingale 127
Sheet, Brownian 39
Size of a OU process 37
Skew Brownian motion 292
Skew-product 194
Skorokhod's lemma 239
Slowly reflecting 307
Space-time Markov process 85
Speed measure 304
Stable
random variable 115
process 116
process, symmetric 116
subspace of Hi 174
Standard
Brownian motion 19, 97
local martingale 213
State space 15
Stationary
process 36
independent increments 96
Stochastic
integral 138, 140, 141
differential equation 366
area 196, 396
process 15
Stopped process 44
Stopping
time 42
optional - 69
Strassen's law of the iterated logarithm
346
Stratonovitch integral 144
Strong
Markov property 102
Markov property, extended — 111
solution 367
Submarkovian 84
Submartingales 51
Subordinator 117
Supermartingale 51
Support of local times 235
Symmetric
local times 234
stable processes 116
Tanaka, Ito-Tanaka formula 223
Terminal time 491
Time
change 180
entry- 43
hitting - 43
inversion 359
reversal 312
stopping - 42
Transient
set 424
process 424
Transition
function 80
probability 80
Trap 97
Tsirel'son's example 392
Upcrossings 60
Variation 5
finite - process 119
quadratic - 28
p-variation of BM 33
Vector
local martingale, semimartingale 147
Ventcell-Freidlin estimates 343
Version 18
Watanabe's process 504
Weak convergence 10
Weak solution to a stochastic differential
equation 367
Wiener
measure 35
space 35
chaos 201,207
Williams
path decomposition theorem 318
description of Ito's measure 499
Zvonkin method 384

Catalogue
Additivity property of squared Bessel processes 440
Approximations of L, and 5, 227, 233
Asymptotic properties for the transition function of the Brownian
motion 100
Central-limit theorem for stochastic integrals 160
Conditioning with respect to last exit times 408
Continuous local martingales on ]0, oo[ 135, 157
Continuous local martingales on stochastic intervals 136
Criterions for an exponential local martingale to be a
martingale 331,332,338
Deterministic semimartingales are functions of bounded variation 133, 145
Differentiation of stochastic integrals 143
Discontinuous local time for
i) a semimartingale 238
ii) a diffusion 420
Exceptional points for the law of the iterated logarithm 59
Exponentials of semimartingales 149
An exponential local martingale which is not a martingale 335
An extremal martingale which is not pure 214
Functions of the BM which are martingales 74
A Fourier transform proof of the existence of occupation densities 166
A Girsanov transform of a BM with a strictly smaller nitration 397
A Girsanov transform of a pure martingale which is not pure 393
Hausdorff dimension of the set of zeros of Brownian motion 247

602 Catalogue
The intervals of constancy are the same for a continuous local martingale and for
its bracket 125
Joint law of (fa, LfJ 265
/^-inequalities for local times 265
Law of large numbers for local martingales 186
A local martingale with strong integrability properties is not necessarily a martin-
martingale 194
A local martingale is Gaussian iff its bracket is deterministic 186
Martingales M with independent DDS BM /f and bracket {M, M)
Minkowski inequality for local martingales 136
A 0 — 1 law for additive functionals of BM 422
The planar Brownian curve: its Lebesgue measure is zero 23, 196
Polar functions for the BM2: they include the functions of bounded
variation 197
Polarity for the Brownian sheet 198
Principal values for Brownian local times 236
A progressive set which is not optional 175
Ray-Knight theorems for local times of Bessel processes 459
A uniformly integrable martingale which is not in H1 75
A semimartingale X = M + A such that .Vх С ,ТМ strictly 259
Semimartingale functions of BM 419
Skew-product representation of Brownian motion 194
A slowly reflecting boundary 421
Solutions to linear SDE's 378, 381
Support of local times 235
Times at which the BM is equal to its supremum 113
A bounded (.^B)-martmgale which is the stochastic integral of an unbounded
process of L2(B) 269
A Brownian filtration sandwiched between •5rB and J^"|B| 208
Kazamaki's criterion is not a necessary condition 384
The time spent by the Brownian Bridge on the positive half-line is uniformly
distributed 493

Grundlehren der mathematischen Wissenschaften
A Series of Comprehensive Studies in Mathematics
A Selection
212. Switzer: Algebraic Topology - Homotopy and Homology
215. Schaefer: Banach Lattices and Positive Operators
217. Stenstrom: Rings of Quotients
218. Gihman/Skorohod: The Theory of Stochastic Processes II
219. Duvaut/Lions: Inequalities in Mechanics and Physics
220. Kirillov: Elements of the Theory of Representations
221. Mumford: Algebraic Geometry I: Complex Protective Varieties
222. Lang: Introduction to Modular Forms
223. Bergh/Lofstrom: Interpolation Spaces. An Introduction
224. Gilbarg/Trudinger: Elliptic Partial Differential Equations of Second Order
225. Schiitte: Proof Theory
226. Karonbi: K-Theory. An Introduction
227. Grauert/Remmert: Theorie der Steinschen Raume
228. Segal/Kunze: Integrals and Operators
229. Hasse: Number Theory
230. Klingenberg: Lectures on Closed Geodesies
231. Lang: Elliptic Curves. Diophantine Analysis
232. Gihman/Skorohod: The Theory of Stochastic Processes III
233. Stroock/Varadhan: Multidimensional Diffusion Processes
234. Aigner: Combinatorial Theory
235. Dynkin/Yushkevich: Controlled Markov Processes
236. Grauert/Remmert: Theory of Stein Spaces
237. Kothe: Topological Vector Spaces II
238. Graham/McGehee: Essays in Commutative Harmonic Analysis
239. Elliott: Probabilistic Number Theory I
240. Elliott: Probabilistic Number Theory II
241. Rudin: Function Theory in the Unit Ball of Cn
242. Huppert/Blackburn: Finite Groups II
243. Huppert/Blackburn: Finite Groups III
244. Kubert/Lang: Modular Units
245. Cornfeld/Fomin/Sinai: Ergodic Theory
246. Naimark/Stern: Theory of Group Representations
247. Suzuki: Group Theory I
248. Suzuki: Group Theory II
249. Chung: Lectures from Markov Processes to Brownian Motion
250. Arnold: Geometrical Methods in the Theory of Ordinary Differential Equations
251. Chow/Hale: Methods of Bifurcation Theory
252. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampere Equations
253. Dwork: Lectures on p-adic Differential Equations
254. Freitag: Siegelsche Modulfunktionen
255. Lang: Complex Multiplication
256. Hormander: The Analysis of Linear Partial Differential Operators I
257. Hormander: The Analysis of Linear Partial Differential Operators II
258. Smoller: Shock Waves and Reaction-Diffusion Equations
259. Duren: Univalent Functions
260. Freidlin/Wentzell: Random Perturbations of Dynamical Systems
261. Bosch/Giintzer/Remmert: Non Archimedian Analysis - A System Approach
to Rigid Analytic Geometry
262. Doob: Classical Potential Theory and Its Probabilistic Counterpart
263. Krasnosel'skii/Zabreiko: Geometrical Methods of Nonlinear Analysis
i?LA AnKin/rvllinQ* Differential Inclusions

265. Grauert/Remmert: Coherent Analytic Sheaves
266. de Rham: Differentiable Manifolds
267. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. I
268. Arbarello/Cornalba/Griffiths/Harris: Geometry of Algebraic Curves, Vol. II
269. Schapira: Microdifferential Systems in the Complex Domain
270. Scharlau: Quadratic and Hermitian Forms
271. Ellis: Entropy, Large Deviations, and Statistical Mechanics
272. Elliott: Arithmetic Functions and Integer Products
273. Nikol'skii: Treatise on the Shift Operator
274. Hormander: The Analysis of Linear Partial Differential Operators 111
275. Hormander: The Analysis of Linear Partial Differential Operators IV
276. Ligget: Interacting Particle Systems
277. Fulton/Lang: Riemann-Roch Algebra
278. Barr/Wells: Toposes, Triples and Theories
279. Bishop/Bridges: Constructive Analysis
280. Neukirch: Class Field Theory
281. Chandrasekharan: Elliptic Functions
282. Lelong/Gruman: Entire Functions of Several Complex Variables
283. Kodaira: Complex Manifolds and Deformation of Complex Structures
284. Finn: Equilibrium Capillary Surfaces
285. Burago/Zalgaller: Geometric Inequalities
286. Andrianaov: Quadratic Forms and Hecke Operators
287. Maskit: Kleinian Groups
288. Jacod/Shiryaev: Limit Theorems for Stochastic Processes
289. Manin: Gauge Field Theory and Complex Geometry
290. Conway/Sloane: Sphere Packings, Lattices and Groups
291. Hahn/O'Meara: The Classical Groups and K-Theory
292. Kashiwara/Schapira: Sheaves on Manifolds
293. Revuz/Yor: Continuous Martingales and Brownian Motion
294. Knus: Quadratic and Hermitian Forms over Rings
295. Dierkes/Hildebrandt/Kuster/Wohlrab: Minimal Surfaces 1
296. Dierkes/Hildebrandt/Kuster/Wohlrab: Minimal Surfaces II
297. Pastur/Figotin: Spectra of Random and Almost-Periodic Operators
298. Berline/Getzler/Vergne: Heat Kernels and Dirac Operators
299. Pommerenke: Boundary Behaviour of Conformal Maps
300. Orlik/Terao: Arrangements of Hyperplanes
301. Loday: Cyclic Homology
302. Lange/Birkenhake: Complex Abelian Varieties
303. DeVore/Lorentz: Constructive Approximation
304. Lorentz/v. Golitschek/Makovoz: Construcitve Approximation. Advanced Problems
305. Hiriart-Urruty/Lemarechal: Convex Analysis and Minimization Algorithms 1.
Fundamentals
306. Hiriart-Urruty/Lemarechal: Convex Analysis and Minimization Algorithms II.
Advanced Theory and Bundle Methods
307. Schwarz: Quantum Field Theory and Topology
308. Schwarz: Topology for Physicists
309. Adem/Milgram: Cohomology of Finite Groups
310. Giaquinta/Hildebrandt: Calculus of Variations I: The Lagrangian Formalism
311. Giaquinta/Hildebrandt: Calculus of Variations II: The Hamiltonian Formalism
312. Chung/Zhao: From Brownian Motion to Schrodinger's Equation
313. Malliavin: Stochastic Analysis
314. Adams/Hedberg: Function Spaces and Potential Theory
315. Biirgisser/Clausen/Shokrollahi: Algebraic Complexity Theory
316. Saff/Totik: Logarithmic Potentials with External Fields
317. Rockafellar/Wets: Variational Analysis
318. Kobayashi: Hyperbolic Complex Spaces
319. Bridson/Haefliaer: Metric Spaces of Non-Positive Curvature

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