SEMI ARTINGALE THEORY
0 STOCHASTIC CALCULUS
Sheng-wu He Jia-gang Wang Jia-an Yan
llll
SCIENCE PRESS
Qf&\ CRC PRESS INC.

Semimartingale Theory and Stochastic Calculus

Semimartingale Theory
and Stochastic Calculus
Sheng-wu He Jia-gang Wang Jia-an Yan
llll
SCIENCE PRESS
Beijing New York
CRC PRESS INC.
Boca Raton Ann Arbor London Tokyo

Sheng-wu He
Department of Mathematical Statistics
East China Normal University
Shanghai 200062, P. R. China
Jia-gang Wang
Institute of Applied Mathematics
East China University of Chemical Technology
Shanghai 200237, P. R. China
Jia-an Yan
Institute of Applied Mathematics
Chinese Academy of Sciences
Beijing 100080, P. R. China
Copyright ©1992 by Science Press and CRC Press, Inc.
Published by Science Press
16 Donghuangchenggen North Street
Beijing 100707, China
* Distribution right throughout the world, excluding
China, Hong Kong and Macao, is granted to CRC Press Inc. Boca Raton
Printed in Hong Kong
All rights reserved. No part of this publication may be
reproduced, stored in a retrieval system, or
transmitted in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without the
prior written permission of the copyright owners.
Library of Congress Cataloging in Publication Data
He. S, (Sheng-wu)
Semimartingale Theory and Stochastic Calculus / by
S. He, J. Wang, J. Yan
p. cm.
ISBN 0-8493-7715-3
1. Semi martingales, 2. Stochastic analysis.
I. Wang, Chia-kang. II. Yan, J. (Jia-an). III. Title.
QA 274.5.H4. 1992
519.2'87-de 2o 91-42567
CIP
ISBN 7-03-003066-4/O-568 Science Press, Beijing
ISBN 0-8493-7715-3 CRC Press Inc., Boca Raton

To Zhen-zhu Tu, Ji-qing You, Lan Gu

Preface
Semimartingales constitute the largest class of integrator-processes
with respect to which stochastic integrals can be reasonably defined.
Stochastic calculus based on the semimartingales is one of the major branches
of modern probability theory. It is not only an important tool for
researches in diverse branches of probability (Markov processes and
diffusion processes, stochastic point processes, statistics of stochastic processes,
stochastic filtering and control etc.), but also has broad applications to
certain branches of mathematics (partial differential equations, harmonic
analysis, differential geometry etc.) and theoretic physics. It is now fil-
tring gradually into engineering, biology, financial mathematics and other
fields as well. In recent years, several monographs on stochastic
calculus have appeared, but most of them stress on a certain topic, and none
of them gives a sufficient discussion on semimartingale theory. It is the
purpose of this book to give a systematic and overall exposition of the
fundamental theory of semimartingales and stochastic calculus, and to
illustrate some of their applications. We hope that it provides a reference
for probabilists, statisticians and engineers, and its main parts also may
be used as a textbook for graduate students. To this end, the materials
for this book have been carefully selected. The text is supplemented and
further developed in the form of problems. Indeed, a large part of these
problems may be used as exercises. By working out these exercises readers
can deepen their understanding of the text and broaden their thinking.
But the others are difficult to be solved. They serve as complements to
the text.
The book contains 16 chapters. The first ten chapters are an
elaborate revision based on the book "An Introduction to Martingale Theory
and Stochastic Integrals" (in Chinese) written by J. A. Yan, one of the
authors, and published in 1981. The last six chapters reflects the new
developments of semimartingales and stochastic calculus in the 1980s.
Naturally, the selection of the material is partly influenced by the authors'
interests of research. Here is a brief survey of the main contents. For the
convenience of the reader, preliminaries, necessary for reading the book,
are given in Chapter 1, of which some are difficult to be found in ordinary
books on measure theory or probability theory. Chapter 2 contains the

Vlll
Preface
main results of classical martingale theory. Chapters 3-5 are devoted to
the exposition of what is commonly called "the general theory of stochastic
processes". This theory is not only an important basis for semimartingales
and stochastic calculus, but also indispensible for studying many specific
types of stochastic processes, such as Markov processes, point processes.
Beginning with Chapter 6, the fundamental theory of semimartingales
and stochastic calculus is developed. Two most important classes of
uniformly integrable martingales are discussed in Chapter 6: martingales
with integrable variation and square integrable martingales. In
Chapter 7 local martingales are introduced and their jumps are characterized.
Semimartingales and quasimartingales and their elementary propertis are
discussed in Chapter 8. Chapter 9, which is devoted to stochastic
integrals and relative topics, is undoubtedly the highlight of the book. It
contains the essence of stochastic calculus, e.g., Ito formula, Doleans-Dade
exponential formula, Lenglart's inequality and local times. A short
introduction to stochastic differential equations is also included. Though the
stochastic differential equation is an important topic of stochastic
calculus, we have no room to develop it fully in the book. Besides, a number of
monographs on this subject are available. Chapter 10 is concerned with
H^martingales and BMO-martingales, in which a series of main
martingale inequalities are established. In Chapter 11, the predictable
characteristics and integral representation of semimartingales are introduced. They
are the generalization of the classical results of processes with independent
increments. Changes of measures presented in Chapter 12 are one of the
key techniques of stochastic calculus. The characterization of
semimartingales as the only class of reasonable integrator-processes is established in
this chapter. Chapter 13 presents the predictable integral representation
of martingales, which is also useful, such as for filtering. Chapter 14
contributes to the problems of absolute continuity and singularity,
contiguity, entire separation and convergence in variation of measures. These
problems have been studied since the 1960s. By semimartingale approach
the satisfactory solutions have been obtained finally. Chapter 16 deals
with the weak convergence theory of semimartingales, and the
preliminaries to weak convergence of stochastic processes are given in Chapter 15.
Stochastic calculus not only provides a completely new method for weak
convergence theory of stochastic processes, but also gives more elaborate
results. Throughout the book, special interests are put on two basic types
of processes: processes with independent increments and step processes,
encountered frequently in applied probability and statistics.
It is supposed that readers have had a basic training in measure
theory and advanced probability theory. However, no particular knowledge of

Preface
IX
stochastic processes is required. A familiarity with the key concepts and
some intuitive grounds of stochastic processes will certainly be helpful. An
enormous literature has been devoted to semimartingales and stochastic
calculus.We have no attempt to list all related papers in references and to
make suitable historical notes. In this respect, the reader may refer to C.
Dellacherie and P. A. Meyer's voluminous book "Probabilites et Poten-
tiel" and J. Jacod and A. N. Shiryaev's "Limit Theorems for Stochastic
Processes".
Sincere thanks are due to P. A. Meyer, who had led authors into the
field of semimartingales and stochastic calculus, and provided a great deal
of support and encouragement with enthusiasm constantly. We would like
to extend our warm appreciation to those, who have offered useful advice
and help in our research work at various stages, in particular to J. Azema,
P. D. Chen, C. S. Chou, C. Dellacherie, M. Emery, G. L. Gong, Z. Y.
Huang, J. Jacod, Z. M. Ma, Y. M. Pan, P. Protter, C. Strieker, S. R.
Wang, Ch. Yoeurp, M. Yor and W. A. Zheng. Many chapters of this book
have been taught many times by the authors in the last decade. We would
like to thank our audience for their contributions to improvements of the
book. The writting of the book is supported by National Natural Science
Foundation of China. We are grateful for having had the grant.
Shanghai and Beijing S. W. He, J. G. Wang, J. A. Yan
December 1991

Table of Contents
Chapter I. Preliminaries 1
§1. Monotone Class Theorems 3
§2. Uniform Integrability 6
§3. Essential Suprema 8
§4. The Generalization of Conditional Expectation 10
§5. Analytic Sets and Choquet Capacity 14
§6. Lebesgue-Stieltjes Integrals 20
Problems and Complements 23
Chapter II. Classical Martingale Theory 27
§1. Elementary Inequalities 27
§2. Convergence Theorems 36
§3. Decomposition Theorems for Supermartingales 42
§4. Doob's Stopping Theorem 45
§5. Martingales with Continuous Time 50
§6. Processes with Independent Increments 63
Problems and Complements 74
Chapter III. Processes and Stopping Times 79
§1. Stopping Times 79
§2. Progressively Measurable, Optional and Predictable
Processes 85
§3. Predictable and Accessible Times 92
§4. Processes with Finite Variation 99
§5. Changes of Time 102
Problems and Complements 106

Xll
Table of Contents
Chapter IV. Section Theorems and Their Applications 110
§1. Section Theorems 110
§2. a.s. Foretellability of Predicatable Times 117
§3. Totally Inaccessible Times 120
§4. Complete Filtrations and the Usual Conditions 124
§5. Applications to Martingales 130
Problems and Complements 132
Chapter V. Projections of Processes 135
§1. Projections of Measurable Processes 135
§2. Dual Projections of Increasing Processes 140
§3. Applications to Stopping Times and Processes 153
§4. Doob-Meyer Decomposition Theorem 157
§5. Filtrations of Discrete Type 160
Problems and Complements 173
Chapter VI. Martingales with Integrable Variation and
Square Integrable Martingales 175
§1. Martingales with Integrable Variation 175
§2. Square Integrable Martingales 177
§3. The Structure of Purely Discontinuous Square Integrable
Martingales 181
§4. Quadratic Variation 185
Problems and Complements 189
Chapter VII. Local Martingales 191
§1. The Localization of Classes of Processes 191
§2. The Decomposition of Local Martingales 196
§3. The Characterization of Jumps of Local Martingales 204
Problems and Complements 207
Chapter VIII. Semimartingales and Quasimartingales — 209
§1. Semimartingales and Special Semimartingales 209
§2. Quasimartingales and Their Rao Decompositions 213
§3. Semimartingales on Stochastic Sets of Interval Type 217
§4. Convergence Theorems for Semimartingales 220
Problems and Complements 223

Table of Contents xiii
Chapter IX. Stochastic Integrals 226
§1. Stochastic Integrals of Predictable Processes
with Respect to Local Martingales 227
§2. Compensated Stochastic Integrals of Progressive Processes
with Respect to Local Martingales 231
§3. Stochastic Integrals of Predictable Processes
with Respect to Semimartingales 234
§4. Lenglart's Inequality and Convergence Theorem for ♦
Stochastic Integrals 237
§5. Ito's Formula and Doleans-Dade Exponential Formula 243
§6. Local Times of Semimartingales 251
§7. Stochastic DiflFerential Equations: Metivier-PellaumaiTs
Method 256
Problems and Complements 261
Chapter X. Martingale Spaces H1 and BMO 265
§1. ^-martingales and BAtO-Martingales 265
§2. Fefferman's Inequality 272
§3. The Dual Space of Hl 274
§4. Davis Inequalities 277
§5. Burkholder-Davis-Gundy Inequality 281
§6. Martingale Space Hp,p > 1 286
§7. John-Nirenberg Inequality 287
Problems and Complements 291
Chapter XI. The Characteristics of Semimartingales ... 293
§1. Random Measures 293
§2. The Integral Representation of Semimartingales 305
§3. Levy Processes 311
§4. Step Processes 320
Problems and Complements 328
Chapter XII. Changes of Measures 332
§1. Local Absolute Continuity 332
§2. Girsanov's Theorems for Local Martingales and
Semimartingales 338
§3. Girsanov's Theorems for Random Measures 347
§4. The Characterization for Semimartingales 353
Problems and Complements 359

XIV
Table of Contents
Chapter XIII. Predictable Representaton Property 362
§1. The Strong Property of Predictable Representation 362
§2. The Weak Property of Predictable Representation 368
§3. The Relation between Two Kinds of Predictable
Representation Properties 378
§4. The Predictable Representation Property of Levy Processes . 388
Problems and Complements 392
Chapter XIV. Absolute Continuity and Contiguity
of Measures 396
§1. Hellinger Processes 396
§2. Absolute Continuity and Singularity 404
§3. Contiguity, Entire Separation and Convergence in Variation . 412
§4. Measures Induced by Levy Processes 427
Problems and Complements 434
Chapter XV. Weak Convergence for Cadlag Processes .. 437
§1. D[0, oo[ and Skorokhod Topology 437
§2. Continuity for Skorokhod Topology 451
§3. Weak Convergence and Tightness 456
§4. Weak Convergence of Step Processes 468
Problems and Complements 477
Chapter XVI. Weak Convergence for Semimartingales ... 481
§1. Convergence to a Quasi-left-continuous Semimartingale 481
§2. Convergence to a Levy Process 497
§3. Convergence to a Continuous Levy process 507
§4. Convergence to a Generalized Diffusion 516
Problems and Complements 525
References 529
Index 539

Chapter I
Preliminaries
Readers are assumed to be familiar with the fundamentals of measure
theory and probability theory in Loeve'1' or Neveu'1', such as measure
extension theorem,' Radon-Nikodym theorem, dominated convergence
theorem, Fatou's lemma, lAspace, Holder's inequality, conditional
mathematical expectation, Jensen's inequality, conditional independence,
product probability space and Fubini's theorem. In this chapter we give some
supplements, necessary for reading this book.
Throughout this book we use the following common notations.
N = the set of all non-negative integers.
7V = 7Vu{+oo}.
R = ] — oo, +oo[ *) (real line).
R+ = [0,oo[.
R = [—00, +00] (extended real line).
B+ = [0,+oo].
Q (resp. Q+) = the set of all rationals in R (resp. -R+).
B(R) ( resp.B(#+)) = the Borel a-field in R (resp. R+).
Let fibea set. A mapping from CI to R (resp. R) is called a (resp.
extended) real-valued function on fi. As restricted on fi, a set always
means a subset of fi. The union and intersection of sets A and B are
denoted by AUB and AC\B (or simply AB) respectively. The complement
of A is denoted by Ac. A\B = ABC is the difference between A and B.
AAB = (A\B) U (B\A) is the symmetric difference of A and B. Empty
set is denoted by 0. The indicator of a set A is denoted by I a-
ja(U) = {h u,eA^
M ' \o, ueAc.
x) ]a,6[= (a, 6), ]a,6] = (a, 6], [a, b[= [a, 6), -00 < a < 6 < +00.

2
Chapter I Preliminaries
The set of all elements uj having the property P is denoted by {u G ft :
P(u;)}, or {uj : P(cj)}, or simply [P], if no ambiguity will be produced. For
example, if / and g are two extended real functions on ft, [/ > g] stands
for {uj G ft : f(uj) > g(uj)}. Let (An) be a sequence of sets. An | A means
(An) is monotone increasing and A = \JAn. Correspondingly, An [ A
n
means (An) is monotone decreasing and A = f)An. The superior limit
n
of (An) is denoted by limni4n or [An i.o.]. The inferior limit of (An) is
denoted by limn An. If limnAn = lin^ An happens, we use the notation
limn An for this set.
A collection of subsets of ft is called a class on ft. Let T be a class on
ft, and A be a set. ^* fl A stands for {AB : B G f}, the trace of ^* on A.
Define
^ = {LUn : An G ^}, Tb = {f]An : An G ^}.
n n
They are the minimal classes, containing T and closed under the formation
of countable union and intersection respectively.
Let C be a class on ft. We denote by a(C) the a-field on ft, generated
by C. Let (Gi)iel be a family of classes on ft. We define <r(Gi,i € I) =
(r(UGi).
iei
Let (£, £) be a measurable space, and / be a mapping from ft to E.
We denote by a(f) the a-field f-\E) = {f~l(A) : A G £}, induced by /
in ft. If p is an extended real function on £*, g G £ (resp. f"1", resp. 6£,
resp. b£~*~) means that g is an ^-measurable (resp. non-negative, resp.
bounded, resp. non-negative and bounded) function.
Let (E{, £i)iei be a family of measurable spaces. For each i G J, fi is a
mapping from ft to E{. Then cr(cr(fi), i G /) is also denoted by o"(/i, i G /).
Let ^i be a class on fti, i = 1,2. Define
Ti®Jr2 = {AxB\ Aefi, Be f2}.
For A C fti x ft2, the projection of A onto fti is defined as
iri(A) = {uj\ G fti : 3uj2 G ft2 such that (c^i,^) G A}.
If (ftj, Ji), i = 1,2, are two measurable spaces, the product a-field of T\
and T2 is denoted by T\ x J2 = <t( Ji ® J2).
Let / and g be two extended real functions. Then / V g stands for
sup(/,p), and fAg for inf(/,p). Hence, /+ = /VO, /" = (-/)V0 = -(/A
0). More generally, V and A stand for supremum and infimum respectively.

§1. Monotone Class Theorems
3
For example, if (fn)neN is a sequence of extended real functions, then
V fn = supn /n, A fn = infn fn- Moreover, if (Ti)iei is a family of a-fields
on ft, then \J Ti = a(\J Ji), A ?i = f) ft-
iei iei iei iei
On the real line iZ, s | t means s —* t, s < t, and s || t means
s —► t, s < t. For a sequence of real numbers (sn), sn | t and sn TT t
imply that (sn) is a monotone increasing sequence in addition. Symbols
| and || have similar meanings. Let (/n) be a sequence of extended real
functions. /n T / (resp. fn | /) means that (/n) is monotone increasing
(resp. decreasing) and / = limn fn.
In general, (ft,^", P) denotes a probability space. The mathematical
expectation of a random variable (abbreviated as r.v.) £ is denoted by
2£[£] if it makes sense. 1^(0,^, P), p > 1, denotes the Banach space of
all p-th power integrable r.v. with p-norm: ||£||p = (E\£\p)l/p.
§1. Monotone Class Theorems
1.1 Definition. Let ft be a set, and C be a class on ft. C is called a
n-class if A,B eC => AB G C. C is called a X-class if
i) ft G C,
ii) A, fl e C, AcB=> £\A G C,
iii) An G C, An|A=>AG C.
C is called a monotone class if
An G C, An T A or An | A => A G C.
Obviously, a A-class is a monotone class. If C is both a 7r-class and a
A-class, or both a field and a monotone class, then C is a a-field.
The following theorem is the monotone class theorem in terms of sets.
1.2 Theorem. Let C,T be two classes on ft, and Ccf.
1) If T is a \-class and C is a it-class, then a(C) C T.
2) If T is a monotone class and C is a field, then a(C) C T.
Proof. 1) The intersection T1 of all A-classes containing C is still a
A-class (called the A-class generated by C). Put
fi = {BGf': VA G C, B n A G 7"}.

4
Chapter I Preliminaries
Obviously, T\ is a A-class and C C F\. Hence, T' = T\. Put
Equally, J2 is a A-class and Cc/i- Hence, T' = T2, and T' is a 7r-class.
Then T' is a a-field, and <r(C) Cf'Cf
2) The intersection ^ of all monotone classes containing C is still a
monotone class (called the monotone class generated by C). Similarly one
can show that T1 is a 7r-class. Put
T" = {B e T1 : Bce T'}.
Then T is a monotone class, and C C T". Hence, T" = T'. This means
^ is a field. Therefore, ^ is a cr-field, and a(B )c/cf. □
As a simple application of Theorem 1.2, we have
1.3 Corollary. 1) Let (fi,^*, P) be a probability space, and £ and rj
be integrable r.v. Suppose that C dT and C is a ir-class. IfE£ = Erj and
for each AeC, E[£Ia] = E[t)Ia[, then
E[£\v(C)] = E[r,\v(C)] a.s. (3.1)
2) Let (fi,^*) be a measurable space, Ccf, and C be a ir-class.
Suppose that /i and v are two bounded signed measures with /i(fi) = v(Q). //
for each A G C, jjl(A) = v(A), then, being restricted on <r(C), /i is identical
with v.
Proof. We only show 1). The proof of 2) is similar. Put
G = {AeT: E[SIA] = E[VIA}}.
Then Q is a A-class and C C Q by the assumption. By Theorem 1.2.1),
<r(C) C 5, and (3.1) follows. □
The following theorem is the monotone class theorem in terms of
functions, corresponding to Theorem 1.2.1), and will be used frequently later.
1.4 Theorem. Let C be a ir-class on CI, and H be a linear space formed
by some real functions on fi. // the following conditions are satisfied:
i) 1 € W,
ii) fn € W, 0 < fn | /, / 25 finite (resp. bounded) ^ f G 7Y,
iii) A G C =► IA G H,
then H contains all a(C)-measurable real (resp. bounded) functions on fi.
Proof. Put T = {A C fi : I a € W}. It is easy to see that T is a A-class,
and C C T by the assumption. Prom Theorem 1.2.1), we have a(C) C ^*.

§1. Monotone Class Theorems
5
Let £ be a a-(C)-measurable real (resp. bounded) function on ft, and
put
n2n k
£n = Z) ;^[fc/2n<£<(fc+l)/2"]-
fc=0 L
Then £n G 7Y, 0 < £n | £+, and £+ G 7Y by the condition ii). By the same
argument, we have £~ G 7Y. Hence f = f+ — f ~" € W. □
Applying monotone class theorems is one of our most useful techniques.
Readers must manipulate them masterly. In general, details about
applying monotone class theorems will be omitted. As an example, we will
show a characterization of o-(/)-measurable functions by using Theorem
1.4. This characterization is very useful in probability theory, and is called
Doob's measurability theorem.
1.5 Theorem. Let f be a mapping from CI to a measurable space
(£, £), and ip be a (resp. extended, resp. bounded) real function on ft.
In order for <p to be a(f)-measurable, it is necessary and sufficient for an
£-measurable (resp. extended, resp. bounded) real function h exists on E
such that </? = /io/, i.e., ip(v) = h(f(uj)).
Proof. The sufficiency is trivial. We are going to show the necessity.
Put
W = {ho f : h is an £ -measurable real function on E}.
Then H is a linear space, and 1 G H. Suppose hn o / g 7Y, 0 < hn o / j ip,
and V> is finite. Put
A = {x G E : sup/in(x) < oo}.
n
Then A G £, and /(ft) C A. Put
x G A,
(supn/in(x),
0,
h(x) =
xeAc.
Obviously, h is an £-measurable real function on E, and ip = hof. Hence,
tp G H. Now H satisfies the condition ii) in Theorem 1.4. Let D G cr(f).
There exists B G £ such that D = f~l(B). Hence, ID = IB o f e H.
By Theorem 1.4, H contains all o-(/)-measurable real functions. This
means that if ip is a o-(/)-measurable real function, then there exists an
£ -measurable real functon h on E such that <p = h o /. Furthermore, if ip
is bounded: \<p\ < c, we take /i' = /i+Ac-/i"Ac, and (p = h' o / remains
true. At last, suppose <p is a o-(/)-measurable extended real function.

6
Chapter I Preliminaries
Then (p' = arctan^ is a a(/)-measurable real function, and \<p'\ < 7r/2.
So there exists an £ -measurable real function h! on E such that (p' = h'of.
Obviously, h = tan h! is an £ -measurable extended real function on E, and
(p = ho f. □
§2. Uniform Integrability
1.6 Definition. Let (fij, Ji,Pt), i G J, be a family of probability
spaces, & G I/1 (ft*, Fi,Pi), i G J. H = {£i,i G /} is called a uniformly
integrable family if
lim sup / \Gi\dpi = °-
When (fij, ^i, Pj) = (fi,^*, P), H is also called a uniformly integrable
family on (fi,^*, P).
1.7 Theorem. Le£ (fti,^i,Pi), i £ I, be a family of probability
spaces, fain} C Ll(nuTuPi), i G J, W = {&,t G /}, A' = {^t G /}.
1) If X is a uniformly integrable family, and for each i G /, |£t| < \t)i\
Pi-a.s., then H is also a uniformly integrable family.
2) Ifii G IP (toil Fi,Pi), i G J, p > 1, and supi€/ £i[|&|p] < oo, then
H is a uniformly integrable family.
Proof. 1) follows immediately from Definition 1.6.
2) Put a = supi€/ £?i[|£i|p]. For any c > 0, we have
■%l>c] J[\ti\>c] & l & l & l
According to Definition 1.6, H is a uniformly integrable family. □
1.8 Theorem. Let (fi,^*, P) be a probability space, £ be an integrable
r.v., and (Gi)iel be a family of sub-a-fields of T. Then (i£[£|£i])i€/ ^s a
uniformly integrable family.
Proof. Set rji = E[\£\\Gi]. For any c> 0 we have
Pirn > c) <-E[Vi) = ±E[\£\], iel
c c
and
S[vi>c] VidP = f[vi>c] \S\dP < 6P(Vi >c) + Jm>6] \s\dP
< 6-E[\t\] + /KM \£\dP.

§2. Uniform Integrability
cp f) ^ 0 such that f„^ rt \f\dP <
For any given e > 0, take 6 > 0 such that J[\^\>s] \Z\dP < «• When c >
—i£[|£|], we have /r >ci 7/idP < £, i G 7. This means that (r/i)^/ is a
uniformly integrable family. By Theorem 1.7.1), we know that (i?i[£|£i])i€/
is a uniformly integrable family. □
1.9 Theorem. Let (fij, Ti, Pi), i e I, be a family of probability spaces,
& G L1(fii,^ri,Pi), i e I. Then H = {&, i e 1} is a uniformly integrable
family if and only if the following conditions are satisfied:
i) a = supi€/jEi[|&|] <oo, •
ii) For any given € > 0, there exists 6 > 0 such that for each A G !F%
with P{(A) < 6 we have
L
\Zi\dPi < e (9.1)
M
i.e., limtf_osupie/sup{j4.P.(i4)<tf} fA \b\dPi = 0.
Proof. Necessity. For any A E Ti and c > 0 we have
/ UdPi < cPi(A) + / IfcldJV i € /. (9.2)
Since H is uniformly integrable, one can take c sufficiently large such that
for each i € /, /[i^i>c] l&M-Pi < ~- h* (9-2) putting A = Q yields the
Li
condition i), and putting 6 = — yields the condition ii).
LiC
Sufficiency. For any given e > 0, choose 6 > 0 such that the condition
ii) holds. When c > a/6 we have
Pi(\^>c)<\Ei[M<^<S.
Then from (9.1) for each i G I we have
\£i\dPi < e.
l[\ti\>4
This means that H is uniformly integrable. □
1.10 Corollary. Suppose both (£i)iei and (r/i)iej are uniformly
integrable families, so is (& + rji)iej.
The following theorem gives a criterion for Z^-convergence, and reveals
the importance of uniform integrability.
1.11 Theorem. Suppose (£n) is a sequence of integrable r.v.y and £
is a real r.v. Then £n —► £ if and only if (£n) is uniformly integrable, and
•Ai*

8 Chapter I Preliminaries
Proof. Necessity. Suppose £n—► £. It is well-known that L1-conver-
gence implies convergence in probability. For any A G T, we have
[\£n\dP< /KldP + l^n-el]. (11.1)
Ja Ja
For given e > 0, choose a positive integer N such that for n > AT, E[\£n —
£|] < e/2, and <5 > 0 such that for A G T
P(A)<6=> j^\dP<S- and jfj&|dP < |, n < TV.
Then by (11.1) we see that for Aef,
P(A) < <5 =► Vn, / |£n|dP < e.
At the same time, we also have supn i?[|£n|] < oo. Hence, by Theorem 1.9
we conclude that (£n) is uniformly integrable.
Sufficiency. Suppose (£n) is uniformly integrable, and £n—► £. By
Fatou's lemma, we have E[\£\] < suPn ^[|£n|] < °°, ie-> £ ls integrable.
Then (£n — £) is uniformly integrable (by Corollary 1.10). For any given
e > 0, from Theorem 1.9 we see that there exists 6 > 0 such that for any
AeF
P(A) < 6 =► / |£n - £\dP < e, n > 1.
Ja
Take AT sufficiently large such that for all n > N we have
P(\Sn-Z\>e)<6.
Hence, for n > AT,
El\Zn - CI] = /Kn^|>e, Kn - C|dP + /Km^|<e, |*» - £|<*P
< e + e = 2e.
This means £n —► £. □
Uniformly integrable families of r.v. will be encountered frequently in
this book. Further materials about uniform integrability can be found in
Problems and Complements of this chapter.
§3. Essential Suprema
1.12 Definition. Let (fi,.F, P) be a probability space, and H be a
non-empty family of r.v. . A r.v. r/ is called the essential supremum of H
if 77 satisfies the following conditions:

§3. Essential Suprema
9
i) For all £ G H, £ < 77 a.s.,
ii) If 7}' is another r.v. satisfying i), i.e., for all £ G 7Y, £ < r/' a.s., then
J) <r)' a.s..
It is easy to see that if the essential supremum of H exists, it must be
unique up to a P-null set. We denote it by esssup^€^£ or ess sup H.
Reversing the symbols of inequality in i) and ii), we obtain the
definition of essential infimum. The essential infimum of H is denoted by
ess inff€?f £ or ess inf H.
The following theorem indicates that the essential supremum and in-
fiinum of a non-empty family of r.v. always exist.
1.13 Theorem. Let H be a non-empty family of r.v.. Then the
essential supremum (resp. infimum) always exists, and there are at most
a denumerable number of elements (£n) ofH such that
ess sup Ti = \J £n ( resp. ess inf H = A£n)-
n n
Moreover, if H is closed under the operation V (resp. A), the sequence
(£n) can be chosen being monotone increasing (resp. decreasing).
Proof We only discuss the case of essential supremum. The second
conclusion is easy. In order to show the first conclusion, we may assume
all elements in H to be uniformly bounded. Otherwise, we may consider
Ti = {arctan£ : £ G H} instead. Furthermore, we can assume that H is
closed under the operation V. At this time, let (£n) C H be a monotone
increasing sequence such that
lim £?[&]= sup E[£\.
71 ten
Put r/ = V £n> and we are going to show that rj is the essential supremum of
n
Ti. To this end, one needs to verify the two conditions in Definition 1.12.
Condition ii) holds trivially, and only condition i) needs to be verified.
Let (GK Put
Then (£„) C H is monotone increasing, and limn £'n = r\ V £. We have
E[V V £] = lim E[?n] < sup E[£] = E[V).
n ten
Because rj V £ > 77, the above inequality means 77 V £ = 77 a.s., i.e., rj > £
a.s.. Condition i) is established. □

10 Chapter I Preliminaries
1.14 Remark. Let (ft,.F, P) be a probability space. Assume C aT,
and C is non-empty. Put
H = {Ic:CeC}.
According to Theorem 1.13, there is (Cn) C C such that
h\cn =VIcn =esssupW,
n
\JCn is called the essential supremum of C, and is denoted by ess sup C.
n
Similarly, there is (Dn) C C such that
7n^n = A^n =essinfW,
n
Pl^n is called the essential infimum of C, and is denoted by essinf C.
n
The way to prove the existence of essential suprema in Theorem 1.13
is also a useful technique, we will use it ever and again in this book.
§4. The Generalization of Conditional Expectation
1.15 Definition. Let (ft, J*, P) be a probability space, Q be a
sub-afield of T. A r.v. £ is called a-integrable with respect to (abbreviated as
w.r.t.) G, if there exist ftn £ Q, ftn | ft a.s. ^ such that each £Jnn is
integrable.
1.16 Theorem. 1) In order for a r.v. £ to be a-integrable w.r.t. Q it
is necessary and sufficient that there exist a Q-measurable finite r.v. rj > 0
a.s. such that £rj is integrable.
2) Let £ be a r.v.. If there exists (Gn) C Q such that \JGn = ft a.s.
n
and each £J<£n *s &-integrable w.r.t. G, then £ itself is a-integrable w.r.t.
o.
Proof. 1) The sufficiency is obvious (one can take ftn = [rj > £]), we
show the necessity. Let £ be a-integrable w.r.t. Q. Then there exists
(ftn) C Q such that ftn \ ft and each £Jnn is integrable. Set
oo 1
V = £l2Hl + E[\£\InJ)Inn-
It is easy to see that rj > 0 is (/-measurable, and £77 is integrable.
1J If necessary, one can replace Q„ by Qn U (Q\ (JQ„) to make Q„ | Q.

§4. The Generalization of Conditional Expectation 11
2) Prom 1) for each n there exists a (/-measurable finite r.v. 7/n > 0
a.s. such that rjn£lGn is integrable. Set
00 i]n A 1
r] = ^'l«{\ + E\r]n\i\IGn\)lG«-
Then rj > 0 a.s., rj € Q, and £77 is integrable. Again by 1) we conclude
that £ is a-integrable w.r.t. Q. □
It is well-known that for any non-negative r.v. £ one can define
conditional expectation i£[£|(7] ( set i£[£|(7] = limni£[£ An|(/] a.s.). However,
even if £ only takes finite values, i£[£|(7] maY be +00 on a set with
positive probability. It is not difficult to prove that for a non-negative r.v. £,
2£[£|(7] is a.s. finite if and only if £ is a-integrable w.r.t. Q.
1.17 Theorem. Let £ be a r.v., a-integrable v.r.t. Q. Set
C = {AeG: E[\£\IA] < +00}.
Then there exists a unique (up to a null set) real r.v. rj £ Q such that for
any A EC,
E[ZIA] = E[r,IA}. (17.1)
77 is called the conditional expectation of £ given G, and is denoted by
Proof. Without loss of generality, we may assume £ to be non-negative.
Suppose that Q,n G (7, fin | fi and each £Jnn is integrable. Set
r)n = E[tInn\g].
Then r)n+ilnn = r\n a.s., r]n \ 77 a.s., where 77 is a (/-measurable real r.v..
For A E C we have
E[£IA] = hm[tIAInn] = lim^M*] = E[r,IA],
i.e., (17.1) holds. By (17.1), r]Ian is the conditional expectation of £/nn
given Q. Hence, rj is determined uniquely up to a null set. □
As mentioned above, i£[|£||(7] < 00 a.s. (i.e., jE[£+|(/] < 00 and
^[£1^] < °° as) ^ a11^ only ^ £ is ^-integrable w.r.t. Q. In this
case, the conditional expectation i?[£|(7] defined in Theorem 1.17 is just
E[(+\G)-E[C\G}.
The properties of conditional expectations of integrable r.v. are well-
known. These properties remain true for conditional expectations of a-
integrable r.v.. Below we list them, and only give the proof for smoothing

12 Chapter I Preliminaries
properties. The proof for others may be proceeded the same as in the case
of integrable r.v..
1.18 Theorem. Let £ andrj be two r.v., a-integrable w.r.t. Q.
1) For all real a and b, a£ + bq is a-integrable w.r.t. G, and
E[a£ + brj\G) = aE[£\G] + bE[r]\G) a.s..
2) //£ < 77 a.s., then E[£\Q\ < E[r]\G] a.s..
1.19 Theorem. Let r.v. £n > 0, n > 1, be a-integrable w.r.t. G-
1) If £n T €> then £ is a-integrable w.r.t. G if and only i/limn i£[£n|£]
< oo a.s.. In this case, we have
E[£\g}=limE[Sn\g}a.s..
2) If lim„£n is a-integrable w.r.t. Q, then
ME[Zn\g] > E\jmtn\G] a.s..
n n
1.20 Theorem. Suppose r.v. £n a.s. converge to a r.v. £, and for
each n, |£n| < rj a.s., where rj is a G-measurable real r.v.. Then
E[\Zn-Z\\g\^0a.a..
1.21 Theorem. Let £ be a r.v., a-integrable w.r.t. G, and rj be a
G-measurable real r.v.. Then £77 is a-integrable w.r.t. G, and
E[tfi\0] = riE[t\g\ a.s.. (21.1)
Proof Assume An G G, An | fi, and each £lAn is integrable. Set
Bn = [\v\ < n]. Then Bn G G and Bn | fi. Put fin = An n Bn. Then
fin G5, fin T fi? and each £rjlnn is integrable. Hence, £77 is a-integrable
w.r.t. G. We have (by (17.1))
E[^\G]Ian = E[tvInn\G} = vInnE[^\G) a.s..
(21.1) follows. □
1.22 Theorem. Let H and G be two sub-a-fields of T, and G C H.
If £ is a r.v., a-integrable w.r.t. G (a-integrable w.r.t. H thereby), then
E[^\7i] is a-integrable w.r.t. G, and
E[Z\G] = E[W\G\ <*•*•> (22.1)
where E[£\H\G] is the abbreviation of E[E[Z\H]\G].

§4. The Generalization of Conditional Expectation 13
Proof. Suppose that ttn G G, ^n T 0? and each £Inn ^ integrable.
Prom (17.1) we know that InnE[^\H] is integrable, then E[£\H] is a-
integrable w.r.t. G- Using smoothing properties of ordinary conditional
expectation, we have
E[iiiln\G] = E\iiUn\n\g]^..
Hence, (by (21.1)) we obtain
m\0)hn = E[(Inn\g] = E[ZInn\H\g) = E[Inn £[£|«]| Q]
= E[E[S\H}\G}IUn a.s..
(22.1) follows. D
The following theorem will be used constantly in this book.
1.23 Theorem. Let £ be a r.v., and A € g. Assume that £IA is
a-integrable w.r.t. g. Put
g' = <r{AnG:Geg}.
Then £IA is a-integrable w.r.t. g', and
E[iiA\g\ = E[nA\g') a.s.. (23.1)
Proof. It is trivial that £IA is a-integrable w.r.t. Gf- Prom Theorem
1.21 we have
E[ZIA\Q] = E[ZIA\G}IA a.s..
Hence, we can take £[£7,410] G G'. Since Q' C G, from Theorem 1.22 we
have
E[iiA\g'\ = E[ziA\g\g'\ = E[tiA\g\ a.s.. □
Remark. If £ is integrable, then for any A &g we have
E[Z\g\IA = E[i\g']IA a.s..
Moreover, if two sub-a-fields G\ and G2 satisfy G\ H A = G2 ^ A and
A G G\ n G2, then for any integrable r.v.£
E[Z\Gi]IA = E[Z\g2]IA a.s..

14
Chapter I Preliminaries
§5. Analytic Sets and Choquet Capacity
In this paragraph we introduce the concept of analytic sets and their
elementary properties. By means of Choquet capacity we show that T-
analytic sets in a measurable space (fi,^*) are universally measurable.
These are prerequisites for section theorems in Chapter IV.
1.24 Definition. Let E be a set, £ be a class on E. If £ contains
empty set 0, we call £ a paving on E, and the ordered pair (E, £) a paved
set.
1.25 Definition. Let (F,T) be a paved set and A be a subset of F.
A is called T-analytic, if there exists a compact metrizable space E and a
subset B of F x E belonging to (T®K(E))Gs such that A is the projection
of B onto F, where tC(E) is the paving consisting of all compact subsets
ofE.
The class of all J*-analytic sets is denoted by A(F). Immediately from
the definition we know that if A G A(T), then there exists B G TG such
that Ac B. In particular, F G A(F) if and only if F G Ta.
1.26 Theorem. Let (F,T) be a paved set. Then
l)TcA(T).
2) A(T) is closed under countable unions and intersections.
Proof. 1) is apparent. We show 2). Let (An) C A(T). According
to the definition, for each n there exists a compact metrizable space En
and Bn G (T ® K,(En))Gt such that An is the projection of Bn onto F.
Denote by E the product topological space Yln En, and by 7r the projection
mapping from F x E onto F. Put
Cn = Bn x J] ^m-
We have
nA, = (>(<?„) = *((-)<?„). (26.i)
n n n
Put Bn = C\Bnk, where for each fc, B^ G (^ <8> /C^))^ Since Bn,fc
x Um^nEm G (^ ® )C(En))a, we have Cn G (F ® K(E))a6, and QCn G
(J* ® K(E))a6' It follows from (26.1) that f)nAn G >t(J*), i.e., A{F) is
closed under countable intersections.

§5. Analytic Sets and Choquet Capacity 15
Now denote by E the one-point compactification of the sum topological
space £„ En, and identify £n(F x En) with F x (£n En). Then
7r(E#„) = lUn. (26.2)
n n
Since £n 5fe,n € (.F® £(£))*, we have
E 5„ = E n 5„,fe = n E 5„,fe e (^ ® £(£)U
n n k k n
and Un^ri € -4(.F) by (26.2), i.e., A(T) is closed under countable unions.
□
♦
1.27 Theorem. Let (E,£) and (F,T) be two paved sets, and (F x
E,F ® £) be their product. Then
A(f) ® A{£) c A(F ® £). (27.1)
Froo/. Let A G .*(£) and B G >l(7"). Take Ai G £a and Bi G T„ such
that Ac4BcBi. It is easy to see that T® A(f) C A(J*® £). It
follows from Theorem 1.26 that
T„ ® ■*(£) C (^ ® .A(£))<r C A{T ® £).
Similarly we have A(^*) ® £a C A(T ® £). Finally we obtain
Bx4 = (B1xA)n(Bx41)G A( J* ® £). □
1.28 Theorem. Let (F,F) be a paved set, and E be a compact metri-
zable space. Then for each A' G A(T®)C(E)), the projection A of A' onto
F is ^-analytic.
Proof. There exists a compact metrizable space G and A" G (F ®
fC(E) ® /C(G))a$ such that A' is the projection of A" onto F x E. But
£ x G is a compact metrizable space, JC(F) ® fC(G) C JC(F x G), and A
is the projection of A" onto F. By the definition we have A G A(T). □
1.29 Theorem. Let (F, J*) be a paved set, and Q be a paving on F
such that T C Q C A(T). Then
A{T) = A(G) = A{A{T)). (29.1)
Proof. Let A G A{A{F)). There exists a compact metrizable space E
and A' G (A(J*) ® K{E))Gs such that A is the projection of A' onto F.
From (27.1) we have
A{F) ® £(F) c A(^) ® A(K(E)) c A(^ ® /C(F)).

16
Chapter I Preliminaries
Hence, A' € A(F®K,(E)). It follows from Theorem 1.28 that A e A(F),
i.e., A(A(f)) C A(F). It is trivial that A(F) C A{Q) C A^F)). Hence,
(29.1) holds. □
1.30 Theorem. Let (F,F) be a paved set, A be a subset of F, and
T n A = {B D A : B € J7}. Then
A(f n A) = A(f) n A, (30.1)
where T n A is considered as a paving on A.
Proof. Let C" € A(FT\ A). There exists a compact metriable space E
and C" € ((T n A) ® fC(E))as such that C is the projection of C onto A.
Furthermore, there exists C" € {F®K{E))a6 such that C = (AxE)nC'.
Hence, C = A f) n{C") € A(T) D /I, where ir is the projection from
F x E onto F. This means A{T n A) C -4(.F) n A. Conversely, let B €
A{!F). There exists a compact metrizable space E and B' € (F®K{E))a6
such that 5 = ir(B'). Since (AxB)nB' e {{f C\ A) ® K{E))a6 and
405 = ir((A x E)f) B'), we have Af)B € A(T n A). This means
A{T) f\Ac A{F n A). Hence, (30.1) holds. □
1.31 Theorem. Let {F,F) be a paved set. Then o~(T) C A{F) if and
only if for any Aef,Ace A{F).
Proof. The necessity is trivial. We show the sufficiency. Put
Q = {A € A(F) : Ac e A(7)}.
We have T C Q. Q is a <r-field (by Theorem 1.26). Then
<r(T) CGC A(F). □
1.32 Theorem. Let (Q,T) be a measurable space, B = B(R),K =
K(R), T x B be the product a-field on (I x R. Then
1) BcA(K),A(B) = A(K),
2)FxBc A{T ®K) = A(T x B),
3) for any A € A(!F ® AC), the projection of A onto Q. is F-analytic.
Proof. 1) Let K e K. It is well-known that Kc € Ka C A{K).
Because cr(JC) = B, we have AC C B C A(K) (by Theorem 1.31). Hence,
A(B) = A(K) (by Theorem 1.29).
2) Let B € T ® AC. We see that 5C € (.F <g> K)a C .4(.F ® K,). Since
<t(:F ® AC) = T x B, we have F®KcTxBc A( T ® K) (by Theorem
1.31). Hence, A(T x B) = A(T®fC) (by Theorem 1.29).

§5. Analytic Sets and Choquet Capacity
17
3) Take (Kn) C /C such that \JKn = R (e.g. Kn = [-n,n]). For each
n
n we have (by Theorem 1.30)
A(r ® /c) n (n x Kn) = a((.f ® K) n (n x Kn))
= ^®(/cnifn)).
Since Kn is a compact metric space, tC(Kn) = K, n i^n, A £ A(J* 0 JC),
the projection of (ft x Kn) n A onto ft is J*-analytic (by Theorem 1.28).
Noticing A = \J[(tt x Kn) fl A], the projection of A onto ft is also ^*-
n
analytic. □
Remark. In Theorem 1.32 R can be replaced by any locally compact
HausdorfF topological space with countable basis.
Now we turn to define Choquet capacity.
1.33 Definition. Let (F,T) be a paved set, where T is closed under
the formation of finite union and intersection. An extended real valued
set function J, defined for all subsets of F, is called a Choquet ^-capacity
on F, if J has the following properties:
i) J is increasing:
ACB=> 1(A) < 1(B), (33.1)
ii) J is continuous from the below:
An T A =► 1(A) = sup J(An), (33.2)
n
iii) J is continuous in T from the above:
An € T, An | A =► /(A) = inf J(i4n). (33.3)
A set A is called I-capacitable, if
1(A) = sup 7(B). (33.4)
B€T6,BCA
1.34 Lemma. Le£ I be a Choquet T-capacity on F. Then each
element of TGs *5 I-capacitable.
Proof. Let A e 7^. K /(A) = -oo, then 7(0) = -oo,0 e 7\ (33.4)
holds, i.e., A is capacitable. Now assume 1(A) > — oo. We have
00
A = fl An, AnGfa, n > 1.
n=l
oo
-An = U A-rvmi Anm G/, 71, 771 > 1.
771=1

18
Chapter I Preliminaries
Since T is closed under the formation of finite union, we may assume that
for each n fixed, (An}m)m>i is an increasing sequence. In order to show
(33.4), it suffices to prove: for any a < 1(A) there exists B G Ts,B C A
such that 1(B) > a.
Assume a < 1(A). Prom (33.2) we have
1(A) = I(A fl Ax) = suPm I(A fl Alm).
There exists an integer mi such that I (A fl A\mi) > a. Then
I(A fl Aimi) = I(A fl Almi n A2) = supm I(A fl AXmi fl A2m) > a,
and there exists another integer 7712 such that I (A C\ A\mi fl A2m2) > a-
Repeating the same argument we obtain a sequence of integers (rrtk)k>i
such that for each k > 1 we have
J(i4nAimin---i4ibmfc)>a.
n 00
Put Bn = H Akmk,B = H #n- It follows from (33.1) that I(fln) > a.
On the other hand, Bn G T,Bn \ B,B G Ji, from (33.3) we see that
1(B) = infn J(fln) > a. Since Bn C An, we get B C A. □
The following theorem is called Choquet 's theorem.
1.35 Theorem. Le£ I be a Choquet T-capacity on F. Then each
A G A(F) is I-capacitable.
Proof. Let A G A(T). There exists a compact metrizable space E
and B G (J* ® K(E))as such that 7r(B) = A, where 7r is the projection
mapping from F x E onto F. Let W be the paving consisting of all sets
of the form U/b=i Ak, Ak G T ® /C(-E),n > 1. It is easy to see that 7Y is
closed under the formation of finite union, and Has = (P ® ^(E))a6- For
each H C F x E define
J(ff) = /(tt(//)).
We are going to show that J is a Choquet 7Y-capacity on F x E. Obviously,
J satisfies properties i) and ii) in Definition 1.33. It remains to verify
property iii).
Let H = UZ=l(Dk x Ck) G W, where Dk G T,Ck G K(E). For
each x G 7r(#), we have ({x} x E) H H = {x} x C, where C ^ 0, and
C = U{jfc:x€Dib}^ € ^(-E*). Now assume that (Bn) C H is a decreasing
sequence of sets, and z G fl^Li ^(^n)- For each n there exists Cn G ^C(-E)
such that
({x} xE)nBn = {x} xCn.

§5. Analytic Sets and Choquet Capacity 19
Since (Bn) is decreasing, so is (Cn). Each Cn is a non-empty compact set
of E, therefore f]n Cn ^ 0. Hence
n n
i.e., x G iv(f)nBn). This means fln71"^") C 7r(nn^n)- The reverse
implication always holds. Thus,
ri7r(Bn) = 7r(nBn). (35.1)
n n
Since 7r(5n) G f,7r(Bn) |, by (33.3) we have
j(nBn) = i(n(nBn)) = i(n*(Bn))
n n n
= infn/(7r(Bn)) = infnJ(BTl),
i.e., property iii) in Definition 1.33 holds for J. Hence, J is a Choquet
7Y-capacity on F x E.
Since B G 7Ya$, from Lemma 1.34 we know that B is J-capacitable.
But from (35.1) we see that C eHs=> 7r(C) G Fs- Hence
I(A) = J(B) = sup J(C)= sup J(tt(C))< sup 1(D).
C€H6,CCB CeH6,CcB DEFsiDcA
1(A) > sup£>€^ £>Ci4 1(D) is trivial. Thus
1(A) = sup 1(D).
Der6,DcA
This means A is J-capacitable. □
As an important application of Choquet's theorem, we will show that
all ^"-analytic sets in a measurable space (fi,^*) are universally
measurable.
Let (fi, T) be a measurable space, and V be the collection of all
probability measures on {£l,T). For each P eV, denote by Tp the completion
of T w.r.t. P. Put
*= n **.
P€T>
^* is called the universal completion of T, the elements in ^* are called
universally measurable sets.
Obviously, T = T. Besides, if T is complete w.r.t. a certain
probability measure P, then T' = T.
1.36 Theorem. Let (ft,T) be a measurable space. Then
A(F) cf = A(f).

20 Chapter I Preliminaries
Proof Let P eV. Define
1(A) = inf P(fl), Ac SI.
v 7 BeT,BcA v 7
It is easy to verify that J is a Choquet ^"-capacity on SI. By Choquet's
theorem, for each A G A(T) we have (noting Ts = F)
1(A) = sup 1(B).
Ber,BcA
Thus AeTp. Because P eV is arbitrary, A G f. This means •A(J') C
T. Therefore we have
f C A(f) C ?= f.
Hence, >t(J*) = J*. □
§6. Lebesgue-Stieltjes Integrals
1.37 Lemma. Let a(t) be a non-negative right-continuous increasing
(extended real-valued) function on i2+. Set
c(t) = inf{s : a(s) > *}, t G #+. (37.1)
TTien c(t) is a non-negative right-continuous increasing function on i2+,
and is called the right-inverse function of a(t). For t G i2+, c(t) < +oo if
and only if t < a(oo) = lim*-^ «(*)• ^e*
a_(£) = a(£—) = lima(s), t > 0,
sTT*
c_(£) = c(£—) = limc(s) = inf{s : a(s) >t} = sup{s : a(s) < t}, t> 0,
a(O-) = a(0), c(O-) = c(0).
TAen we Ziave
a_(c_(*)) < a-(c(0) < *, * € ii+, (37.2)
and
a(c(t)) > a(c-(t)) >t,t< a(oo). (37.3)
In particular, if a is continuous, then for all t < a(oo) we have
a(c(t)) = a(c-(t)) = t.

§6. Lebesgue-Stieltjes Integrals 21
At last, the relation between c{t) and a(t) is symmetric, i.e., a(t) is the
right-inverse function of c(t):
a(s) = inf{t : c(t) > s}, s G R+. (37.4)
Besides, we have
a(s) = sup{t : c(t) < s}, s G R+. (37.5)
Proof. Left to readers as an exercise. □
The following lemma is called Lebesgue 's lemma. It reduces Lebesgue-
Stieltjes integrals w.r.t. a certain increasing function to ordinary Lebesgue
integrals. It will be used in Chapter V.
1.38 Lemma. Let a(t) be a non-negative right-continuous increasing
real function, and f(t) be a bounded or non-negative Borel function. Then
I f(s)da(s) = I f(c(s))I[c<oo](s)ds, (38.1)
•/[0,oo[ «'[0loo[
/ f(s)da(s)= f f(c.(s))I[c_<oo](s)ds, (38.2)
J[0,oo[ -/[0,oo[
where c(t) is defined by (37.1). (By convention I f(s)da(s) = /(0)a(0).)
[0]
Proof. Assume f(t) = I[oyU](t),u G il+. From (37.5) we have
/ f(s)da(s) = a(u) = sup{s : c(s) <u}=l Lc<v](s)ds
J[0,oo[ •/[0,oo[ ~
f(c(s))I[e<QO](s)ds,
J\o,
'[0,oo[
i.e., (38.1) holds for such /. Using the monotone class theorem (Theorem
1.4), (38.1) holds for any bounded or non-negative Borel function /. At
last, the set {s : c(s) ^ c(s—)} is at most countable, (38.2) follows from
(38.1). □
The following lemma is the formula of integration by parts for
Lebesgue-Stieltjes integrals.
1.39 Lemma. Let f(t) and g(t) be two right-continuous functions
with finite variation on i2+ (i.e., which can be represented as the difference
of two non-negative right-continuous increasing real functions). Then for
0<a<b< +00
f(b)g(b) = f(a)g(a) + t f(s)dg(s) + f g(s-)df(s). (39.1)
Ja J a

22 Chapter I Preliminaries
(By convention fif(8)dg(a) = /]a6] f(s)dg(s).)
Proof We have
(f(b)-f(a))(g(b)-f(a)) = J J df(x)dg(y)
]a,b] x]a,b]
= j J df(x)dg(y) + J J df(x)dg(y)
a<x<y<b a<y<x<b
= f" dg(y) I" df(x) + t df(x) f dg(y)
Ja Ja Ja ^]a,x[
= fb[f(y)-f(a))dg(y)+ fb[g(x-)-g(a)}df(x)
Ja Ja
= f"f(y)dg(y)+ f"g(x-)df(x)- f(a)[g(b)- g(a)]
Ja Ja
-9(a)[f(b) - f(a)}.
(39.1) follows immediately. □
The formula (40.1) in the following theorem is called Kunita- Watanabe
inequality.
1.40 Theorem. Let a(i) be a right-continuous function with finite
variation on i2+, b(t) and c(t) be two non-negative right-continuous
increasing functions on i2+. If |a(0)| < ^6(0)^(0), and for all 0 < s <
t < +oo,
|o(t) - a(s)\ < y/b(t)-b(8)y/c(t)-c(s),
then for any Borel functions f and g on il+
/ l/(*)*W||Aitol<(/ f{s)db{s))^{( g2(s)dc(s))1/2.
J[0,oo[ •'[0,oo[ •'[0,oo[
(40.1)
Proof We will use the following elementary fact: let x,y,z be three
real numbers and x > 0, z > 0, then for all rationals A, X2x + 2Xy + z > 0
if and only if |y| < y/xz.
Put fi(t) = /[ot] \da(s)\ + b(t) + c(t), and o! = -p V = — ,d = -p
For a given rational A set
u(t) = X2b(t) + 2Xa(t) + c(«).
By the assumption and the above-mentioned elementary fact we know that

Problems and Complements 23
v is a non-negative right-continuous increasing function on 12+. Hence
^ = \2b' + 2Aa' + c' > 0, d/i-a.e. (40.2)
d/i
Since Q is countable, (40.2) holds for all A G Q. Again using the above-
mentioned elementary fact, we obtain
|a'| < y/Vd, d/i-a.e. (40.3)
Now for any two Borel functions / and g on i2+, by using Schwarz
inequality and (40.3) we have
/ \f(s)g(s)\\da(s)\ = f \f(s)g(s)\\a'(s)\dn(s)
7[0,oo[ ^[0,oo[
<f \f(s)\y/b^)\g(s)\y/d{r)d^s)
•>[0,oo[ V V
< ( / f2(s)b'(s)dfx(s))1/2( i g2(s)c'(s)d»(s))1/2
V./[0,oo[ ' V^[0,oo[ '
= ([ f(s)db(s))1/2(f gHs)dc(s))1/2. D
Problems and Complements
1.1 Let C be a class on ft, A(C) and m(C) be the A-class and monotone
class generated by C respectively. Then
1) \(C) = a(C) if and only if
A,BeC=>ABe A(C),
2) m(C) = a(C) if and only if
AeC=>Ac G m(C); A,5 G C => AB G m(C).
1.2 Let 7Y be a family of real (resp. bounded) functions on ft. Then
there exists a a-field T on ft such that H is the collection of all real
(resp. bounded) measurable functions on (ft, T) if and only if the following
conditions are satisfied:
i) H is a linear space,
ii) 1 G W,
iii) 0 < fn | /, (/n) C 7Y, / is a real (resp. bounded) function => / G 7Y,
iv) f,geH=> fAgeH.

24
Chapter I Preliminaries
1.3 Let H be a family of bounded functions on fi satisfying the
following conditions:
i) H is a linear space,
ii) 1 € «,
iii) 0 < /n T /, (/n) C W, / is bounded =►/€«.
Let C be a sub-family of 7Y, stable under the multiplication. Then 7Y
contains all <r(/ : / G C)-measurable bounded functions.
1.4 Let (Ei,£i)i£j be a family of measurable spaces. For each i G
J, fi is a mapping from fi to i£j. If <p is a cr(fi,i G immeasurable (resp.
extended) real function on fi, then there exists a countable subset J of I
and a measurable (resp. extended) real function h on dlieJ ^*»rii€J^*)
such that <p = ho fj, where fj is the mapping from fi to Yliej Ei defined
as fj(u) = (fi(u))ieJ.
1.5 Let (fi,^*, P) be a probability space, and Q be a sub-a-field of
T. Let (£, £) be a measurable space, £ be an E-valued (/-measurable r.v.,
and /(cj, x) be an T x f-measurable bQunded function. Denote
fx(u>) = f(u>,x), T)(u>) = f(u>,Z(u>)).
Then there exists a Q x £-measurable bounded function h(uj, x) such that
i) for all a; € E, E[fx\Q](uj) = h(u,x) a.s.,
ii)E[ri\G](u>) = h(u,,t(e>)) a.s.
1.6 Let H be a uniformly integrable family of r.v. on a probability
space (fi, T, P). Then the closed convex hull of 7Y in Ll(il, T, P) is also
uniformly integrable.
1.7 Let (£n) be a sequence of integrable r.v., and assume £n —► £,
jE[|fn|] -+ £7[|£|] < oo. Then (fn) is uniformly integrable and £[|£n-£|] "*
0.
1.8 Let (£n) be a sequence of uniformly integrable r.v.. Then
lim£?[- sup |&|] = 0.
71 W l<ifc<Ti
1.9 Let (fi*, Ji,Pj),i G J, be a family of probability spaces, and
£; G L1(fii,^rt,Pi). In order that (Ci)ie/ t>e a uniformly integrable family
it is necessary and sufficient that there exist a non-negative Borel function
G{t) on i2+ such that
i) limt-+oo ^ = +oo,
u)8upS€/£?i[G(|6|)]<+oo.

Problems and Complements
25
1.10 Let (fi,^*, P) be a probability space, Q be a sub-a-field of T,
and A G T. Then
1) [f^lS] > 0] = essinf{£ eG:BDA},
2) [^[/Aie] = 1] = esssup{£ eG:BcA}.
1.11 Let (£n) be a sequence of r.v. on (fi,^*, P). Define
slim£n = ess inf {77 G J*: limnP(£n > rj) = 0},
71
sUm£n = esssup{r/ G J*: limn P(£n < 77) = 0}.
n
Then
1) Uffin&i < slinvfn < slimn£n < limn£n a.s.,
2) Cn -^ £ <*=> 5hmn^n = 5 hn^ £n = £ a.s.,
where £ is a real r.v..
1.12 Suppose that H C Ll(tl, T, P) satisfies inf {£[£], £ GH}> -oo.
Then the following assertions are equivalent:
1) E[essmfH) = inf {£[£],£ 6 W}.
2) ess inf H is integrable, and for each sub-a-field Q of T
jB[essinf W|C] = essinf{jE[£|0],£ e W}.
3) For any fi, & € W and e > 0, there exists £3 G 7Y such that
£[(6-£iV6)+]<e.
1.13 Let £ be a real r.v. on (ft, J*, P), and (/ be a sub-a-field of T.
Let (^(x) be a real convex function. If both £ and y>(£) are a-integrable
w.r.t. Q, then
¥>(£?[£!<?]) < E[<p(0\G], a.s..
This is Jensen's inequality.
1.14 Let (£n) be a sequence of r.v. on (ft,.F, P), and (/ be a
sub-afield of ^*. If for any e > 0 there exists a (/-measurable finite positive r.v.
77 such that
esssupjE[|fn|%n|>l?]|0] < e a.s.
n
(in this case, (£n) is called conditionally uniformly integrable given Q),
then
E[Kmtn\G\ < HmE[zn\g] < ^e[u\0] < E[H^zn\g].
n n n n
In particular, if (£n) is uniformly integrable, then
E[)mZn} < Mm £?[£„] < B5^[e„] < E\jfa£n].
n n n n

26
Chapter I Preliminaries
1.15 Lret (F, !F) and (G, Q) be two paved sets, and / be a mapping
from F into G such that f~l(G) C A(T). Then
r\A{G)) C A{T).
1.16 Let (F,T) be a paved set, and J be a Choquet ^"-capacity on
F. Then J is also a Choquet ^v-capacity on F.
1.17 Let a(t) be a non-negative real right-continuous increasing
function on i2+, and c(t) be its right-inverse function. Then
1) c(t) is strictly increasing on [0, a(oo)[ if and only if a(0) = 0 and
a(t) is continuous on i2+. In this case, we have a(c(t)) = t,t< a(oo).
2) c(t) is continuous on [0, a(oo)[ if a(t) is strictly increasing on i2+.
In this case, we have c(a(t)) = t, t G i2+.
1.18 Let a(t) be a non-negative real continuous increasing function
on R+. Then for any non-negative Borel function /(£) on [a(0),a(oo)[ we
have
/•oo /-a(oo)
/ f(a(t))da(t) = / /(t)dt.
./O Ja(0)
1.19 Let / be a non-negative Borel functin on R+ and locally inte-
grable, i.e. integrable on any finite interval. Set
a(t) = I f(s)ds,t>0.
Jo
Let c(t) be the right-inverse function of a(t). Denote
A = {t : f(t) = 0},B = {t: c(t) G A}.
Then the Lebesgue measure of B is equal to zero.
1.20 Let a(t) and b(t) be two non-negative real right-continuous
increasing functions on iZ+, and 6(0) > 0. Then for t > 0 we have
a(t) _ a(0) fl da(t) fl a(t)db(t)
&W " b(0) + Jo W1) ~ X b(t)b(t-)'

Chapter II
Classical Martingale Theory
In this chapter we present the major results of classical martingale
theory, such as maximal inequality, upcrossing inequality, Doob's inequality,
convergence theorems, Riesz decomposition theorem and Doob's stopping
theorem. We deal with the discrete time case in §1-4, and the continuous
time case in §5. In order to deepen readers' understanding we illustrate
some examples of applications at times. In §6 an introduction to processes
with independent increments is given.
§1. Elementary Inequalities
In this chapter all discussions are proceeded on a fixed probability
space (tt,T,P). In §1-4 we suppose in addition that an increasing
sequence of sub-a-fields (Tn, n G N) is given: for all n G N
We call {Tn,n G N) or (Pn)n>o a filtration. It can be denoted simply by
F or (Tn). Usually, we denote
J~oo = V ^n*
n>0
A sequence of real r.v. (Xn,n G N) or (Xn)n>o is called a stochastic
sequence, and is also denoted simply by X or (Xn). A stochastic sequence
X = (Xn)n>o is called F-adapted, if for each n, Xn is ^-measurable.
2.1 Definition. An F-adapted stochastic sequence (Xn,n G N) is
called an F-martingale (resp. F-supermartingale,Tesp. F-submartingale),

28
Chapter II Classical Martingale Theory
if for each n > 0, Xn is integrable and
ElXn+ilfn] = Xn (resp. < Xn, resp. > Xn) a.s. .
In this case, for all m > n > 0,
E[Xm\Tn] = Xn (resp. < Xn, resp. > Xn ) a.s.,
EXm = EXn (resp. < EXn, resp. > EXn).
The term "martingale" is originated from a French acronym for the
gambling strategy of doubling one's bets until a win is secured. Let Xn
be a gambler's fortune at time n. The martingale property means that
the gambler's average fortune on the next game is just his current fortune.
Hence, the game is fair. In fact, the class of martingale sequences is one
of the most important types of stochastic sequences and indispensable for
modern probability theory and statistics.
For a stochastic sequence X = (Xn,n £ N) set
F°(X) = (J*(X),neN), J*(X)=<r(X(hXu---,Xn), neN.
Obviously, F°(X) is a filtration, and is called the natural filtration of X.
It is also the smallest filtration, w.r.t. which X is adapted.
By the smoothing property of conditional expectation, any F-martin-
gale (resp. F-supermartingale, resp. F-submartingale) is also a
martingale (resp. supermartingale, resp. submartingale) w.r.t. its natural
filtration.
Since the filtration F is fixed in the following discussions, for
convenience we often omit the suffix "F-". For examples, an F-martingale
is simply called a martingale, an adapted sequence means an F-adapted
sequence.
From the definition we see that if X = (Xn) is a supermartingale (resp.
submartingale), then —X = (—Xn) is a submartingale (resp.
supermartingale). A stochastic sequence is a martingale if and only if it is both a
supermartingale and a submartingale.
In the following we give some examples of martingales, supermartin-
gales and submartingales. Readers can check them directly.
2.2 Examples. 1) Let £ be an integrable r.v.. Put Xn = E[£\Tn].
Then (Xn) is a martingale.
2) Let (£n) be an adapted sequence of integrable r.v.. For each n >
0, £n+i is independent of Tn. (Consequently, (£n) are independent.) If for

§1. Elementary Inequalities 29
each n > 1, E[£n] = 0 ( resp. < 0, resp. > 0), then (Xn = f) &,n G N)
is a martingale (resp. supermartingale, resp. submartingale).
3) Let (£n) be an adapted sequence of integrable non-negative r.v.. For
each n > 0,£n+i is independent of Tn. If for each n > 1, E[£n] = 1 (
n
resp.< 1, resp. > 1), then (Xn = Y\ £i,n e N) is a martingale (resp.
i=0
supermartingale, resp. submartingale).
4) (The generalization of 2)). Let (£n) be an adapted sequence of
integrable r.v.. If for each n > 0, .E7[fn+i|.Fn] = 0 (resp. < 0, resp. > 0),
n
then (Xn = ^2 £i,n € N) is a martingale (resp. supermartingale, resp.
i=o
submartingale).
5) Let (£n) be an i.i.d. (i.e., independent identically distributed)
stochastic sequence, and
P(& = 1)=P, P(£o = -l) = 9 = l-p, 0<p<l.
Put
Sn=£ti, Xn=(l)S\ neN.
i=0 KP/
Then (Xn) is a martingale w.r.t. its natural filtration.
2.3 Theorem. 1) Let X = (Xn) and Y = (Yn) be two martingales
(resp. supermartingales). Then X + Y = (Xn + Yn) is a martingale (resp.
supermartingale), and X A Y = (Xn AYn) is a supermartingale.
2) Let X = (Xn) be a martingale (resp. submartingale), and f be a
continuous convex (resp. continuous increasing convex) function on R. If
each f(Xn) is integrable, then f(X) = (f(Xn)) is a submartingale.
Proof. 1) is evident. We show 2). On the one hand, we have
f(Xn) = f{E[Xn+1\Fn]) (re8P.</(£?[Xn+i|^n])) a.s. . (3.1)
On the other hand, by Jensen's inequality
f(E[Xn^\Tn}) < E[f(Xn^)\Tn] a.s. . (3.2)
Combining (3.1) with (3.2) yields (f(Xn)) which is a submartingale. □
2.4 Corollary. 1) If(Xn) is a submartingale, so is (X+). Moreover, if
for each n, Xn log"1" Xn is integrable, then (Xn log"1" Xn) is a submartingale,
where log+ x = (logx)I[hoo[(x).

30
Chapter II Classical Martingale Theory
2) Let (Xn) be a martingale or non-negative submartingale, and X >
1 be a constant. If for each n, \Xn\x is integrable, then (\Xn\x) is a
submartingale.
Now we start to introduce the maximal inequality and upcrossing
inequality of martingales. To this end, we need the concept of stopping
time.
2.5 Definition. An iV-valued random variable T is called a stopping
time (F-stopping time) or optional time, if for each n, [T = n] G Tn, or
equivalently, for each n, [T < n] G Tn.
For a stopping time T put
TT = {A G Too : VneN, A n [T = n] G Tn},
it is called the a-field of events prior to T. Obviously,
TT = {A e T^ : VneN, A n [T < n] e Tn}.
It is easy to see that a stopping time T is ^-measurable. The constant
time T = n is a stopping time, and Tt — Tn. If T is a stopping time,
then for each n > 1, T + n is also a stopping time.
In practice, a filtration (^n) describes the history of some random
phenomenon, Tn represents the information observed up to time n. The
characterization of a stopping time T consists in that the event "T has
occurred up to time n" depends only on the history up to time n, not on
any information of the future. For instance, suppose that Xn represents
the fortune of a gambler at time n, and Tn = ct(Xq, • • •, Xn). The gambler
has the right to choose the time to stop the gamble. But the decision
wether or not the gamble be stopped at time n must be determined on the
base of Tn, i.e., the information observed by him up to time n. Obviously,
at time n he does not know anyone of future's outcomes Xn+i,Xn+2>
Hence, the time chosen by him to stop the gamble must be a stopping
time. This is the reason why we use the term "stopping time" or "optional
time". The following theorem offers a class of most useful and frequently
encountered stopping times associated with stochastic sequences.
2.6 Theorem. Let (Xn)n>o be an adapted stochastic sequence, and
B G B(R). Let S be a stopping time. Put
T(uj) = inf {n : n > S(u) and Xn(u) G B).
Then T is a stopping time (by convention inf 0 = +oo).

§1. Elementary Inequalities 31
In particularj T = inf {n > 0 : Xn G B} is a stopping time.
Proof. For each n G N
[T = n]= (J \[S = k]( n [XmeBc])[xneB]}eTn.
k=0 l v k<m<n ' J
Hence T is a stopping time. □
2.7 Example. Let us consider independent repeated trials, each of
which admits two possible outcomes, success or failure. Explicitly, define
{1, if the n-th trial succeeds,
n> 1.
0, if the n-th trial fails,
Let (Tn) be the natural filtration of (Xn). Denote
Ti = inf{n > 1 : Xn = 1}, Tn+i = inf{n > Tn : Xn = 1}, n > 1.
Then Tn is the waiting time for n-th success. Since (Xn) is i.i.d., it is
not hard to prove that (TUT2 - Tu • • • ,Tn - Tn_i, • • •) is i.i.d., and Tx is
distributed geometrically.
2.8 Theorem. Let (Xn)n>o be an adapted stochastic sequence, £ be
an Too-measurable real r.v., and T be a stopping time. Put X^ = £, and
Xt(uj) = XT(u)(uj), u eft.
Then Xt is Tt-measurable.
Proof. For B G B{R) and n G N
[XT eB)= [JJ[Xk g B] n [T = fc]) g T^
k€N
[XT e B] n [T = n] = [Xn G B] H [T = n] G J*n.
This means [Xp GB]6 ^t? ie., Xp is ^-measurable. □
2.9 Theorem. Let S and T be two stopping times, and (Sk) be a
sequence of stopping times. Then
1) /\Sk and \/ Sk are stopping times,
k k
2) A e Ts =* A n [S < T] e TT, a n [s = T] e ?T,
3)S<T=>fsCfT,
4) for Aefs
SA = SIA + (+oo)IAe
is a stopping time, and !FsA fl A = Ts H A.

32 Chapter II Classical Martingale Theory
Proof. 1) follows from the following equalities:
[A«b<n]=U[Sfc<n],
k k
[\/Sk<n]=r\[Sk<n].
k k
2) Let A G J*5. Then A n [5 < T] G J*oo, and for each neN
An[S <T]n[T = n] = (An[S <n])n[T = n]eTn.
Hence, A H [5 < T] G Ft> By the same argument we have A H [S = T] G
Jr.
3) follows from 2).
4) is trivial. □
2.10 Theorem. Let (Xn) be a martingale (resp. supermartingale), S
and T be two bounded stopping times, and S <T. Then
E[XT\TS] = Xs (resp. < Xs) a.s.. (10.1)
Proof. We give the proof for the supermartingale case. Suppose T <n.
n n
Then \Xt\ < ]£ |-Xj|, \Xs\ < ]C \Xj\, whence Xt and Xs are integrable.
j=0 j=0
For Aefs and j G N
An[S = j]n[T>j]eFj.
We first suppose T — 5 < 1. In this case, by the supermartingale property
we have
/ (XS - XT)dP =£ I (Xj - Xj+x)dP > 0.
J A j=o JAn[S=j]n[T>j]
In general situation put Rj = T A (5 + j), j = 1, • • •, n. Then each Rj is
a stopping time, and S < R\ < • • • < Rn = T, R\ — S < 1, Rj+i — Rj < 1
(j = 1, • • •, n - 1). Let A G Ts- F°r each j, 1 < j < n, A G J/t, (Theorem
2.9.3)). Using the conclusion proved above, we obtain
/ XsdP > [ XRldP>---> ( XTdP. (10.2)
J A J A J A
Because Xs G Fs (by Theorem 2.5), (10.1) follows from (10.2). □
2.11 Corollary. Let (Xn) be a supermartingale, T be a stopping time.
Then
E[\XTAk\)<E[X0) + 2E[Xj;l (11.1)
E[\XTI[T<oo]\]<3supE[\Xn\}. (11.2)

§1. Elementary Inequalities 33
Proof. Because (X~) is a submartingale, by Theorem 2.10 we have
E[\XTAk\) = E[XTAk] + 2E[XjAk) < E[X0) + 2E[Xk~).
This is (11.1). Furthermore,
E[\XTAkI[T<oo]\] < E[X0] + 2E[Xk~\ < 3sup£7[|Xn|]. (11.3)
n
Letting k —> oo in (11.3), (11.2) follows from Fatou's lemma. □
Theorem 2.10 is a special case (the bounded stopping time case) of
Doob's stopping theorem, whose general formulation can be referred to
Theorems 2.35 and 2.38. The inequality (12.3) in the following theorem
is usually called the maximal inequality for supermartingales.
2.12 Theorem. Let (Xn) be a swpermartingale, and k G N. Then
for any A > 0 we have
\P(supXn > A) < E[X0] - ( XkdP, (12.1)
Kn<k ' ./[supXn<A]
n<k
XP( inf Xn <-X)< f {-Xk)dP, (12.2)
Kn<k / J[infX„<-A]
n<k
\p(sup\Xn\ > A) < E[X0] + 2E[X^]. (12.3)
Xn<k '
Proof Put T = inf {n G N : Xn > A} A k. Then T is a bounded
stopping time, Xt > Aon [supXn > A] and T = k on [supXn < A]. By
n<k n<k
Theorem 2.10,
E[X0] > E[XT] = I XTdP + I XTdP
J[s\ip Xn>\] J[sup Xn<X]
n<k n<k
> AP(supXn > A) + /[supXn<A] XkdP.
This is (12.1). (12.2) can be shown in the same way. (12.3) follows
immediately from (12.1) and (12.2). □
2.13 Corollary. Let (Xn) be a martingale. If E[X%] < +oo, then for
any A > 0
P(sup\Xn\>\)<±E[X2k). (13.1)
n<k *
(This is Kolmogorov's inequality.)

34 Chapter II Classical Martingale Theory
Proof. By Jensen's inequality, for each n < k
E[X2n) = E[(E[Xk\rn])*\ < E[Xl\ < oo.
Hence, (—-X"^)n=ofif...fjb ls a supermartingale. Applying inequaUty (12.2)
to (-Xl) and A2 yields (13.1). □
In the sequel we will show the very important wpcrossing inequality for
supermartingales and submartingales. To this end, we should introduce
some necessary notations.
Let X = (Xn)n>o be an adapted stochastic sequence, [a, b] be a finite
closed interval. Put
T0 = inf{n > 0; Xn < a},
7\ = inf{n : n > T0, Xn> &},
T2j = inf{n : n > T2j-\, Xn < a},
T2j+i = inf{n : n > T2j, Xn > &},
Then (Tjt)fc>0 is an increasing sequence of stopping times. If T2j-i(uj) <
+oo, sequence (Xo(uj), X\(uj), • • •,Xt2j-i(v)) upcrosses interval [a, b] j
times. Denote by U%[X, k] the number of upcrossings of [a, b] by sequence
(Xo,-X"i, • • • , Xjt). It is apparent that
[U*[X, k] = j] = [T2i_x < k < r2j+1] € Tk.
Consequently, U*[X, k] € Tk.
2.14 Theorem. If (Xn) is a supermartingale, then for N > 1 and
k>0
P(U£[X,N} > k + 1) < ^E[(XN - a)-I{u!,[x<N]=k]}, (14.1)
E[Uba[X,N)] < ^~aE[{XN - a)~). (14.2)
// (Xn) is a submartingale, then for N > 1 and k > 1
P{Uha[X,N]>k)< ^E[(XN - a)+ImXtN]=k]), (14.3)
E[U>[X,N\] < ^l-E[(XN-a)+]. (14.4)

§1. Elementary Inequalities 35
Proof. Let (Xn) be a supermartingale. Then by Theorem 2.10, for
Ar>0
0 > E[XT2k+lAN - XT2kAN]
= E[(XT2k+1AN - XT2kAN)(I[T2k<N<T2k+l] + V>7Wi])]
> E[(XN - a)I[T2k<N<T2k+l] + (b- a)I[N>T2k+l]]. (14.5)
Since [U*[X,N] > k + 1] C [N > T2M) and [T2k < N < T2M) C
[Ub[X,N] = fc], (14.1) follows immediately from (14.5). Summing the two
sides of (14.1) for k > 0 gives (14.2).
Now let (Xn) be a submartingale. Then by Theorem 2.10, for k > 1
0 > E[XT2k_XAN - Xr2kAN]
= EKXt^^aN - -XrT2fcAiv)(/[T2ib_1<N<T2ib] + hu^k})]
> E[(b - ^)/[T2ib_1<iV<T2fc] + (b- a)I[N>T2k]]
= E[(a - X^Ip^^^^] + (b- aKjjvxr^]]. (14.6)
Since [Uba[X,N] > k) C [N > T2fc-i] and [T2k.x <N< T2k] C [Uba[X,N]
= fc], (14.3) follows from (14.6). Summing the two sides of (14.3) for k > 1
yields (14.4). □
Finally, we show Doob 's inequalities in the following theorem.
2.15 Theorem. Let (Xn) be a non-negative submartingale. PutX* =
supXn. Then
n
E[X*} < -^-(l + sup£7[Xnlog+Xn]), (15.1)
e — 1 v n '
||X*||p<gsup||Xn||p, (15.2)
n
where p > 1 and q > 1 are a couple of conjugate indices: - + - = 1.
P Q
Proof. For k G N set X£ = supXn.
n<k
Let $(A) be a right-continuous increasing function on i2+ with $(0) =
0. By Fubini's theorem and (12.2) we have
«[*(**)] = / / , d*(WP = I P(X*k > \)d*(\)
JU J[0,X'} J[0,oo[
(15.3)

36 Chapter II Classical Martingale Theory
Putting $(A) = (A - 1)+ in (15.3) yields
E[(X*k - 1)] < E[(X*k - 1)+] < E[Xklog+ X*k\. (15.4)
X
Because logx < — (x > 0), for any a > 0, b > 0 we have
e
a log"1" b < a log"1" a + -.
e
Hence
E[Xklog+ XI] < E[Xklog+ Xk] + -E[Xt\. (15.5)
From (15.4) and (15.5) we obtain
E[X*k] < -Ar(l + syipE[Xk\og+Xk}). (15.6)
c - 1 it
Because X^ | X*, letting fc —> oo in (15.6) yields (15.1) by Fatou's lemma.
If put *(A) = AP, p > 1 in (15.3), then
£[TO1 < -^^[^(^r1] = qE[xk(xiy-1].
p- 1
Using Holder's inequality (note that (p — \)q = p), we get
E[(X*kr] < q(E[(Xkf})^(E[(X*kr})l/g. (15.7)
In order to show (15.2), we may suppose sup ||Xn||p < oo. Then
n
\\X*k\\p < II E *n| < E \\Xn\\p < OO.
11 n=0 "P n=0
Dividing the two sides of (15.7) by (EKX^f})1^, we obtain
||^fc||p<?l|^fc||p<?8up||X„||p. (15.8)
n
Because X£ | X*, letting k —> oo in (15.8) yields (15.2) by Fatou's lemma.
□
2.16 Corollary. Le£ (Xn) be a martingale, p > 1 and q > 1 be a
couple of conjugate indices. Then
SUp|Xn| < gSUpHXnllp. (16.1)
1 n "P n
§2. Convergence Theorems
2.17 Theorem. Let (Xn) be a supermartingale. Ifs\ipnE[X~] < oo
(or, equivalent^, supnE[\Xn\] < oo, since E[\Xn\] = E[Xn] + 2E[X~])1

§2. Convergence Theorems 37
then Xn a.s. converge to an integrable r.v. X^ as n —► +00. Moreover,
if (Xn) is non-negative, then for each n £ N
ElX^F^KXn a.s.. (17.1)
Proof. Let a, 6 £ Q, a < b. Denote by U^(X) the number of upcross-
ings of [a,b] by X = (X„)„>o, i.e., Ub(X) = Urn Uba[X,N). FVom (14.2)
we have
E[Uba(X)] < -1— sxipE[(XN - a)-} < -?—(a+ + 8upE[X„]) < oo.
o — a n b — a TV
Consequently, U%(X) < oo a.s.. Set
Wa b = liminf Xn < a, limsup Xn > 6 ,
L Ti —OO ^qq J
W = U Wa.,6.
a,6€Q,a<6
Since Wa,6 C [0*(X) = +00], then P(Wa>b) = 0, and hence P(W) = 0. If
u> ^ W, then limXn(a') exists, and is denoted by X00(a;); if a; 6 W, put
•^cxj(w) = 0- Hence, Xn —► Xoo a.s.. By Fatou's lemma
^[|X00|]<sup£;[|X7l|]<oo,
n
i.e., Xqq is integrable.
If (Xn) is non-negative, then for any m> n
E[Xm\Tn] < Xn a.s. .
Letting m —> oo, (17.1) follows from Fatou's Lemma. □
2.18 Theorem. Let (Xn) be a martingale (resp. supermartingale). If
(Xn) is uniformly integrable, then there exists an integrable r.v. Xqq such
a.s.,Ll
that Xn ► Xoo, and for each n £ N
^[Xool^nl = Xn {resp. < Xn) a.s. . (18.1)
Proof. Since (Xn) is uniformly integrable, sup£?[|Xn|] < oo (Theorem
n
L1
1.9). By Theorem 2.17, Xn -* X^ a.s.. By Theorem 1.11, Xn—► X00.
Now it is easy to deduce (18.1).
(18.1) is the general form of a uniformly integrable martingale. □
2.19 Corollary. Let £ be an integrable r.v.. Put £n = f?[^|J*n],
V = E[£\f00]. Then gw ' "' > V-

38 Chapter II Classical Martingale Theory
Proof, Because (£n) is uniformly integrable (Theorem 1.8), by Theo-
a.s.jL1
rem 2.18, £n ►^oo. Let A G (J^n- Then A e Tn for some n, and
n
E[ZoJa] = E[£nIA} = E[£IA] = E[VIA}.
Because &» and rj are .^-measurable, and ^x, = a I (J^n)» by Corollary
1.3.1) Coo = V a.s.. D
2.20 Corollary. Let (Xn) be a martingale or non-negative
submartingale, and p > 1. 7/sup£?[|Xn|p] < oo, then (Xn) is uniformly integrable,
n
a.s., lp
Xn ► Xqq , and
11X001^ = 8UP ||Jfn||p. (20.1)
n
Proof. Prom Theorem 1.7.2) we see that (Xn) is uniformly integrable.
By Theorem 2.18, Xn —► X^ a.s. Applying Doob's inequality (15.2) to
non-negative submartingale (|Xn|) yields X* = sup|Xn| G LP. Because
n
\Xn — Aoo|p < (2X*)P, by the dominated convergence theorem we have
Xn —► Xoo. Hence, (20.1) follows. □
The following theorem is a generalization of Corollary 2.19.
2.21 Theorem. Let (£n) be a sequence of integrable r.v., and£n —> £»
a.s.. If there exists an integrable r.v. £ such that for each n |£n| < |£| a.s.,
Proof Set um = inf £n,vm = sup£n. Then |um| < £, \vm\ < £ a.s.,
n>™> n>m
a.s.jL1 a.s.jL1
and um ►^oo, vm ► foo. On the other hand,
E[um\Tn] < E[£n\Fn] < E[vm\Fn], n>m.
By Corollary 2.19 we have
^[timl^oo] < liminf E[£n\Fn} < E[vm\Too} a.s., (21.1)
^[^ml^oo] < limsupjE?Kn|^n] < ^Kl^oo] a.s.. (21.2)
n—►oo
Letting m —> oo in (21.1) and (21.2) yields
limoE[Zn\fn) = E[Z00\f00) a.s..

§2. Convergence Theorems
39
Moreover, since (i5[|f ||^n]) is uniformly integrable and
\E[tn\Fn)\ < E[\£n\ \Fn] < E[\£\ \?n\ O.8.,
then (£?[£„ |.Fn]) is uniformly integrable. Therefore, by Theorem 1.11
Now we turn to study the convergence of supermartingales with the
index set -N ={•••, -2, -1,0}.
Let (Tn)ne-N be a sequence of sub-a-fields of T such that for all n G
—iV, Tn-\ C Tn. An (^rn)n€_7v-adapted stochastic sequence (Xn)ne-N
is called a martingale (resp. supermartingale), if for each n G —N,Xn is
integrable and
ElXnlTn-i] = Xn-i (resp. < Xn-i) a.s..
2.22 Theorem. Let (Xn)neN be a supermartingale. If lim 15[Xn]
n—►—oo
a.s., L1
< +oo, £/ien (Xn) is uniformly integrable and Xn ► X^^.
Proof. Denote by U%[X, —N] the number of upcrossings of [a, b] by
(X_jv, X-n+i, • • •, Xo). Then by (14.2) we obtain
EUba[X, -N) < -r^—E[(X0 - a)~}.
b — a
PutU*(X)= lim U*[X,-N]. We have
N—>+oo
£tf06(X) < -r^—E[(X0 - a)-} < +oo.
b — a
Since t/^(x) is the number of upcrossings of [—6, —a] by the sequence
(—Xo, — X-i, • • •, —X_n, • • •), from the proof of Theorem 2.17 we can
assert that Xn —► X-oo a.s.. ( Note that this conclusion holds
unconditionally, but it needn't be |X_oo| < oo a.s..)
When n —► —oo, jE[-Yn] | A > —oo. According to the assumption,
A < +oo. We are going to show that (Xn)ne-N is uniformly integrable.
Because (E[Xo\Pn])ne-N is uniformly integrable, it suffices to show that
(Xn — £?[Xo|^>i]) is also uniformly integrable. Therefore, we may suppose
that (Xn) is a non-negative supermartingale. For any given e > 0, take a
natural number k sufficiently large such that A — £?[X_jt] < -. For c > 0

40 Chapter II Classical Martingale Theory
and n < — k by the supermartingale property we have
/ XndP = E[Xn] - f XndP < E[Xn] - f X.kdP
J[Xn>c] J[Xn<c] J[Xn<c)
= E[Xn) - E[X_k] + / X-kdP.
J[Xn>c]
Because A > E[Xn] > E[X-k], for n < -Jfc we have E[Xn] - E[X-k] < -■
Li
1 A
On the other hand, because P(Xn > c) < -E[Xn] < —, when c is
c c
sufficiently large, for all n £ — N we have
k
X-kdP < 6-
[Xn>c] 2
and
/ XjdP<e, j = 0,-1, ••-,-&.
J[Xj>c]
Hence, when c is large enough we have
sup / XndP < e,
n J[Xn>c)
i.e., (Xn) is uniformly integrable. Now that Xn —> X^^ a.s., by Theorem
L1
1.11 we obtain Xn —► X-^. □
2.23 Corollary. Let £ be an integrable r.v., (Gn)neN be a decreasing
sequence of sub-a-fields of T. Put £n = 15[f |(/n]. Then
a.s., L1
n
Proof For all n G -iV, put Tn = G-niVn = £-n, then (r]n)ne-N is
a uniformly integrable martingale w.r.t. (^rOne-iV- By Theorem 2.22 we
a.s.yL1 a.s.,Ll
have r)n ► ^-oo? as n —> — oo, i.e., £n ► ^-oo? as n —► oo.
For A e f| Qn, we have
n
lin^EitnlA] = Efr-oolA].
But for all n, E[^nIA] = E[£IA], thus E^-^Ia) = E[£IA]. Because
*?-oo € PI Qn, then ij-oo = E[£\ fl £„] • °
n n
Corollaries 2.19 and 2.23 together are usually called Levy's theorem.
Below we use the martingale convergence theorem to show the strong
law of large numbers. It is one of the most brilliant examples of early
applications of martingale theory, given by J. L. Doob in 1944.

§2. Convergence Theorems 41
2.24 Theorem. Let (£n)n>i be an i.i.d. sequence of integrable r.v..
Put Xn = £ &,n > 1. Then
n
Proof. By the assumption we have
£[&|X„] = £[6l*n] a.s., i = 1, • • • ,n,n > 1.
Thus, for n > 1
— = - £ E[£i\Xn] = E[h\Xn] = E[h\Xn,Zn+1,Zn+2,■ ■ ■}
n n i=i
= E[£i\ XniXn+i,Xn+2,--] a-s- •
Xn a.s.yL1
Set Gn = &(Xn,Xn+i, • • •). Then by Corollary 2.23 we obtain ►
Z, where Z = E[^\f]Gn]- Because Z e fW£n,£n+i,"-)> by Kol-
n n
mogorov's 0-1 law Z is a.s. equal to a constant. Since E[Z] = E[£i],
Z = JE7[fi] a.s.. D
The following is a simple but important application of the martingale
convergence theorem to measure theory.
2.25 Lemma. Let (fi,^*) be a separable measurable space, T be
generated by (An)n>Q. Put Tn = <t(Aq, • • • ,An). Denote by Vn the collection
of all atoms of Tn i.e., Vn is the finite partition of ft, which generates
Tn. Let P and P' be two probability measures on (fi,^*) such that P' is
absolutely continuous w.r.t. P. Its Radon-Nikodym derivative is denoted
dP'
by ——. Put
y dP
(by convention, - = 0). Then (Xn) is a uniformly integrable (Fn)-mar-
dP'
tingale, and lim Xn = —— P-a.s..
n->oo dP
dP'
Proof. Put £ = -™-. It is well-known that
E[Z\fn]= £ E^IA = Xn P-a.s..
AePn r{A)

42 Chapter II Classical Martingale Theory
Thus (Xn) is a uniformly integrable martingale, and by Corollary 2.19 we
have
lim Xn = E[£\ V *■„] = E[£\7] = £ P-a.8. . □
2.26 Theorem. Le£ (tt,T) 6e a separable measurable space, (E,£)
be a measurable space, (Px)xeE and (Px)xeE be two measurable families
of probability measures (i.e., for each A G T, x i—► PX(A) and x i—►
P'X{A) are £-measurable functions on E) such that for each ig£, P'x
is absolutely continuous w.r.t. Px. Then there exists a non-negative real
£ x T-measurable function X(x,lj) on E x CI such that for each x G E,
X(x,-) is the Radon-Nikodym derivative of Px w.r.t. Px.
Proof. Suppose that (An)n>o generates T. We use the notations in
Lemma 2.25. Put
AeVn ^x\A)
Then Xn is £ x ^-measurable. For each x G E, by Lemma 2.25, Xn(x, •)
dP'
Px-a.s. converge to —. Now it needs only to define
Qi± x
I 0,
lirn^ Xn(x,cj), if this limit exists and is finite,
X{x,uS) =
for other cases.
§3. Decomposition Theorems for Supermartingales
In this paragraph we study the structure of supermartingales. The
main results are Doob decomposition, Riesz decomposition and Krickeberg
decomposition of supermartingales.
2.27 Definition. A stochastic sequence (Xn,n G N) is called F-
predictable, if Xq is ^b-naeasurable and for each n > 1, Xn is Tn-i-
measurable.
A stochastic sequence (Xn,n G N) is called increasing, if for each
n G iV,0 < Xn < Xn+i a.s. In this case, define X^ = lim Xn. An
n—*oo
increasing sequence (Xn,n G N) is called integrable, if jE7[Xoo] < oo.

§3. Decomposition Theorems for Supermartingales 43
2.28 Theorem. Let X = (Xn) be a supermartingale. Then (Xn) has
the following unique decomposition:
Xn = Mn-An, (28.1)
where (Mn) is a martingale, and (An) is a predictable increasing sequence
with Aq = 0. Decomposition (28.1) is called Doob decomposition of the
supermartinagle X.
Proof If (28.1) is the decomposition satisfying the requirements in the
theorem, then from the predictability of (An) and the martingale peoperty
of (Mn) we obtain
An+l - An = E[An+l - An\?n] = E[Xn - Xn+i|^n]
= Xn- E[Xn+i\Tn]'
Thus, since Aq = 0, we have
An^ZiXj-EiXj+^j}), n>l. (28.2)
This means that the decomposition satisfying the requirements is unique.
On the other hand, if define (An) by (28.2), and put Mo = Xo,
Mn = Xn + An = M0 + J^(Xj+x - E[Xj+l\Tj]), U > 1,
then Xn = Mn — An is just the decomposition satisfying the requirements.
□
2.29 Definition. A non-negative supermartingale (Xn) is called a
L1
potential, if lim i^LXVj = 0, i.e., Xn —►O.
n—►oo
Prom Theorem 1.11 we see that any potential is a uniformly integrable
supermartingale.
2.30 Definition. Let X = (Xn) be a supermartingale. If there exists
a martingale Y = (Yn) and a potential Z = (Zn) such that
Xn = Yn + Zn,
then we say that X has Riesz decomposition: X = Y + Z.
If a supermartingale X = (Xn) has Riesz decomposition, then its Riesz
decomposition is unique. In fact, if Xn = Yn + Zn and Xn = Yn' + Zn are
two Riesz decompositions, then (Yn — Y„) is a martingale and

44
Chapter II Classical Martingale Theory
By Theorem 1.11 (Yn — Y„) is a uniformly integrable martingale. By (18.1)
it must be that for each n, Yn = Y^ a.s. Hence Zn = Z'n a.s..
2.31 Theorem. 1) A supermartingale (Xn) has Riesz decomposition
if and only if lim 15 [Xn] > — oo.
n—>oo
2) Let (Xn) be a non-negative supermartingale, and Xn = Yn + Zn be
its Riesz decomposition. Then (Yn) is a non-negative martingale.
3) Let (Xn) be a uniformly integrable supermartingale, and Xn —►-Xqq.
Put
Yn = ElXoolFn], Zn = Xn — Yn.
Then Xn = Yn + Zn is the Riesz decomposition of (Xn).
Proof. 1) The necessity is trivial. We will show the sufficiency. Assume
lim 15 [Xn] > — oo, and Xn = Mn—An is the Doob decomposition of (Xn).
n—►oo
Then (An) is integrable: £[4x] < oo. Put Yn = Mn - jE^l^n], Zn =
ElAoolfn] — An. Thus, Xn = Yn + Zn is just the Riesz decomposition of
(Xn).
2) Let (Xn) be a non-negative supermartingale, and Xn = Yn + Zn be
its Riesz decomposition. Then by 1) we have
Yn = Mn- EiA^Tn] = Jim E[Mp\fn] - hm^ E[Ap\Fn)
= lim E[MP - Ap\fn] = lim E[Xp\Fn] > 0.
p—►oo ^ fi j p-+oo ri
3) is obvious. □
2.32 Theorem. Let (Xn) be a supermartingale (resp. martingale).
Then the following conditions are equivalent
1) supnf;[A'~] < oo ( resp. supjE[|Xn|] < oo,),
n
2) (Xn) has the following decomposition (called Krickeberg
decomposition) :
Xn = Ln- Mn, (32.1)
where (Ln) is a non-negative supermartingale (resp. martingale), and
(Mn) is a non-negative martingale. In addition, if I) holds, we can make
the above decomposition have minimality in a sense that if Xn = L'n — M'n
is another such decomposition, then for each n, Ln < L'n and Mn < M'n.
Proof. 2) => 1) is obvious. We will show 1) => 2). Because (—X~)
is a supermartingale and lim E[—X~] > — oo ( by 1)), by Theorem 2.31

§4. Doob's Stopping Theorem 45
(—X~) has Riesz decomposition:
~~ Xn =Yn + Zn,
where (Yn) is a martingale, and (Zn) is a potential. Put Ln = Xn — Yn,
Mn = —Yn. Then (Ln) is a supermartingale, and (Mn) is a martingale.
At the same time, we have
Ln = X+ + Zn > 0, Mn = X- + Zn > 0.
Therefore, 1) => 2) is proved. Let Xn = L'n—M'n be another decomposition
satisfying the requirements. Then M'v > X~ = Mp — Zp, and
M'n = lim^ElM^n] > Kin E[MP - Zv\Tn\ = Mn.
Immediately, L'n = Xn + M'n > Xn + Mn = Ln. □
§4. Doob's Stopping Theorem
In §1 in order to establish the elementary inequalities of martingales
and supermartingales we have already proved Doob's stopping theorem (or
so-called optional sampling theorem) for bounded stopping times. Here
we will generalize this result to more general cases. There are two kinds of
generalizations. The first is to a class of closable martingales and
supermartingales (see Definition 2.33). At this time Doob's stopping theorem
holds for all stopping times. The second is to general martingales and
supermartingales. In this case Doob's stopping theorems holds only for
certain stopping times. The latter has important applications in statistics.
2.33 Definition. A martingale (resp. supermartingale) (Xn,n G N)
is called right-closable, if there exists an integrable r.v. X^ G ^oo such
that for each n G N^ElX^^n] = Xn (resp. < Xn) a.s.. In this case
(Xn, n G N) is called a right-closed martingale (resp. supermartingale),
and Xqq is the right-closing element of (Xn, n G N).
Immediately from the martingale convergence theorem we see that
for a right-closable martingale the right-closing element is uniquely
determined. For a right-closable supermartingale there exists a maximal
right-closing element (see the remark after Theorem 2.34). Obviously by
the definition, a right-closable martingale is just a uniformly integrable

46 Chapter II Classical Martingale Theory
martingale. It must be pointed out that a martingale may be a right-
closed supermartingale, but not a right-closed martingale.
The following theorem gives a necessary and sufficient condition of
right-closability for supermartingales.
2.34 Theorem. In order that a supermartingale (Xn)n>o be right-
closable it is necessary and sufficient that (X~)n>o be uniformly integrable.
Proof. Necessity. Let X^ be a right-closing element of (Xn). Then
for each n
Xn > ^[Xool^n] > -E[X-\Fn] a.s.,
X-<E[X-\Tn] a.s..
Because (^[X^l^]) is uniformly integrable, so is (X~).
Sufficiency. Assume that (X~) is a uniformly integrable supermartin-
a.s.
gale. Then supl5[Xn] < oo, and by Theorem 2.17 Xn ► X^ , where
n
Xqq is an integrable r.v.. We will prove that X^ is a right-closing element
of (Xn). For A G Tn we have
/ XndP > f Xn+mdP = f X^mdP - f X"+mdP, m > 1. (34.1)
J A J A J A J A
L1
Because Xn+m —►-Xqo, as m —► oo (by Theorem 1.11),
j^Lx^dp=Lx~dp- <34-2>
However, X„+m —> X+, a.s., as m —► oo. Then by Fatou's lemma
/ X+dP < Um / X+^dP. (34.3)
J A ra-KX> J A
It follows from (34.1)-(34.3) that E[IAXn] > E^aX^], i.e., E^^Tn] <
Xn a.s.. This means that Xoq is a right-closing element of (Xn). □
Remark. Let (Xn) be a right-closable supermartingale. From the
above proof we see that lim Xn = X^ exists, and Xqq is a right-closing
element of (Xn). X^ is, indeed, the maximal right-closing element. In
fact, if £ is another right-closing element of (Xn), then
£ = £?[f|^oo] = lim E[£\Fn] <HmIn = X^ a.s. .
n—►oo n—>oo
The following theorem is Doob 's stopping theorem for right-closed
martingales and supermartingales.

§4. Doob's Stopping Theorem 47
2.35 Theorem. Let (Xn,n G N) be a martingale (resp.
supermartingale), S and T be two stopping times, S < T. Then Xs and Xt are
integrable, and
E[XT\FS] = Xs (resp. < Xs) a.s.. (35.1)
Proof. Let (Xn n G N) be a martingale. Put Sn = S7[s<n] +
(+oo)/[5>n]. Because the set {0,1, • • • , n, +00} is an isomorphism,
preserving order, of {0,1, •, n, n + 1}, by Theorem 2.10 we have
XSn = Sixers.] a.s..
Since ^n [S = Sn] = FSn n [5 = Sn] ( Theorem 2.9.2)), by Theorem 1.23
EiX^Fs) I[s=Sn] = EiX^FsJIis^]
= Xs„I[s=sn] = xsl[s=s„] a-s- •
Since [5 = 5n] | ft, we obtain
E[X00\fs] = Xs a.s. .
Especially, this means Xs is integrable. The same equality holds for T.
Hence,
E[XT\FS] = £[£[Xoo|*t]|*s] = EiX^s] = XS a.s. .
Now let (Xn, n G N) be a supermartingale. Put Yn = jE7[Xoo|,Fn],
Zn = Xn — yn, Voo = Xqq and Z^ = 0. Then (Yn, n G N) is a martingale,
(Zn, n e N) is a non-negative supermartingale. Because E[Zsn] < E[Zq]
(Theorem 2.10), it follows from Fatou's lemma that Zs is integrable.
Hence, Xs = Ys + Zs is integrable. Put Tn = TI[T<n] + (+oo)J[T>n].
By Theorem 2.10 we have
ZSn>E[ZTn\Fsn] a.s. ,
Zsl[s=sn) > E[ZTn \FSn}I[s=sn] = E[ZTn \Fs)I[s=sn] a.s.. (35.2)
Since Zrn T ^T? letting n —► +oo in (35.2) yields
Zs > E[ZT\FS] a.s. .
We have already shown Ys = E[Yr\Ts] a.s.. Therefore,
Xs > E[XT\?s] a.s. . □
The following theorem is a strengthened form of Theorem 2.35.

48 Chapter II Classical Martingale Theory
2.36 Theorem. Let (Xn,n G N) be a martingale (resp. supermartin-
gale), S and T be two stopping times. Then
E[Xt\Ts] = XtaS (resp. < XTaS) a.s. . (36.1)
Proof. By Theorem 2.9.2) XtI[t<s] ls ^-measurable, and by (35.1)
E[Xt\Fs] = E[XtI\t<s] + XsvtI[t>s) \^s]
= XTI[T<S] + XsI[T>S] (resP- <)
= XtaS a.s. . □
2.37 Corollary. Let £ be an integrable r.v., S and T be two stopping
times. Then
E[i\Ts\TT\ = E[S\?Sat] *■*■■
The following theorem is Doob's stopping theorem for general
martingales and supermartingales.
2.38 Theorem. 1) Let (Xn)n>o be a martingale, S and T be two
finite stopping times. Suppose that Xt is integrable. Then we have
E[XT\Fs] = Xtas a.s. (38.1)
if and only if
Jtar^ E[XnI[T>n\ | fs] = 0, a.s.,
or, equivalently,
05 E[XnI[T>n]\Fs] =0 (or Urn E[XnI[T>n]\fs) = 0) a.s..
n-»oo l - J n->oo l " J
In particular, if Urn jE[|-Yn|Ip>ni] = 0, (38.1) holds.
n—►oo
2) Let (Xn)n>o be a supermartingale, S and T be two finite stopping
times. Suppose that Xt is integrable, and
jSm^ E[XnI[T>n] \FS] > 0 a.s.
Then
E[XT\fs) < XTAS a.s. . (38.2)
In particular, if Urn E[X~I\r>rA = 0> (38.2) holds.
n—►oo

§4. Doob's Stopping Theorem 49
Proof. 1) For each n, XTAn € Fn, by Corollary 2.37 and Theorem 2.10
we have
E[XTAn\fs] = E[XTAn\^SAn] = XtaSAti,
E[XT\Fs] = Km^EiXTlpKn^s] = h^E^TAn - XnI[T>n]\rs]
= lim (XtaSati - E[XnhT>n\\Jrs\)
n—*oo l — J
= Xtas - Una E[XnhT>rAJrs\-
n—>oo l — J
This means that the limit lim E[XnI\T>n]\Fs] always exists, and (38.1)
n—►oo l — J
holds if and only if it is a.s. equal to zero. The other assertions are trivial.
The proof of 2) is similar, and is omitted. □
The following theorem, as a consequence of Theorem 2.38, is very
useful.
2.39 Theorem. Let (Xn)n>o be a martingale (resp. supermartingale),
T be a stopping time, and E[T] < oo. // there exists a constant C such
that for each n G N
E[\Xn+i - Xn\ \Tn] < C a.s. on[T>n + 1], (39.1)
then E[\Xt\] < oo, and
E[XT] = E[X0] (resp. < E[X0]). (39.2)
Proof. By Theorem 2.38, in order to show (39.2) it suffices to show that
Xt is integrable and lim i2[|Xn|iW>ni] = 0. To this end, put Yq = |Xo|,
n—►oo
Yj = \Xj - Xj-il, j > 1. Then
rT-i OOrTl -i 00
E E Yj]= E E E YjI[T=n]] = E E[YjIlT>j])
Lj=0 J n=0 Lj=0 J j=0
oo
= ZE[E[Yj\fj-l}I[T>j]} + E[Y0}
j=l
OO
<CEP(T> j) + E[Y0) = CE[T) + E[\X0\) < oo.
3=1
T
Because \Xt\ < E Yj, E[\Xt\] < oo. At the same time, we have
E[\Xn\I[T>n]) <e[Eq \Yj\Ifr>n]\ - 0, n -> oo. D
Below we use Theorem 2.39 to show the famous Wald's equation, which
is very useful in statistics.

50 Chapter II Classical Martingale Theory
2.40 Theorem (Wald's equation). Let (£n)n>i be an i.i.d. stochastic
sequence, i£[|£i|] < oo. Put Tn = <r(£i, • • • ,£n). Suppose that T > 1 is a
stopping time, and E[T] < oo. Then
e[e^]=E[^]E[T]. (40.1)
In addition, if E[^f] < oo, then
E[( E & - TEfo})2} = Dl^ElT], (40.2)
where D[£\] = E(% — (E£\)2 denotes the variance of £\.
n
Proof. Set Xn = £ £j — nl£[£i],n > 1. Then (Xn)n>i is a martingale,
and
£?[|Xn+1 -Xn||^n] = £?[|en+l " JE?Kl]| |^n]
By Theorem 2.39 we have E[XT] = E[XX] = 0, i.e., (40.1) holds.
Considering martingale (yn), where Yn = X2 — nD[£i], we can show (40.2)
similarly. □
Finally, to end this paragraph we summarize the main results about
supermartingales with discrete time as follows:
Let (Xn,n G N) be a supermartingale. Then
a.s.jL1
(Xn) is uniformly integrable <=$> Xn ►-Xoo
(X~) is uniformly integrable <=> (Xn) is right-closable
*-«,r i I •<=>■ (-^n) has Krickeberg decomposition
sup£[X~] < oo { ^ n0;.s. v • • ♦ ki
n [ =» Xn —y Xoo, Aoo is integrable
lim 15 [Xn] > —oo <=3> (Xn) has Riesz decomposition.
§5. Martingales with Continuous Time
We continue to proceed our discussion in a fixed probability space
(fi, T, P). But in §5 and §6 a filtration with continuous time F = (^i? * €

§5. Martingales with Continuous Time
51
it+) (or F = (Tt)t>o) is given, i.e., (Tt, t G R+) is an increasing family of
sub-a-fileds indexed by #+: for all 0 < s < t, Ts C Tt- Put Too = Vt>oTt
and
^i+ = n ?» t>o,
s>t
Tt-=y Ta = a(\J^X t>0.
3<t X5<t '
It is natural to define To- = To, ^oo- = ^oo- A filtration F is called
right-continuous, if for each £ > 0, Tt = Tt+> Obviously, the filtration
F+ = (Tt+,t € R+) is right-continuous.
A family of real r.v. indexed by i2+ (Xt, t G i2+) (or (Xt)t>o) is called
a stochastic process1^ or simply a process, and is also denoted simply by
X or (Xt). Apparently, a stochastic process X = (Xt(uj)) is indeed a real
function defined on fi xR+ such that for each t Xt(uj) is ^-measurable. For
each uj G fi, X.(cj) is a function on i2+ and is called a trajectory (or pa£/i, or
sample function) of X. If all trajectories of X are continuous (resp. right-
continuous, resp. left-continuous) functions on i2+, stochastic process X
is called continuous (resp. right-continuous, resp. left-continuous). If all
trajectories of X are right-continuous with finite left hand limits, X is
called a cadlag process2\ We denote by X_ = (Xt~) the left hand limit
process, where Xo- = Xq by convention, and denote by AX = (AXt) the
jump process of X: AXt = Xt — Xt-, t>0, i.e., AX = X — X-.
A stochastic prcess X ia called F-adapted if for each t > 0, Xt is
^-measurable. Put
F°(X) = (J^(X)), ??(X) = a{X3,s<t}, t>0.
F°(X) is called the natural filtration of X. Obviously, X is always F°(X)-
adapted, and if X is F-adapted, then for each t > 0 ^(X) C Tt.
For any n > 1 and *i, *2> • " »*n € ii+,
^*i,*2,•••,t»(a?lia?2>"""iZn) = P(Xtl < Xi,Xt2 < a?2,-"",Xtn < Zn)
is called a finite-dimensional (n-dimensional) distribution of X. The
collection of all finite-dimensional distributions of X is called the family of
1 * In general, any family of r.v. is called a stochastic process. A stochastic sequence
is also called a stochastic process with discrete time, and a stochastic process indexed
by an interval is called a stochastic process with continuous time.
2) The term "cadlag" is the French abbreviation of "continu a droite avec des limites
a gauche", and now has been accepted into English literature on stochastic processes.

52
Chapter II Classical Martingale Theory
finite-dimensional distributions of X. Put
(RR+,BR+) = Y[(Eu£t), (Et,£t) = (R,B(R)), t>0.
t>0
By Kolmogorov's extension theorem, the family of finite-dimensional
distributions of X determines a probability measure on (RR+, BR+), denoted
by Px or C(X), called the distribution law (or simply law) of X. In
principle, the probabilistic properties of a stochastic process are determined by
its distribution law. In fact, to determine the law of a process usually we
give its family of fimite-dimensional distributions. It is worth noting that
neither C(R+), the set of all continuous functions on i2+, nor D(R+), the
set of all cadlag functions on i2+, belong to 5**+. If process X is
continuous (resp. cadlag), we only conclude that the outer measure of C(R+)
(resp. D(R+)) under law Px is one. In this case one can define a
probability measure on (C(#+),B*+ nC(R+)) (resp. (DiR+^B** nD(R+)))
as follows: for all A G #**+,
li(A fl C(J*+)) = PX(A) (resp. fi(A n £>(#+)) = PX(A)).
We also call /i the /at/; of X, and denote it by Px.
In general, if £ C #**+,£ = BR+ C\ E, and P is a probability measure
on (E,£). Define a process X, called coordinate process or canonical
process, as follows:
*t(¥>) = <Pu <P = (<Pt) € £?, t > 0.
Then P is just the law of X.
In this paragraph first we study the properties of trajectories of super-
martingales with continuous time. Then, comparing with supermartin-
gales with discrete time, we establish corresponding results for right-
continuous supermartingales with continuous time. But Doob
decomposition for supermartingales will be discussed in Chapter V.
2.41 Definition. An P-adapted stochastic process X = (Xt)t>o is
called an F-martingale (resp. F-supermartingale, resp. F-submartingale),
if for each t > 0, Xt is integrable, and for all 0 < s < t
E[Xt\Fa] = Xs (resp. < Xs, resp. > Xs) a.s..
Obviously, any P-martingale (resp. P-supermartingale, resp. P-sub-
martingale) is a martingale (resp. supermartingale, resp. submartingale),
w.r.t. F°(X). As in the discrete time case, since the filtration P is

§5. Martingales with Continuous Time
53
fixed, the suffix "F-" will be omitted. Thus a martingale means an F-
martingale, unless clarity dictates otherwise.
Similarly, we can define martingales, supermartingales and submartin-
gales indexed by i2+ or i2+\{0}.
Now we start to study the properties of trajectories of
supermartingales. To this end, we generalize the upcrossing inequality of
supermartingales with discrete time to the case of continuous time.
Let X = (Xt)t>o be an adapted process, and u be a finite subset of
it+. Suppose u = {t\, • • •, tn} and t\ <t2 < - - <tn. Denote by U%[X, u]
the number of upcrossings of [a, b] by {Xtx, Xt2, • • •, Xtn }. For any subset
D of i2+, define
U*[X, D] = sup{C/^[X, u] : u is a finite subset of D}.
Let D = {*i, £2, • • • ? } be a denumerable set. Set un = {£1, t2, • • • > *n}-
Obviously, we have
C7a6[X,D]=nlim)C/a6[X)un].
2.42 Theorem. Le£ X = (Xt)t>o be a supermartingale , D be a
denumerable dense subset of i2+. Then for any r < s ( r,s G -R+), a < 6
(a, b E R) and X> 0 we have
XP( sup |Xt| > A) < £[Xr] + 2E[X~], (42.1)
Vt€Dn[r,a] '
EUba[X, D H [r, 3]] < -r^—E[(Xa - a)"]. (42.2)
0 — a
Furthermore, if almost all trajectories of X are right-continuous, in the
above inequalities D fl [r, 5] can be replaced by [r, s].
Proo/. Suppose D fl [r, 5] = {^, t2, • ,tn-}. ?ut ^n = {*ir • •, *n}-
The inequahties (42.1) and (42.2) with D fl [r, 5] replaced by un follow
from (12.3) and (14.2) respectively (note that (Xf) and ((Xt — a)~) are
submartingales). Then letting n —> 00 yields (42.1) and (42.2). The last
conclusion is trivial. □
2.43 Theorem. Let (Xt) be a supermartingale, and D be a
denumerable dense subset of i2+. Then for almost all u, for any t £ i2+ (resp.
t £ il+\{0}), lim Xs(u) (resp. lim Xs(u)) exists and is finite.
seD,s[it 3eDy3^t
Furthermore, if almost all trajectories of X are right-continuous, then for

54 Chapter II Classical Martingale Theory
almost all u, for any t G i2+\{0},Xt-(uj) = lim Xs(uj) exists and is
seR+,8iit
finite.
Proof. Let t G -R+, a,b e R, and a < b. Put
Htfa,b = W : sup \Xs(u)\ = oo or U*[X.(uj),D n[0,t]] = oo}.
s€Dn[Oyt]
Then i/t)a)6 G Tt. From Theorem 2.42, we know P(Ht^b) = 0. Set
Ht= U JW, # = U Ht=\JHn.
a<b,a,b€Q t€R+ n=l
Then Ht G Ji, #t T H, and P(#) = 0. If w $ H, for each t G #+ (resp.
£ G i2+\{0}), Um Xs(u) (resp. lim Xa(u;)) exists and is finite.
seD,siit seD^it
If almost all trajectoeies of X are right-continuous, obviously we have
lim X3(uj)= JmX.(u>) facte R+\{0}. □
The following theorem is called Follmer's lemma ( see Follmer [1]).
Its diflFerence from the classical results is without the requirement that To
contains all P-null sets of T.
2.44 Theorem. Let (Xt) be a supermartingale (resp. martingale),
and D be a denumerable dense subset of R+. Then there exists an F+-
adapted process (Xt) such that
1) (Xt) is right-continuousj and for almost all u, for any t G i2+
Xt(u>) = ton X,M; (44.1)
seDysiit
2) for almost all ujy for any t G i2+\{0}, Xt-(uj) = lim X3(lj)
s€R+,sWt
exists and is finite. In addition, we have
Xt-(«>)= Km Xs(u); (44.2)
3) for all t G i2+ we have
Xt > E[Xt\Ft] (resp. Xt = E\Xt\Ft]) a.s.; (44.3)
4) (Xt) is an F+-supermartingle (resp. martingale).
Proof. We continue to use the notations in Theorem 2.43. For all
t G i2+ put Ht+ = fl Hs, then Ht+ G Ji+. If u & Ht+, there is a t\ > t
3>t
such that u) & Htx. Thus, lim X8(u) exists and is finite. Put
seD,siit
lim Xs(u), v<£Ht+,
Xt(uj) = { seDMlt (44.4)
0, u) G Ht+.

§5. Martingales with Continuous Time 55
Apparently, (Xt) is an .F+-adapted process. We are going to show that
(Xt) satisfies the above-listed properties.
1) If t G R+ and u G //*+, then for all s > t, uj G Ha+. Hence,
JCs(uj) = 0, for s > t, and X.(u) is right-continuous at point t. Let
uj & Ht+. Since Ht+ = f| #r+> there is an ro > t such that for all
r>t
r G]£, ro] we have uj £ #r+- For given e > 0 take <S G (0, ro — t) such that
|Xt(cj) - Xs(u)\ < e when s e D,s > t, and s -t < 6. Thus, when r > £
and r — t < <5, we have
|Zt(u,)-Xr(u>)| = lim |Xt(w) - Xs(u,)| < e.
seD,sllr
This means that X.(uj) is right-continuous at point t. Therefore, (Xt) is
right-continuous. At last, if uj & H, then for all t G i2+ (44.1) follows
from (44.4).
2) If t > 0 and cj £ H, then lim Xa(u;) exists and is finite. (44.2)
can be shown as in 1).
3) We discuss only the supermartingale case. Let rn G D and rn [[t.
For any A G ^i we have
f XtdP> f XTndP. (44.5)
L1
Because (Xrn) is uniformly integrable (Theorem 2.22), XTn —► Xt follows
from 1). Letting n-^ooon the right-hand side of (44.5) gives
/ XtdP > f XtdP,
Ja Ja
i.e., (44.3) holds.
4) We discuss only the supermartingale case. Let s < £, s, t G i2+, and
sn G D,sn < t, sn || 5, tn G D, tn || t. For any A G J*a+. We have
f X3ndP> [ XtndP. (44.6)
./a Ja
Because (X3n) and (Xin) are uniformly integrable (Theorem 2.22), letting
n —> oo in both sides of (44.6) yields
/ X3dP > f XtdP,
Ja Ja
i.e., (Xt) is an F^-supermartingale. □
2.45 Definition. Let (Xt) and (Yt) be two stochastic processes. We
say that (Xt) is a modification of (Yt), if for each £ G i2+, Xt = V* a.s.. We

56
Chapter II Classical Martingale Theory
say that (Xt) and (Yt) are indistinguishable, if for almost all u trajectories
X.(u) and Y.(u) are identical.
Obviously, two indistinguishable processes are modifications of each
other. But the converse is not true in general. However, if two right-
continuous (or left-continuous) processes are modifications of each other,
then they are indistinguishable. Later, we will not distinguish two indis-
tingushable processes, i.e., they are regarded as the same.
2.46 Theorem. // (Xt) is a right-continuous F-supermartingale
(resp. martingale), then (Xt) is also an F-supermartingale (resp.
martingale), and almost all trajectories of (Xt) are cadlag.
Proof. This is a consequence of Theorem 2.44. Under the assumption
of the theorem, (Xt) and (Xt) are indistinguishable. □
2.47 Theorem. Suppose F = (Ft) is right-continuous and (Xt) is an
F-supermartingale. In order that (Xt) have a right-continuous adapted
modification it is necessary and sufficient that t »—► JE7[Xt] be a right-
continuous function on R+.
Proof. Let D be a denumerable dense subset of i2+. Because F
is right-continuous, the process (Xt) defined in Theorem 2.44 is an F-
supermartingale, and from (44.3) for each t > 0 we have Xt > Xt a.s..
Let tn e D,tn || t. Because (Xtn) is uniformly integrable (Theorem 2.22),
we have
E[Xt] = lim E[Xtn).
Thus, Xt = Xt a.s., or equivalently, E[Xt] = E{Xt] (since Xt > Xt a.s.),
if and only if
E[Xt}=l\moE[Xtn). (47.1)
Because s »-► E[X3] is a decreasing function, (47.1) is equivalent to saying
that this function is right-continuous at point t. Hence, if function s »—►
E[XS] is right-continuous on i2+, then supermartingale (Xt) is a right-
continuous adapted modification of supermartingale (Xt). Conversely,
if (Xt) has a right-continuous adapted modification (Yt), then E[Xt] =
E[Yt]. By the same way we know that t i-> E[Yt] = E[Xt] is right-
continuous. □
2.48 Corollary. If F is right-continuous, then any F-martingale has
a right-continuous adapted modification.

§5. Martingales with Continuous Time
57
If two processes are modifications of each other, they have the same
family of finite-dimensional distributions, i.e., in the sense of law they
axe of no difference. Therefore, we may assume that F = (Tt) is right-
continuous, and discuss right-continuous martingales or supermartingales
only. At present, we can generalize all fundamental results on martingales
and supermartingales with discrete time to the continuous time setting,
apart from Doob decomposition theorem for supermartingales. The
corresponding Doob-Meyer decomposition theorem for supermartingales will
be presented in Chapter V §4 .
The following theorem is Doob's inequality, it can be deduced directly
from Theorem 2.15.
2.49 Theorem. Let (Xt) be a non-negative right-continuous submar-
tingale and X* = sup Xt- Then
t>o
E[X*] < -^-(l + sxiPE[Xtlog+Xt]), (49.1)
\\X*\\p<qsup\\Xt\\q, (49.2)
t>o
where p > 1 and q > 1 are a couple of conjugate indices.
The convergence properties of martingales or supermartingales are
listed below. Their proofs are completely similar to those for the discrete
time case, and are omitted here.
2.50 Theorem. Let (Xt) be a right-continuous supermartingale. If
s\ipE[Xf] < oo (or equivalently, sup£?[|X^|] < oo), then (Xt) a.s. con-
t t
verges to an integrable r.v. X^ as t —► oo. Furthermore, if (Xt) is
non-negative, then (Xt,t G i2+) is a supermartingale.
2.51 Theorem. // (Xt) is a uniformly integrable right-continuous
supermartingale (resp. martingale), then
a.s.,Ll
Xt — ► Xqq , as t —► oo
and (Xt,t G i2+) is a supermartingale (resp. martingale).
2.52 Corollary. Assume (Ft) is right-continuous. Let £ be an
integrable r.v., and (&) be a right-continuous adapted modification of martin-

58
Chapter II Classical Martingale Theory
gale(E[£\Ft]). Then
a.s.yL1
6 ►tfKl^oo], as t^oo.
2.53 Corollary. Let (Xt) be a right-continuous martingale (or non-
negative submartingale) and p > 1. If supE[\Xt\p] < oo, then
t>o
a.s.yif
Xt ►^oo, OS t —► OO,
and H-Yoollp = sup||Xt||p.
t>o
2.54 Theorem. Let (Xt)t>o be a right-continuous (Tt)t>o-supermar-
tingale. If To = .7-0+ and supE[Xt] < oo, then Xt a.s. and L1 converge
t>o
to an To-measurable integrable r.v. Xo as t j 0, and (Xt)t>o is an (Ft)-
supermartingale.
Now we study Riesz decomposition of a supermartingale. For the time
being, we cannot establish Doob decomposition of a supermartingale. So
the line of proof here is different from that for discrete time case. As
to Krickeberg decomposition concerned, the formulation and proof are
completely similar to those for the discrete time case, and are omitted
here.
2.55 Definition. Let (Xt) be a non-negative right-continuous
supermartingale. (Xt) is called a potential, if lim E[Xt] = 0.
t—►oo
Suppose that X = (Xt) is a right-continuous supermartingale. If there
exists a right-continuous martingale Y = (Yt) and a potential Z = (Zt)
such that
Xt = Yt + Zu (55.1)
we say that X has Riesz decomposition: X = Y + Z. It is easy to see that
if X has Riesz decomposition, then the decomposition is unique.
2.56 Theorem. Assume that (Ft) i>s right-continuous and (Xt) is a
right-continuous supermartingale.
1) (Xt) has Riesz decomposition if and only if lim i^X*] > —oo.
t—+oo
2) Suppose that (Xt) has Riesz decomposition (55.1). If (Xt) is non-
negative, so is martingale (Yt).
3) If (Xt) is uniformly integrable, then
L1
Xt—► Xqq, as t —> oo.

§5. Martingales with Continuous Time
59
Let Zt = Xt — Yt and (Yt) be a right-continuous adapted modification of
martingale (JE7[Xoo|Jf]). Then (Zt) is a potential.
Proof. 1) The necessity is trivial. We are going to show the sufficiency.
Put
YtiS = E[Xt+s\Ft], t,seR+.
For s > r,
YtyS = E[E[Xt+3\ft+r]\ft] < E[Xt+r\ft} = YttT, a.s.
L1
Define Yt = lim Yt n a.s.. Then Yt n —► Yt, as n —> oo. For t > s,
n—►oo ' '
E[Yt\Ts] = lim E[Yt<n\fs}= limE[Xt+n\Ts]
n—►oo ' n—►oo
For each t G i2+ choose Yt G Tt, then (Yt) is a martingale. Since (Ft)
is right-continuous, by Corollary 2.58 (Yt) has a right-continuous adapted
modification, denoted also by (Yt). Set Zt = Xt — Yt. Because for all
n, Ytyn < Xt a.s., we have Yt < Xt, a.s.. Hence, (Zt) is a non-negative
supermartingale, and
E[Zt] = lim E[Xt - Yt<n] = lim E[Xt - Xt+n]
n—►oo ' n—►oo
= E[Xt] - lim E[Xt],
a—►oo
lim E\Zt\ = lim f;[Xt] - Urn jELYJ = 0,
t-*oo t-too s-*ao
i.e., (Zf) is a potential.
2) can be easily seen from proof 1). 3) is trivial. □
Now we turn to establish Doob's stopping theorem of martingales
(resp. supermartingales) with continuous time. The discussion will be
restricted to closable martingales (resp. supermartingales). Of course, we
need to introduce the concept of stopping time. But the detailed
discussion about stopping times will be given in Chapter III §1.
2.57 Definition. An i2+-valued r.v. T is called an F-stopping time
or optional time if for each t > 0, [T < t] G Tf
For each stopping time T put
Ft = {A e Too'- VtGfl+, A[T<t]eTt}- (57.1)
This is a a-field, called the a-field of events prior to T.

60
Chapter II Classical Martingale Theory
An F+-stopping time T is called an F-stopping time in the wide sense.
The a- field of events prior to T w.r.t. F+ is denoted by Tt+, ie.,
J*T+ = {AeJrOQ: Vt € R+, A[T <t]e 7i+}. (57.2)
2.58 Theorem. Let (Xt,t G i2+) 6e a right-continuous martingale
(resp. supermartingale), and S <T be two stopping times. Then Xs and
Xt are integrable, and
E[XT\TS] = Xs (resp. < Xs) a.s. . (58.1)
Proof. We give the proof only for the supermartingale case. For each
1 2
natural number n put Dn = {0, —, —, • • •, +oo}. Then \Xt, t G Dn) is
Li Li
an (Tt,t G Z}n)-supermartingale. Put
00 Jb
Sn = Si F7[^<s<£] + (+0°)7t*=+~]'
00 k
Tn = Si ¥Ii^<T<^} + (+°°)/P,=+oo]-
5n and Tn are (Tt,t G Dn)-stopping times (see Theorem 3.7.2)), and
Sn I S, Tn I T. By Theorem 2.35 we have
^[XTn|^J<X5na.s.,
where Xrn and Xsn are integrable. In particular, for any Ag^C Fsn
(see Theorem 3.4.2))
I XTndP< I XSndP. (58.2)
But
lim Xrn = Xt, lim Xsn = X$,
n—►oo n—►oo
and Xt G Tt,Xs G J5 (see Theorem 3.12). In order to deduce (58.1)
from (58.2), it remains to show that (Xsn) and (Xrn) are uniformly
integrable. By Theorem 2.35 we know for each n > 1
E[XSn_l\rSn\<XSn a.s..
Set
Y-n = XSn, G-n = ?sn, n>l.
Then (Y"n)n<-i is a (£n)n<_i-supermartingale. Since -E[V-n] = E[Xsn] <
E[Xo] (Theorem 2.35 ), (Yijn^-i = (-Xsn)n>i is uniformly integrable
(Theorem 2.22), so is (XTn). □

§5. Martingales with Continuous Time 61
The following theorem is a strengthened form of Doob's stopping
theorem. Its proof is completely similar to that of Theorem 2.36 and is omitted
here.
2.59 Theorem. Let (Xt,t G i2+) be a right-continuous martingale
(resp. supermartingale), and S,T be two stopping times. Then
E[XT\Ts] = XTAS (resp. < XTas) a.s.. (59.1)
2.60 Corollary. Suppose that (Tt) is right-continuous. Let £ be an
integrable r.v., and S,T be two stopping times. Then
E\Z\Ft\Fs] = E[S\Ts\Ft] = E[t\TSAT] a.s. . (60.1)
Proof. Let (Xt) be a right-continuous adapted modification of
martingale (EfclTt]), and Xoo = Eltlfoo]. Then (60.1) follows from (59.1).
□
2.61 Corollary. Let (Xt)t>o be a right-continuous supermartingale.
Then for any stopping time T
E[\XT\I[T<oo]]<3supE[\Xt\}. (61.1)
t>o
Proof. Let a > 0. Put X$ = Xt^X^ = Xa. Then (X?,t G R+) is a
supermartingale and X% = Xtacl- Noting that (Xf) is a submartingale,
by Theorem 2.58 we have
E[\XTAa\ I[T<oo]] < E[\XTAa\] = E[XTAa] + 2E[X^a]
< E[X0] + 2E[X~] < 3sxipE[\Xt\\.
t>o
Letting a —> oo yields (61.1). □
The follwoing theorem is a simple application of Doob's stopping
theorem.
2.62 Theorem. Let (Xt) be a non-negative right-continuous
supermartingale. Put
Tn = inf{t: Xt < -}, T = supTn. (62.1)
n n
Then T is a wide-sense stopping time, and for almost all u G [T < oo]
Xt(u>) = 0, t > T(w),

62
Chapter II Classical Martingale Theory
for all ue [T>0], t < T(u),
Xt(u>)>0, limXa(a;)>0,
(T is the time when (Xt) attains to zero).
Proof. For any t G i2+
[Tn < t) = U [*r < -] € Fu
r<t,r€Q+ n
thus [Tn < t] G .7-*+, i.e., Tn is a wide-sense stopping time, so is T.
Put Xqq = 0, then (Xt, t G i2+) is a right-continuous supermartingale.
Prom Theorem 2.46 we know that (Xt, t G i2+) is an F+-supermartingale.
By Theorem 2.58 we have
E[Xrvt] < E[Xrn] < -•
n
Since n is arbitrary, E[Xrvt] = 0 and X^Vt = 0 a.s.. In particular,
XtI[t>T\ = XrvtI[t>T] = 0 a-s-- (62.2)
Because all trajectories of (Xt) are right-continuous, it is deduced from
(62.2) that for almost all u G [T < oo], we have Xt(uj) = 0 for all t > T(u).
On the other hand, we have
oo
[T > t) = U [Tn > t\.
n=l
Let t < T(u). Then there exists n such that Tn(u) > t. By the definition
of Tn, for all s < t,X3(uj) > —, and limX,(u;) > —. □
n a|t n
2.63 Definition. A filtration F = (Ft) is called complete, if To
contains all P-null sets. If a filtration F = (Tt) is both complete and
right-continuous, we say that F = (Ft) satisfies the usual conditions.
If F satisfies the usual conditions, (Xt) is an F-supermartingale, and
£?[Xf] is right-continuous on i2+, then from Theorems 2.46 and 2.47 we
know that (Xt) has an adapted cadlag modification, which is also an F-
supermartingale. In particular, if F satisfies the usual conditions, each
F-martingale has an adapted cadlag modification, which is also an F-
martingale.

§6. Processes with Independent Increments
63
Let (Xt) be a non-negative right-continuous F-supermartingale. Put
Si = inf{t :Xt = 0oi Xt- = 0},
52 = inf{t : Xt = 0},
53 = inf{* : Xt- = 0},
Tn = inf it : Xt < -), T = supTn.
Then by Theorem 2.62 we have
S\ = S2 = S3 = T a.s. .
If F satisfies the usual condition, it is not difficult to see that Si, S2, S3
and T all are F-stopping times.
An arbitrary filtration can always be completed. First, we complete
the probability space (fi, F, P), i.e., we suppose that the space (ft, T, P)
is complete: T = Tp. Then denote by M the a-field, generated by all
P-null sets. For any filtration F = (^t)t>o put
Fp = (Jiv"JSOt>o-
Then Fp is complete, called the completion of F. Obviously, Fp satisfies
the usual conditions. It is called the usual augmentation of F and is
denoted by F = (JS0t>o- It is not hard to show that for each t > 0
Tt+VAf= n(^VAT) = ^,
3>t
i.e. F = (?t+\/N)t>v.
For any process X = (Xt)t>o, we denote by FP(X) the completion
of its natural filtration F°(X), and FP(X) is called the complete natural
filtration of X. And denote by F(X) the usual augmentation of F°(X),
and F(X) is called the usual natural filtration of X. Obviously, if two
processes are modifications of each other, they have the same complete
natural filtrations.
§6. Processes with Independent Increments
2.64 Definition. A process (Xt) is called stochastically continuous
on i2+, if for each t G i2+, X3 converge to Xt in propability as s —► t.
We say that a process (Xt) has stationary increments, if for all s <
t,s,t G .R+, the law of X* — X3 depends only on t — s.

64
Chapter II Classical Martingale Theory
An adapted process (Xt) is called a process with independent
increments (w.r.t. F = (ft)), if for all s < t, s, t G -R+, Xt — X3 is independent
of Ts- In this case, for any 0 < to < t\, • • • < tn
Xto,Xtl -XtQ,-',Xtn- Xtn_x are independent. (64.1)
The independence property (64.1) (in particular, the increments of
disjoint intervals are independent) is equivalent to the fact that (Xt) is a
process with independent increments w.r.t. its natural filtration F°(X).
If the filtration is not specified, a process with independent increments
is always meant w.r.t. its natural filtration. A process with independent
increments in called homogeneous, if it has stationary increments.
It is easy to see that if (Xt) is a process with indpendent increments
w.r.t. F, so is (Xt) w.r.t. Fp (the completion of F). Moreover, if (Xt)
is stochastically continuous, then (Xt) is also a process with independent
increments w.r.t. F (the usual augmentation of F). For simplicity, we
call a stochastically continuous process with independent increments as
Levy process, whether or not it is homogeneous. It will be the object of
study in this paragraph. Below we suppose the filtration satisfies the usual
conditions.
For any Levy process X denote by <Ps,t(u) the characteristic function
of increment Xt — X3(s < t):
ipSyt(u) = E[exp{iu(Xt - Xs)}].
By the independence of increments, we have
iprAu)<Pst(u) = ¥V,t(*0, r < s < t. (64.2)
By the stochastic continuity of X we know that (fi3yt(u) is continuous in
5, t and u.
2.65 Lemma. Let X be a Levy process. Then for all u G R, s,t G
R+,s < t
^s,t(w)^0.
Proof. Set to = inf{£ > s : <p3tt(u) = 0}. Since <p3,3(u) = 1, it must be
to > s. It suffices to show £o = oo. In fact, if £o < °°, then <£s,t0(u) = 0.
From (64.2),
ip3,t(u)ipttto(u) = 0, s < t < t0.
Because <ps,t(u) ¥" °, we have <pt,t0(u) = 0. Letting 11| t0 yields ipt0yt0(u) =
0. This contradicts ^t0,t0(w) = 1- n

§6. Processes with Independent Increments 65
2.66 Theorem. Let X be a Levy process. Put
Z3,t(u) = [pa9t(u)]-1 exp{iu(Xt - X,)}, s < t. (66.1)
Then (Z3j(u))t>3 is an {Tt)t>s-martingale.
Proof. Let s <t0 <t. Then
E[Z3yt(u)\Tt0) = [^t(u))-lexp{iu(Xt0 - X3)}E[eiu^-^o)\Tto]
= ^{xtQ-xa)^^ = {u) u
Martingale (Z3}t(u))t>3 will play an important role in the study of
processes with independent increments. For simplicity, Zqj and <^o,t are
denoted by Zt and <pt respectively.
2.67 Lemma ( Ottaviani's inequality). Suppose that independent r.v.
fi>f2> • • • >fn satisfy the following conditions:
P(\Zk + ''' + U > a) < a, k = 1,2, • • •, n,
where a > 0 and a G (0,1) are constants. Then for any b > 0
P( sup |£1 + ... + £fc|>a + 6)<—?— P(|£1 + ... + £n|>6). (67.1)
v i<fe<n ' 1-a
Proof. Put
4b = [|£i + ---+&|>a + &],
Bfc = [|& + ---+£n|>a], fc = l,2,---,n, Bn+i = 0,
C = [|£i + --- + £„|>6].
Then
U (AkBck+l) C C,
fc=i
£ P{A\ ■ ■ ■ Ack_!AkBck+1) < P( (J AfcB£+1) < P(C).
fe=i v fc=i '
On the other hand, Bk+l is independent of {A\ • • • Ak_lAk). Thus,
P(C) > t P(A{ • • • Al^A^l^) = £ P(A\ • • • A%^Ak)P(Bi^)
>(l-a)±P(A<i...Ack_lAk) = (l-a)P( (J Ak).
(67.1) follows immediately. □

66 Chapter II Classical Martingale Theory
2.68 Theorem. Every Levy process has an adapted cadlag
modification, which is also a Levy process.
Proof. Let X = (Xt) be a Levy process. First, we show that for every
c> 0
P(sup{|Xt| : t G [0,c] flQ} < oo) = 1. (68.1)
In fact, for each t G [0, c]
P{\XC — Xt\ > n) I 0, as n —► oo.
By stochastic continuity of X, for each n and to G [0, c],
lim P(\XC - Xt\ >n)< P(\XC -Xto\> n).
t—►to
Hence, by Dini's theorem we have
Take n\ such that
lim sup P(\XC - Xt\ > n) = 0.
n—oo o<f<c
sup P(\Xc-Xt\>ni)<±
0<t<c *
By Lemma 2.67 we obtain
P( sup \Xt\>n + ni) <2P(\Xc\>n). (68.2)
Then (68.1) follows from (68.2).
Now applying Follmer's lemma to martingale (Zf(u))t>o, we know that
there exists fio € T with P(fio) = 1 such that for each cj G fio
1) for all c > 0, sup |At(u;)| < oo ,
t€[o,c]no
2) for all u G Q and * > 0 (resp. * > 0) lim eiuXr^ ( resp.
r€Q+,rUt
lim e*uXr(u))\ exists and is finite. Furthermore, we can conclude that
r€Q+,rTT*
for each u G fio, for all £ > 0 lim Xr(u;) (resp. for all t > 0,
r€Q+,r||t
lim Xr(u;)) exists and is finite. In fact, suppose there exist two se-
r€Q+,rTT*
quences (rn) and (r'n) in Q+ such that rn [[t,r'n [[t and
limXrn(u;) = a ^ b = ]imXTfn(uj).
Obviously, a and b are finite. For all u G Q, elua = elub. This is impossible
when 0 < u < r.
|6-a|

§6. Processes with Independent Increments 67
Define
Yt(«>) = I
Jim Xr(uj), u) e fi0,
reQ+Hlt t > 0>
0, u) £ fio?
Then Y = (Yt)t>o is a cadlag process. By stochastic continuity of X, for
each t > 0
P(xt = yt) = l,
i.e. Y is a cadlag modification of X.
Since the filtration satisfies the usual conditions, it is easy to see that
Y is adapted. Hence, Y is also a Levy process. □
The following theorem provides another connection between processes
with independent increments and martingales.
2.69 Theorem. Let X = (Xt) be a process with independent
increments, and for all t > 0,E[\Xt\] < oo. Put mt = E[Xt],t > 0. Then
(Xt — mt) is a martingale.
Moreover, if dt = D[Xt] < oo,t > 0, then ((Xt — mt)2 — dt) is a
martingale.
Proof Since Xt — Xs, s < t, is independent of Ts, we have
E[Xt — Xs\Ts] = E[Xt — X3] = mt — ma, a.s. ,
E[Xt — mt\Ts] = Xs — raa, a.s..
Hence (Xt — mt) is a martingale.
Now suppose dt = D[Xt] < oo. Without loss of generality, we suppose
mt = 0. Then for s < t
E[\Xt - XS\2\TS] = E[\Xt - Xs\% a.s. . (69.1)
Since X3 and Xt — Xs are independent, we have
dt = D[Xt) = D[X3) + D[Xt - X3] = d3 + E[\Xt - Xs\2]. (69.2)
On the other hand,
E[\Xt - Xs\2\f3] = E[X?\fs) - 2X.E[Xt\Ft] + X2S
= E[X?\fs]-X2a a.s.. (69.3)
From (69.1)-(69.3) we obtain
E[X2-dt\fs} = X23-ds a.s..
Hence (X2 — dt) is a martingale. □

68 Chapter II Classical Martingale Theory
For a homogeneous Levy process X = (Xt) the characteristic function
(Ps,t(u) °f increment Xt — Xs, s < t, depends only on t — s:
<PsAu) = <Pt-s(u),
and (<ft(u))t>o satisfies the following functional equation:
<Pt+s(u) = ipt(u)tp3(u), t,s£ R+.
Because (ft(u) is continuous in t, we have
<Pt(u) = [ViM]'
and <fii(u) is infinitely divisible.
Furthermore, if the expectations of (Xt) exist, then
rrit = mo + (mi — mo)£, t > 0.
If the variances of (Xt) exist, then
dt = d0 + (di - d0)*, t > 0.
2.70 Theorem. Let X = (Xt) be a cadlag homogeneous Levy process,
and T be a finite stopping time. Put
Yt = Xx+t — Xt, t >0.
Then
1) Y = (Yt)t>o is indpendent of Ft,
2) Y is a process with independent increments w.r.t. (TT+t),
3) Y has the same law as X — Xq.
Proof Since (Zt = —;—-:elu^Xt~x°^) is a cadlag martingale, for any
<Pt(u)
bounded stopping time S by Theorem 2.58 we have
E[Zs+t\fs] = %s, 0"S.,
E[e^s+t-xs)^s] = <*>±<M = ^(tt) a.s„ (7°-1}
For any A € FT and n, A[T < n] 6 ^tati- Applying (70.1) toT An
we get
EilAlTKn]^*} = EilAp^e^XT^+'-XT.n)}
= E[IA[T<n]<pt(u)} = Mu)P(A[T < n]). (70.2)
Letting n —► oo in (70.2) yields
E[IAeiuY<) = E[IA)<pt(u). (70.3)

§6. Processes with Independent Increments
69
Putting A = fi in (70.3), we obtain
E[eiuY<] = ^(ti),
E[IAJ»Y'] = E[IA]E[J«Y<].
Hence Y% is independent of Tt, and Y% has the same law as Xt — Xo-
For 0 < s < t, applying the assertion proved above to stopping time
T + s, we know that Xr+t — Xt+s is independent of Ft+s- Therefore, Y =
(Yf) is a process with independent increments w.r.t. (Tr+t), independent
of Tt, an(i has the same law as X — Xq. □
Remark. In Theorem 2.70 if the stopping time T is equal to +oo,
then the process Y is only defined on [T < oo]. Replacing (fi,^*, P) by
([T < oo], TH [T < oo], P(-)/P(T < oo)), the assertions remain true.
2.71 Definition. A stochastic process W = (Wt)t>o is called a
Wiener process or Brownian motion (w.r.t. (^i)), if the following
conditions are satisfied:
1) Wo = 0,
2) W is a process with independent increments (w.r.t. (^i)),
3) for all s < t, s, t G R+, Wt - W3 has normal distribution JV(0, a2(t -
s)),a2>0.
If a = 1, the Wiener process is refered to as standard. Obviously, a
Wiener process is a homogeneous Levy process.
It has long been observed that particles suspended in a liquid are in
a state of constant highly irregular motion, so-called Brownian motion,
named after the British botanist who discovered it first. The Wiener
process is a reasonable mathematical model for Brownian motion. In
fact, the Wiener process has been found useful in fitting many random
phenomena of diverse domains.
2.72 Definition. A stochastic process X = (Xt)t>o is called Gaussian
or normal if all finite-dimensional distributions of X are Gaussian.
Obviously, the law of a Gaussian process X is completely determined
by its mean function
rot = E[Xt], t > 0
and covariance function
C(t, s) = E[(Xt - mt)(X3 - ma)], t, s > 0.

70 Chapter II Classical Martingale Theory
It is easy to see that the Wiener process is a Gaussian process, and
E[Wt] = 0, t > 0,
(72.1)
E[WtWs] = (T2(tAs), t,s>0.
Conversely, if W = (Wt) is a Gaussian process, and its mean function
and covariance function are specified as in (72.1), then W is a Wiener
process w.r.t. F(W).
2.73 Theorem. Let W = (Wt) be a Wiener process. Then W has an
adapted continuous modification, which is also a Wiener process.
Proof. Since a Wiener process is a homogeneous Levy process, by
Theorem 2.68 W has a cadlag modification X = (Xt). Put
oo oo oo oo p2n 1
h= u u u n u[i*^-*i^i>-].
p=lm=ll=ln=lj=l 2* 2« m
It is not difficult to see that if u G Hc,X.(u) is continuous on i2+. On
the other hand, for the fixed p and m
/ P2n 1 \ P2n 1
p( u [\x* -x^\ > -) < £ P{\x* -x^\ > -)
<m*P£E[\x^-x^\*}
= m^(P2«)(3^) (*£ - X<_. ~ AT(0, £))
= 3pm4a* • — —> 0, as n —► oo.
Consequently, for any I
oo p2n 1
p(n u[i^jL-^i=ii>-) = o.
Therefore, we know
P(#) = 0.
Now define
yt(cj) = { ^ *>o.
J Xt(w), u; G i/c
\ 0, (JGff,
It is clear that y = (If) is a continuous modification of W = (Wt). Since
the filtration satisfies the usual conditions, Y is adapted. □

§6. Processes with Independent Increments
71
Remark. In view of Theorem 2.73, henceforth we add another
requirement to the definition of Wiener process: it is a continuous process.
2.74 Theorem. Let W = (Wt) be a Wiener process. Then for each
t>0,
on n q f2
£ (W±t - Wi^i t)2 ■ a\ as n^oo.
Proof. Put
vn=Y,{w^t-wi^t)\
7=1 2" 2*^
Since {(W±t - W^t) : j = 1, • • •, 2n} are i.i.d. and
we have
m.)=J: mwj,, - ^.n=2" p=°%
dk) = £ D[\w^t - wvjn=2" • 2(^t)2 - f£-
Since D[iy —► 0, we obtain V^ —► a2t.
On the other hand,
00 1 °° >. « °° 7?^
E P(|vn - E\vn}\ > -) < e «2^n] = *4'2 £ ^n < 00.
n=l " n=l n=l *
a.s.
By Borel-Cantelli Lemma we obtain Vn ► a2t. D
Remark. For each t and n, let Dn : 0 = £njo < £n,i < • • • < £nfc„ = *
be a partition of [0,t] such that max |£nj — tnj-i\ —► 0 as n —> 00. In the
same way one can show for a Wiener process W
Moreover, if Dn+i is a refinement of Dn for each n, then we have

72 Chapter II Classical Martingale Theory
as well. In fact, define V_n = £ (WtnJ - W^-.J2, n > 1, then (Vn)n<-i
is a martingale w.r.t. a due filtration. The details are left to readers as
an exercise.
2.75 Definition. A stochastic process N = (Nt)t>o is called a
(homogeneous) Poisson process with parameter (or rate) A > 0 ( w.r.t. (^i)),
if the following conditions are satisfied:
1) N0 = 0,
2) AT is a process with independent increments (w.r.t. (^)),
3) for all s < t,s,t £ i2+, Nt — N3 has a Poisson distribution with
parameter X(t — s).
2.76 Theorem. Let N = (Nt)t>o be a Poisson process. Then N has
an adapted cadlag modification, whose trajectories are all increasing step
functions with jump size one and take values only in the set of non-negative
integers.
Proof. Since Poisson process AT is a homogeneous Levy process, by
theorem 2.68 N has a cadlag modification X = (Xt). Put
a=( n [xreN])n( n [xt-xs>o]).
Obviouly, P(A) = 1, and for wGi, X.(u) is a cadlag increasing function
and takes values only in N.
Put
oo oo oo p2n
ff=uun u[i^-^i>2].
p=ll=ln=lj=l 2* 2*"
Then for any fixed p
p (£«** - x^ * 20 * %pv+ ~x^^
= p2n(l - e-A2_n - A2-ne-A2_") ^ 0, as n -» oo.
Hence P(17) = 0. It is not difficult to see that for u> € AHC, X.(u>) is a
cadlag increasing step function with jump size one and takes values only
in JV.
Now define
\ 0,
w € AHC,
Yt(u) = J t>0.
u><? AHC,

§6. Processes with Independent Increments 73
Then Y = (Yt) is the required modification of N. □
Remark. By the same reason as in the case of Wiener process, from
now on we add another requirement to the definition of Poisson process,
i.e., all trajectories of a Poisson process are cadlag increasing step functions
with jump size one and take values only in N.
2.77 Theorem. Let N = (Nt) be a Poisson process with parameter
A. Put
Tn = inf{*>0: Nt = n}, n > 1. (77.1)
Then
1) for each n, Tn is a finite stopping time,
2) T\ has the exponential distribution with parameter A,
3) (Ti,T2 — Ti, • • • ,Tn — Tn_i • • •) is an i.i.d. sequence.
Proof. For any n > 1 and t > 0
[Tn <t} = [Nt > n}.
Hence Tn is a stopping time. Moreover,
P(Tn < t) = 1 - £ ^Le-Xt -> 1, as t -> oo.
i=o J'
Therefore, Tn is finite. In particular,
P(Ti <t) = l-e~At, ^>0,
i.e., T\ has the exponential distribution with parameter A.
Define
Yt = NTl+t - NTl, t>0.
By Theorem 2.70 Y = (Yt) is a Poisson process with parameter A w.r.t.
(^Ti+t)? and independent of Ttx- In addition,
T2-Ti= ini{t > 0 : Yt = l}.
Therefore, T2 — T\ is independent of T\ (since T\ G ^Ti)? and identically
distributed with T\. In the same way, by induction one can show that (Ti,
T2 — Ti, • • •, Tn — Tn_i, • • •) is an i.i.d. sequence. □
Remark. Actually, a Poisson process N = (Nt) can be represented
as follows:
Nt = n, when Tn < t < Tn+i,
where Tn,n > 1, is defined in (77.1), and T0 = 0.

74
Chapter II Classical Martingale Theory
Suppose that (5n)n>i is an i.i.d. sequence of r.v., and their common
distribution is the exponential distribution with parameter A > 0. Define
f0 = 0, fn = 5i + ---S'n, n>\
and
Nt = n, fn<t<fn+i, n>0.
Then N = (Nt)t>o is a Poisson process with parameter A w.r.t. F(N),
since N has the same law as N.
Generally, let (Tn)n>o be a sequence of r.v. such that
l)0 = T0<T1<...Tn<...,TnToo,
2) for each n > 0, Tn < oo => Tn < Tn+i.
Define
Xt = n, when Tn < t < Tn+i.
X = (Xt) is called a counting process or poin^ process. Counting processes
are used to model events occurring in time, such as arrivals of customers
in a queue, calls coming into a telephone exchange, accidents on roads etc.
Hence, Xt may be considered as the number of customers arriving in [0, £],
and Tn is called the n-th arrival time (Tn = oo means the n-th customer
never occurs). The Poisson process is the simplest but most important
and useful type of counting processes.
Problems and Complements
2.1 Let (Xn)n>\ be an i.i.d. sequence of r.v.. Assume that X\ is
distributed uniformly on (0,1). Let d G (0,1). Set
T = inf{n > 1 : Xn> d},
5 = inf{n> 1 : Xx + --Xn> 1}.
Evaluate E[T], E[XT], E[S] and E[XS\.
2.2 Let (Xn)n>\ be an i.i.d. (^rn)-adapted sequence of r.v. such that
for each n > 1, Xn+\ is independent of Tn, and T be a finite (^rn)-stopping
time. Then
1) (Xr+n)n>i is independent of Tt,
2) (Xr+n)n>i has the same law as (Xn)n>i.

Problems and Complements
75
2.3 Let (Xn)n>\ be an (^rn)n>i-adapted sequence of non-negative r.v.
such that for each n > 1, Xn+\ is independent of Tn. Then
J5[sup-Xn] < 2sup{E[XtI[t<oo]] : T is an (J*n)-stopping time}
n>l
(prophet's inequality).
2.4 Let (Xn)n>o be a martingale. Then (|-yn|)n>o ls also a martingale
if and only if
1) for each n > 1, XnXo > 0, a.s.,
2) for each n > 0, [Xn = 0] C [Xn+i = 0], a.s..
2.5 Let X = (Xn) be an (^>i)-martingale (resp. supermartingale),
and T be an (^rn)-stopping time. Then the sequence stopped at time T
XT = (X„) (defined by X„ = Xtati) is als° an (^rn)-martingale (resp.
supermartingale).
2.6 Let a<b,a,b,e R and N > 1.
1) If (Xn) is an (^rn)-supermartingale, then
P(t;a6[x,iV]>i|jr0]<
■£—^{E[(XN - a)"/lt/6[X)N]=0]|^b] - (*o - a)-},
E[Uba[X,N}\?0] < -*-{E[(XN - a)-|jT0] - (X0 - a)"}.
o — a
2) If (Xn) is an (,Fn)-submartingale, then
(X0 - a)+ < E[(XN - a)+/[t/»[X)N]=0]|Jb],
E[U*[X, N]\T0] < ^{E[(XN - a)+|jT0] - (X0 - a)+}.
2.7 Let (Xn) be a martingale, and i£[sup |Xn+i — Xn\] < oo. Then for
n
almost all cj, either lim Xn(u) exists and is finite, or limsupXn(u) = +oo
n—°° n—oo
and liminf Xn(u) = — oo hold simultaneously.
n—►oo
2.8 Let (An) be an (^r7l)-adapted sequence of events. Then
li^ An = \ZP[An\Fn-i) = +ool a.s.
n—►oo L n J
(generalized Borel-Cantelli Lemma).
2.9 Let (Xn) be a non-negative supermartingale. Then (Xn) is a
potential if and only if there is a non-negative martingale (Yn) such that
for each n, Yn < Xn a.s. must be equal to zero: for all n, Yn = 0 a.s. .
2.10 Let (Xn) be an (^rn)-adapted sequence of integrable r.v.. Then
(Xn) is a potential if and only if there exists an (J*n)-adapted integrable

76 Chapter II Classical Martingale Theory
increasing sequence (An) such that for all n
Xn = ElA^Tn] - An a.s. .
Moveover, if (An) is required to be predictable and Aq = 0, then (An) is
determined uniquely.
2.11 Let (Yn)n>o and (Zn)n>o be two non-negative martingales. Then
Xn = Yn - Zn, n > 0
is the Krickeberg decomposition of martingale (Xn) if and only if E[Yq] +
E[Z0] = sap E[\Xn\].
n
2.12 Let (Xn)n>o be an (^rn)-adapted sequence such that for each
n, Xn+\ is independent of .Fn, sup £711^1] < oo, and i£[Xn] = 0,n > 0.
n
Let T be an (J*n)-stopping time and E[T] < oo. Then
E[X0 + Xx + • • • + XT] = 0.
2.13 An adapted sequence (Xn) is a right-closable supermartingale
(resp. martingale) if and only if
i)supf;[|ArTl|] <oo,
ii) for all finite stopping times S <T, E[XS] > E[XT] (resp. E[XS] =
E[XT\).
2.14 Let X = (Xn)n>o be an (^rn)n>o-adapted sequence. Then the
following statements are equivalent:
1) There exists a sequence of stopping times Tjt | oo such that for each
fc, XTk = (XTkAn) is an (^rn)-martingale.
2) Xq is integrable, for each n > 0, Xn+\ is a-integrable w.r.t. Tn,
and
E[Xn+1\Tn] = Xn a.s..
2.15 Let W = (Wt)t>o be a Wiener process. Then all the following
processes are Wiener processes:
WtW = -Wt, t>0,
Wi2) = Wt+a -Ws, t > 0, for a fixed s > 0,
\ 0, t = 0,
Wi4) = Wa- Wa-U 0 < t < a, for a fixed a > 0.
2.16 Let W = (Wt) be a standard Wiener process.

Problems and Complements 77
1) Show that
Bt = Wt-tWl, 0<t<l,
is a Gaussian process with zero mean function and covariance function
C(t,8) =
(t(l-s), t<s,
(l-t)st t>s.
(Such a Gaussian process is called a Brownian bridge.)
2) Show that the conditional law of (Wt,0 < t < 1) given W\ = 0 is
identical with the law of a Brownian bridge.
2.17 Let W = (Wt) be a Wiener process, and T be a stopping time
such that E[T] < oo. Then E[WT] = 0 and E[W$] = a2E[T}.
2.18 Let W = (Wt) be a standard Wiener process and a < 0 < b.
Define
T = inf{* > 0 : Wt = a ot Wt = b}.
Evaluate E[T] and P(WT = a).
2.19 Let W = (Wt) be a Wiener process and T be a stopping time.
Define
[ Wu t< T,
Xt = I t > 0.
I 2WT-Wt, *>T,
Then X = (Xt) is still a Wiener process and has the same law as W
(reflexion principle). By making use of the reflexion principle, for fixed
t > 0, max Ws, \Wt\ and max Ws — Wt have the following common density
0<s<t 0<s<t
function:
2 =e-2/(2^)/]0>oo[(x).
\TtKah
2.20 Let (Tn)n>o be a sequence of r.v. such that
0 = To < Ti < • • • < Tn < • • •, Tn T oo,
and N = (Nt) be defined as follows:
Nt = n, when Tn < t < Tn+i.
The following statements are equivalent:
1) AT is a Poisson process with parameter A (w.r.t. its natural
filtration).
2) For any t > 0 and n > 1, given Nt = n, 7\, • • •, Tn have the same
distribution as the order statistics corresponding to n independent r.v.

78 Chapter II Classical Martingale Theory
uniformly distributed on [0,£], and Nt has the Poisson distribution with
parameter Xt.
3) For any non-negative Borel function / on i2+,
E[zf(Tn)]=\rf(t)dt,
ln=l J JO
and Ti,T2 - Tu ■ ■ ■,Tn - T„_i, • • • are i.i.d..

Chapter III
Processes and Stopping Times
Prom now on we introduce "the general theory of stochastic processes"
and its preliminary applications in three consecutive chapters.
The basic starting point of this chapter is a measurable space (ft, T)
with a given filtration F = (Ft)t>o- No probability measure appears
throughout this chapter.
§1. Stopping Times
To begin with, we recall the definition of stopping time, given in
Chapter II §5.
3.1 Definition. An J?+-valued r.v. T defined on (ft,^*) is called an
F-stopping time, if for each t > 0,
[T < t] e Ft;
and an F-stopping time in the wide sense, if for each t > 0,
[T<t]e Tu
or equivalently,
TAte Tt.
Obviously, F-stopping times in the wide sense and F+ -stopping times
are the same things (recall that F+ = (^i+)). In particular, F-stopping
times are also F-stopping times in the wide sense. In what follows, all
stopping times are relative to F, unless otherwise stated.
It is not hard for readers to show the following theorem.

80
Chapter III Processes and Stopping Times
3.2 Theorem. 1) If S and T are two stopping times (resp. in the
wide sense), so are S AT and S V T.
2) Let (Sn) be a sequence of stopping times (resp. in the wide sense).
Then V Sn is a stopping time (resp. in the wide sense), and /\ Sn is a wide-
n n
sense stopping time, moreover, if (Sn) is stationary, i.e for each uj G £l,
there exists a natural number n^, such that Sn(uj) = SnfjJ(u>) when n > n^,
then f\Sn is a stopping time.
n
3.3 Definition. Let T be a stopping time. Put
TT+ = {AeToo--Vte #+, A[T<t]e Ji+}, (3.1)
^T = Mefoo:VtG fl+, A[T <t]e Tt}, (3.2)
TT- =f0va{A[t <T]:AeTt, te #+}. (3.3)
Then Tt+,Tt and Tt- are all a-fields. Tt is called the a-field of events
prior to T, and Tt- the a-field of events strictly prior to T.
It is easy to see
TT+ = {Aefoo'.Vte #+,A[T <t]e Tt}, (3.4)
TT- = v{A[t < T] : Ae Tt-, t G #+}, (3.5)
TT- = TQM a{A[t < T] : Ae 7i+, t G #+}. (3.6)
If T is just a stopping time in the wide sense, the definitions of Tt+
and Tt- remain available. But Tt defined by (3.2) is no longer a a-field
(for some t G -R+, [T < t] & Tt, then ft £ Tt)- Therefore, for a wide-sense
stopping time T only Tt+ and Tt- make sense. But (3.4)—(3.6) remain
true in this case. In fact, for any ii+-valued r.v. T we can define Tt-
still by (3.3), and (3.5), (3.6) are valid.
Obviously, for every stopping time T we have Tt- C Tt C Tt+\ for
every wide-sense stopping time T we have Tt- C Tt+-
It is also easy to see that if T = t G R+ then T is a stopping time, and
Tt = TuTt- = Tt-,TT+ = Tt+ (recall that To- = To,Too- = T^).
The main properties of a-fields Tt,Tt- and Tt+ are listed in the
following theorem.
3.4 Theorem. In the following statements, T, 5, Si, • • •, Sn, • • • denote
stopping times, and U,R,R\, — •, Rn, • • • denote wide-sense stopping times.
1) R is Tr--measurable.

§1. Stopping Times ei
2) 5 < T =» Ts C Tt\R < U =* Tr- C Jru-,^R+ C ^t/+.
3)^5n^r = ^5AT.
4) >l € :FsvT =► ^[5 < T] € *r, 4[S < T] € JFr, A[S = T]e Fsat-
5) FsV?T = ?SvT = MUB : ^ € TS,B € ^r,^5 = 0}.
6)A£jrR+^A[R<U]eJru-
7) A 6 ?<*>=> A[R = oo] € ^"r-.
8) R<U,andR<U on [R < oo] =*• ^r+ C Jf/_.
9) S < T, and S < T on [T>0]^5SC TT-
10) IfR = MRn andU = /\Rn, then
n n
Tr- = VFRn-1 FU+ = A^*Rn + -
n n
11) If S = \JSn and for each n,Sn < S on [0 < Sn < oo], tfien
n
Proo/. 1) For each t G #+, [fl > t] G J*h- by (3.3), and [R = 0] =
[#>0]c. Hence ReTR-.
2) Let A G J5. For each t G i*+
A[T <t] = (A[S < t])[T <t]e Tt.
This means A e Tr- Hence Ts C Jr- Similarly, we have .7^+ C .Fim-.
Let A G ^i+, then A[t < R] G J*t+. From (3.6) we have
A[t < R] = (A[t < R])[t <U]e Tu-
Hence Tr- C T\j-.
3) In view of 2) it suffices to show Ts H ?T C ^sat- Let A G ^5 fl ^T-
For each £ G -R+
A[S A T < t] = (A[S < t])(A[T < t]) G Tt.
This means A G ^*5at- Hence Ts[\Tt C ^*5aT.
4) Let A G Tsvt- From 1) and 2) we know that both 5 and T are
^5vT-measurable. Hence A[S < T] G Tsvt- For each £ G 21+
(A[S < T])[T <t] = (A[S < T])[S VT<t]eTt.
This means A[S < T] G Tt- By the same argument we have A[S < T] G
^"t- Thus A[S = T] G ^r. By the symmetry between 5 and T,
A[S = T]Gf5nfT= ^5at.
5) Let C G J*5vt, 4 = C[T < S] and B = C[S < T]. Then by 4) we
know that A G Ts, B G TT, and AB = 0, A (J B = C. Therefore,
^5vT C {A U B : A G Ts, B G fr^B = 0} C Ts V ^T.

82 Chapter III Processes and Stopping Times
Owing to 2),fsV^rC Fsvt is trivial. Hence 5) is established.
6) Let A e Fr+. For each t G ii+, A[R <t]e 5i+- By (3.6) we have
A[R<U]= (J (A[R<r][r<U])efu-
reQ+
7) Let G = {A e Too: A[R = oo] e Fr-}. Then Q is a a-field. Let
A e Tn- We have
oo
A[R = oo] = 0 (4* < #]) € ^r_,
fc=n
i.e. 4 € (?• Hence 5dU^,5d <KU^n) = -F°°' and ^ = T*>-
8) Let A <E JrR+. Then 4[i? < U] € ^t/_ (by 6)), A[i? = oo] € ^r_ c
Tv- (by 7)), and
A = (A[R < U}) U (A[R = oo]) € Ji/-.
9) Let A € JF5. Then A[S = 0] 6 T0 C Jr-, and
A = (A[5 < T]) U (A[T = 0]) € .Fr-
10) Let t € i*+ and A € J7*. Then A[* < i?„] € ^rRn-, and
A[t<R}= U(^<fln])eV^„-
n=l n
By Definition 3.3 we have
^-cva-
n
The converse implication is always true.
Let A e n^iln+- For each * G R+ by (3-4) we kave
n
A[t/ < *] = U(^K < t]) € ^t.
n
This means A 6 ^ch-. Hence
n
The converse implication is always true, too.
11) By 9) we have V^*5„ C Fs-- By 10) we have
n
n n
Hence fS- =V^5„- °
n
3.5 Corollary. Le£ 5 and T be stopping times, R and U be wide-sense
stopping times.
1) [5 < T], [5 < T] and [5 = T] all belong to TSat; [S <T]e ?T-

§1. Stopping Times 83
2) r.v. £ € FsvT ^ £.I[S<T\ € ?T,£I[S<T\ € Ft,£I[S=T\ € FsaT-
3) r.v. ^ 6 ^r+ =► ^[fl<i/] € JV_.
4)
^5vTn[5<r] = ^rn[5<T],
rsvrn[s<T\ = rTnLs<T\,
Fsvt n [5 = T) = ^5at n[5 = T] = fsn[s = T] = fTn[5 = r].
5)
JrSATr\[S<T} = Fsn[S<T),
rsATn[s<T\ = rsflLs<T\.
6) // (Sn) 25 a stationary sequence of stopping times, then
^vS„ = V^5n = { U An- AneTSn,n = l,2,---,AkAj=H),k^j},
n n <■ n=l '
(5.1)
*a*. = Q**.- (5.2)
n
Proo/. We only give the proof of 6). Put S = \/ Sn. Because (Sn) is
n
oo
stationary, we have (J [Sn = S] = ft. Let A e Ts- Put
Ti=l
Ai = A[S = 5i], An = A( (1 [5j < S])[S = S»], n > 2.
Since [Sj < S] G Ts, by Theorem 3.4.4) we have An G ,Fsn, n > 1.
oo
Obviously, AkAj = 0, k ^ j, and A = (J An. (5.1) is established.
n=l
Put T = ASn. Let A G fl^Sn- By Theorem 3.4.4), for each n,
A[Sn = T] G £T. Hence A = (JUpn = T\) G Tr- (5.2) is established.
□
Remark. In 4) and 5) FsvTiFsaTiFt and ?s °an be replaced by
^(Svt)-i F(SaT)-i Ft- and FS- respectively.
3.6 Theorem. Suppose To = ^b+- ^e* (^n) 6e a monotone sequence
of wide-sense stopping times, and T = lim Tn.
n—>oo
1) // (Tn) 25 decreasing, and for each n, T <Tn on [0 < Tn] then
^r+=n^rn-. (6.1)
n
2) // (Tn) is increasing, and for each n, Tn <T on [0 < T], then
JrT- = \lfTn+- (6.2)

84
Chapter III Processes and Stopping Times
Proof. 1) Put Gt = Ft+,t > 0. Since Go = ^0+ = Fo, we also have
Gt- = ft-it > 0. Now T and Tn are ((zt)-stopping times. By Theorem
3.4.9) we have
Pt+ = Gt C (?Tn- = ^t„--
On the other hand, by theorem 3.4.10)
tt+ c n^r„- c n^Tn-i- = ^t+.
n n
Hence (6.1) is established.
2) By the same argument, we have Frn+ C Tt-- Then by Theorem
3.4.10)
n n n
Hence (6.2) is established. □
3.7 Theorem. 1) If S is a stopping time, r.v. T G Ts and T > S,
then T is a stopping time. If S is a wide-sense stopping time, r.v. T G
^5+, T > S and T > S on [S < 00], then T is a stopping time, too.
2) Let S be a wide-sense stopping time. For each n > 1 put
Sn=T, ^I,k=±<s<±, + ( + 00)/[S=+Oo]. (7.1)
TYien 5n,n > 1, are stopping times, and Sn j S.
3) // S and T are two stopping times, so is S + T.
Proof 1) For the first case, for each t > 0, [T < t] G ^5, and by the
definition of Ts we have
[T<*] = [T<*][S<*]e Ji.
Hence T is a stopping time. For the second case, for each t > 0, [T < t] G
^5+, and by Theorem 3.4.6) we have
[T<t] = [T<t)[S<t]eTt-CJrt
(note that T > 0 by the assumption). Hence T is a stopping time, too.
2) By Theorem 3.4.1) we know Sn G Ts— Obviously, Sn > S and
Sn > S on [5 < 00]. Thus by 1) Sn,n > 1, are stopping times. Sn [ S is
trivial.
3) Since S + T>S\/T and S + T efSvT, by 1) S + T is a stopping
time. □

§2. Progressively Measurable, Optional and Predictable Processes 85
3.8 Definition. Let T be a non-negative function on ft and A C ft.
Put
TA = TIA + (+oo)IAc
TA is called the restriction ofT on A. Obviously, T <TA.
3.9 Theorem. 1) Let T be a stopping time and A G .Too- Then TA is
a stopping time if and only if A G Tt-
2) Let T be a stopping time and A G Tt> Then
TTAnA = TTn A, TTa ^Ac = TOQr\ Ac, (9.1)
tTa- nA = rT-nA, tTa- n Ac = T^ n Ac. (9.2)
In particular, if Ft = Ft-, then for any A G Tt
TTa =Fta-
Proof. 1) is evident.
2) Let B G Tta- Then AB G Tta, and for each t > 0
AB[T <t]= AB[TA <t]e Tt.
This means AB G TT and BnA = (AB)nA G TTnA. Hence TTAnA C
TT D A. But T < 7^, TT C Ttu- Thus ^n>l = fTn A. Noting that
(TyO^c = +oo and making use of the assertion established just before, we
obtain
tTa nAc = ?{Ta)ac nAc = ToonAc.
Let B G Tt. Then B[t < TA] G ?ta-, B[t < T] <E J*T-. Since
(B[t < TA\) nA = (B[t < T\) n A g J*T- n A,
J*T/I- n A C Jt- n A. But TT- C Tta— Thus TTa- nA = TT- n A.
Similarly, we have
Jtu_ n Ac = f(TA)AC. n Ac = j-oo n Ac.
Now assume J*T = ^r- and A G Jr- Let 5 G ^tu- By (9.1),
ABGfr = ?t- C Ttu- By (9.2), ACB g ^-f!^ C ^- Thus
B = (AB) \J(ACB) G TTa- Hence ^ = Tta- d
§2. Progressively Measurable, Optional and Predictable
Processes
In this paragraph we study three most useful classes of measurable
processes: progressively measurable, optional and predictable processes.

86
Chapter III Processes and Stopping Times
3.10 Definition. Let X = (Xt)t>o be a stochastic process. X is
said to be measurable, if Xt(ui), as a function of (uj,t), is T x B(R+)~
measurable, X is said to be progressively measurable (or simply
progressive), if for each t > 0, restricted on fix [0, t], X is Tt x#([0, £])-measurable.
Obviously, every progressive process is measurable and adapted. But
the converse is not true in general.
3.11 Theorem. Right-continuous (or left-continuous) adapted
processes are progressive.
Proof. Let X = (Xt) be a right-continuous adapted process. For any
given t > 0, define a sequence of processes on ft x [0, t] as follows:
XW(u>) = X0(u)I[s=0] + £ Xkt (0;)//^^ w ,* € [0,*].
ThenX(n\n > 1, areTtxB([0,^-measurable, and Urn xin)(uj) = Xs(u)
on ft x [0, i\. Hence restricted on ft x [0, t], X is Tt x B([0, ^-measurable,
i.e., X is progressive. The proof for left-continuous adapted processes is
similar. □
3.12 Theorem. Let (Xt) be a progressive process. Then for every
stopping time T, XtI\t<oo) ^s Ft-measurable.
Proof. For each t > 0, TAt G Tt. Therefore, Xtm, as the composition
of measurable mapping from (ft,^f) to (ft x [0,t],^i x B([0, t])) : u) i->
(u),T(uj) f\t) and measurable mapping from (ft x [0,t],^i x B([0, t])) to
(R, B(R)) : (uj,s) k-> X3(uj), is Jf-measurable. (This assertion is valid
even for a wide-sense stopping time.)
Let A G B(R). For each t > 0
[XtI[t<oo] € A)[T <t] = [XTM e A)[T <t)e Tt.
Since XtI[t<oo] = ljm^XTAn^r^n]? we have Xt/[t<oo] € J*oo- Hence
[A"t/[T<00] e A] e Ft, ie., A"t/[t<oo] is ^r-nieasurable. □
Remark. Let (-Xt,£ G i2+) be a progressive process, and X^ be a
real .Foo-measiirable r.v.. Then for any stopping time T, Xt = XtI\t<oo] +
-XqoJ[t=oo] is ^-measurable.
3.13 Definition. A subset B of ft x i2+ is called a stochastic set, if its
indicator 7b is a stochastic process: Is = ((lB)t)t>o, where (lB)t = Ibh
and Bt is the section of B at t. Aset B e Fx B(R+) is called a measurable

§2. Progressively Measurable, Optional and Predictable Processes 87
(stochastic) set. A subset B of Q, x R+ is said to be progressive, if Is is
progressive. All progressive sets constitute a sub-a-field of T x B(i?+),
called progressive a-field, and denoted by £.
It is easy to see that a process is progressive if and only if it is £ -
measurable.
3.14 Definition. Let U and V be two i2+-valued functions defined
on ft, and U < V. Define
[U,V] = {(cj,<) G ft x #+ : tf(w) < t < V(cj)},
[tf, V[= {(w,t) G ft x #+ : C/(cj) < t < V(cj)},
][/, F] = {(u,, t) G ft x R+ : tf (cj) < t < V(cj)},
]tf, V[= {(w,t) G ft x tt+ : tf(cj) < t < V(u>)}.
Note that when V = +oo, we have [E/, +oo] = [E/, +oo[. When U and
V are r.v., [[/, Vj, [£/, V[, • • • are called stochastic intervals. [UJ is defined
as [17, £/], and is called the graph of U.
3.15 Definition. The a-field on ft x i2+, generated by all cadlag
adapted processes, is called optional a-field, and denoted by O. The a-
field on ft x i2+, generated by all left-continuous adapted proceses, is
called predictable a-field, and denoted by V. A stochastic set or process
is said to be optional (resp. predictable), if it is 0-( resp.'P-) measurable.
Prom Theorem 3.11 we know that all optional or predictable processes
are adapted.
If X = (Xt)t>o is a cadlag adapted process, then the left limit process
X- = (Xt-)t>o is predictable. Some elementary optional or predictable
sets and processes are given in the following theorem.
3.16 Theorem. 1) Let S and T be a couple of stopping times, and
S <T. Then all intervals [5, T], [S, T[, • • • and the graphs of S and T are
optional. Moreover, if £ is a real Fs-measurable r.v., then X = £I[s,t[ ^s
an optional process.
2) Let S and T be a couple of wide-sense stopping times, and S <T.
Then ]S,T\ is predictable. Moreover, if £ is a real Fs+ -measurable r.v.,
then £/|s,t] *5 a predictable process.
Proof. 1) It is easy to see that I[syT[ ls cadlag and adapted. Hence
[S,T[ is optional. Put Tn = S + -. Then [S] = n[S,Tn[ is optional. So
n n

88 Chapter III Processes and Stopping Times
is [T\. Thus [S,T],]S,T] and }S,T[ are all optional.
By Corollary 3.5.2) we know that Xt = £I[s<t]I[t<T] € Tt- Then
X = £I[s}T[ ls adapted and cadlag. Immediately, X is optional.
2) Since I]s,T] is left-continuous and adapted, ]5, T] is predictable. By
Corollary 3.5.3) we know that for t > 0,Xt = £I[s<t]I[t<T] € Ft--
Obviously, Xo = 0. Hence X = £I]s,T\ ls a left-continuous adapted process,
and is predictable immediately. □
3.17 Theorem. Denote by T the collection of all stopping times.
Then
0 = a{[S,oo[:SeT}.
Proof. Put C = {[5, oo[: 5 G T}. Since CcO,we have a(C) C O. It
suffices to show O C cr(C).
Let (Xt) be a cadlag adapted process. We are going to show that (Xt)
is 0-(C)-measurable. For each given e > 0, put Tq = 0, and define (T^)n>1
by induction as follows: for n > 1
T*+1(u;) = inf {t: t > 1t(u), \XT<(w)(u>) - Xt(u)\ > e
or |Xr«M(u;)-Xt_(a;)|>£}. (17.1)
We are able to show all T* are stopping times by induction on n. Noting
that the set on the right hand side of (17.1) is closed under the limit on
the right in i2+, for every r £ i2+ we have
PS+i =r]C [Tt < r)([\Xn - XT\ > e) U [\Xn -XT.\> e)) C [TnE+1 < r].
(17.2)
Because
U [T*+1 = r] = U [Tne+1 < r] = [Tn£+1 < t],
r<t r<t
from (17.2) we obtain
PS+i < *] = U {[Tn£ < r)([\Xn - Xr\ > e] \J[\Xn -Xr-\> e})}
r<t
= fl U {K<r][|Jri5-Xr|>e(l-l)]}l
m=l r€Qt m
where Qt = (Qn[0»*])U{*}- Assume T* is a stopping time, by Theorem
3.4.4) we know [T^x <t] G Tt, i.e., T^_|_1 is a stopping time, too.
Obviously, (T*) is increasing. When T^x(uj) < oo, T^x(uj) > T^(u)
and |XTe+i(u;)(a;)-XT5(u;)(a;)| >eor \XTe+i{u})_(uj)-XTe{u))(uj)\ > e must

§2. Progressively Measurable, Optional and Predictable Processes 89
hold. Since X.(u) has finite left limits on ]0, oo[, {T^{u)))n>\ has no finite
accumulation point. Hence T„(uj) | +oo. Put
oo
xe = £nxnhn,T^+li
For all t € [T'H, T*+1(u;)[, \XTe(lA})(u) - Xt(u)\ < e. Thus for all
(u>,t) € fi x R+, \Xf(u) - Xt{u)\ < e. This means
KmXf(u) = Xt(u;).
It is easy to show that Xt^I\t^t£ \ ls ^(C)-measurable (to approximate
Xt% by Tt^-measurable simple functions). Hence for each e > 0, X£ is
0-(C)-measurable, and so is X. □
3.18 Definition. A stochastic set B is called a thin set, if B =
oo
U Pnl> where Tn,n > 1, are stopping times. Obviously, a thin set is
n=l
optional.
3.19 Theorem. // a progressive set B is contained in a thin set, then
B is also a thin set.
oo
Proof. Let B C (J [Tn], where each Tn is a stopping time. Put
n=l
Ln = {u:{u,Tn{uj))eB}.
Then ILn = IB(Tn)I[Tn<oo]. By Theorem 3.12, Ln G TTn. Hence
(Tn)Ln = Tn/Ln + (+oo)/Lc
is a stopping time. Evidently, we have
oo
B = U [(T„)lJ. □
n=l
3.20 Theorem. Let (X$) be an optional process. Then there exists a
predictable process (Yt) such that
A={(u,t):Xt(uJ)^Yt(uJ)}
is a thin set.
Proof. Denote by H the collection of all optional processes for which
the assertion holds. Obviously, "H is a linear space. Put
C = {[S,T[: S<T, S, TeT}
(recall that T is the set of all stopping times). If 5 < T, U < V, S, T, U, V e
T,
[S, T[n[U, V{= [{S V U), (5V[/)V(TA V)[.

90
Chapter III Processes and Stopping Times
This means that C is a 7r-class (Definition 1.1). Let S,T E T, and S <T.
Put X = I[s,T[- Then Y =]S,T] is a predictable process, and
[X # Y] C [5] U [T].
Hence the indicators of sets in C belong to H.
Let XW € H, and 0 < X^ 1 X < +00. Take predictable processes
y<B> such that all [X<n) ^ y<B>] are thin sets. Put
F = limsuPy("), Y = YIm<oo].
n—►oo u ' J
Then y is predictable, and
[X^Y]C U [*(n) ^ ^(n)]- (20.1)
n=l
The right side of (20.1) is still a thin set. By Theorem 3.19, X G H. Since
a(C) = O, by the monotone class Theorem (Theorem 1.4) we know that
H is just the collection of all optional processes. □
3.21 Theorem. Put
d = {Ax {0} :Ae T0}U{Ax]s,t] :0<s <t,s,teQ+,Ae U «?>},
r<s
C2 = {Ax {0} :Ae f0}U{i4x [s,*[: 0 < 3 < t,s,t G Q+,A G U •?>},
C3 = {A x {0} : A G Jb} U {]S,oo[: S G T}.
77ien a(Ci) = a(C2) = (t(C3) = V. In particular, V C O.
Proof. First of all, Ci C V is obvious. Thus <r(Ci) C P. On the other
hand, for each left-continuous adapted process (Xt) define
Xtn = X0I[t=o] + £ X k 1]k _^fc+ir
k=Q 2n l2n<* - 2n J
Then lim JQn' = Xt. It is easy to see that (X^) is a-(Ci)-measurable
for each n > 1, and so is (Xt). Hence V C <t{C\). We obtain <r(Ci) = V.
Secondly, let A G TT,r < s. Apparently,
OO OO f g \
Ax]s,t]= (J fl Ax[s + ,*+—[>
n=lm=l 71 m
oo ex) 1 / _ o
A x [s,t[= fl U Ax]r + (1 - -)(* -r),t ].
n=lm=l n Tl
Hence Ci C <r(C2), and C2 C <r(Ci). Thus we obtain a(C2) = v(Ci) = V.
Obviously, we have <r(C3) C V and C\ C <r(C3) (Ax]s,£] ^Syi,*^],
0<s<t,Ae (J «?>). Hence <r(C3) = V.
T<3
Finally, since all sets in C2 are optional, we have V C O. □

§2. Progressively Measurable, Optional and Predictable Processes 91
3.22 Corollary. Let T be a stopping time. Define
f(u) = (uj,T(uj)) on[T<oo].
Then
/-1(0) = Jrn[T<oo], (22.1)
J"\V) = TT- H [T < oo]. (22.2)
Proof. We give the proof of (22.2) only. The proof for (22.1) is similar.
Let Aq G F0. Then f~l(A x {0}) = A[T = 0] G TT- f)[T < oo]. Let
S e T. Then /"Hl^ooD = [5 < T < oo] G TT- f][T < oo]. By
Theorem 3.21 f~l(V) C Jt- HP1 < oo]. Conversely, let A G Jb- Then
A[T < oo] = f~l(A x R+) G f~l(V). Let A G 7i. Then
(A[t < T])[T < oo] = r^Ax^ooJ) G Z"1^).
Hence J*T- f)[T < oo] C f~l(V). (22.2) follows. In fact, (22.2) holds even
for any wide-sense stopping time T. □
3.23 Corollary. 1) Let T be a stopping time. Then for any op-
tional process (Xt), XtI[t<oo] ^s Ft-measurable. Conversely, if£ is a real
Ft-measurable r.v., then there exists an optional process (Xt) such that
6f[T<oo] = XtI[T<oo]-
2) Let T be a wide-sense stopping time. Then for any predictable
process (Xt),XTl[T<oo] i>s Ft--measurable. Conversely, if £ is a real
Ft--measurable r.v., then there exists a predictable process (Xt) such that
£I[T<oo] = XtI[T<oq\-
Proof. Using the mapping f(u) = (u,T(u)) on [T < oo], for any
process (Xt), XtI\t<oo]-> being restricted on [T < oo], results from the
composition of X and /. Now the assertions follow from Corollary 3.22
and Doob's measurability theorem (Theorem 1.5) immediately. □
3.24 Corollary. Let T be a stopping time, and X = (Xt) be an
optional (resp. predictable) process. Then the stopped process of X atT:
X = (Xt )t>o = (XTAt)t>o
is still optional (resp. predictable).
Proof. In fact, the stopped process XT can be written as
XT = XI[0T] + XtI]t,oo[

92 Chapter III Processes and Stopping Times
We already know that i[o,r] and XtI]t,oo[ = (^tJ[t<oo]K]7\oo] are
predictable (Theorems 3.12 and 3.16.2)). Hence if X is optional (resp.
predictable), so is XT. In fact, if X is predictable, even for any wide-sense
stopping time T, XT is predictable. □
§3. Predictable and Accessible Times
3.25 Definition. An i2+-valued r.v. T, defined on fi, is called a
predictable time, if [T, oo[ is a predictable set.
Obviously, a predictable time is a stopping time, and a constant
stopping time is predictable. Besides, let T be a wide-sense stopping time.
Since ]T, oo[ is a predictable set, T is a predictable time if and only if [T]
is predictable.
3.26 Definition. Let T be a wide-sense stopping time, (Tn)n>i be an
increasing sequence of wide-sense stopping times, and for all n,Tn < T.
Let A Cft. We say that the sequence (Tn) foretells T on A, if on A[T > 0]
we have Tn < T for all n and lim Tn = T. We say briefly that (Tn)
n—►oo
foretells T if (Tn) foretells T on the whole ft.
A wide-sense stopping time T is calleld foretellable, if there exists an
increasing sequence of wide-sense stopping times which foretells T.
3.27 Theorem. Let T be a foretellable wide-sense stopping time. If
[T = 0] G To, then T is a predictable time. In paricular, i/^o-i- = ^o,
then all foretellable wide-sense stopping times are predictable times.
Proof. Let (Tn) be an increasing sequense of wide-sense stopping times
which foretells T. Then
[T,oo[= ([T = 0] x {0}) U (mrn,oo[) G V.
Hence T is a predictable time. □
3.28 Corollary. V = a{[S,T{. S and T are foretellable predictable
times, and S <T).
Proof. In Theorem 3.21 C2 are composed of two classes of elements.
The elements of the first class have the form A x {0}, where A G Tq. We
have
Ax{0}= f\l0A,(-Ul
n=i n

§3. Predictable and Accessible Times 93
The stopping times 0,4 and (-)a axe foretelled by sequences (k A 0,0fc>i
n
and (k A ( ) ) respectively. The elements of the second
V \n n + k'AJk>\
class have the form A x [s, t[, where 0 < s < t and A G (J ?v We have
A x [s,i[= [5^,^[. Take n sufficienlty large such that A € ^ i- Then
n
the stopping times sa and ^ axe foretelled by sequences ^(n + k)Ays-
r) ) and((n + fc)A(7 r) ), respectively. □
The main properties of predictable times axe listed in the following
theorem.
3.29 Theorem. 1) Suppose (5n) is a sequence of predictable times.
Then VSn is a predictable time. Moreover, if (Sn) is stationary, f\Sn is
n n
a predictable time, too.
2) Let S be a predictable time, and T be a stopping time. Then
AeTs-^ A[S <T]e TT-, A[S = t}£ Ft-
In paricular, [S = T] G Tt- •
3) If S and T are predictable times, then Ts- fl^T- = F(SaT)--
4) // S and T are predictable times, then
A G F(svT)- =>
A[S <T]e Tt-, A[S <T]e TT-, A[S = T]e ^(5aT)-,
F(SvT)- =Ts-VFt- = {AUB: AeJrs-,BeFT-,AB = Q}.
5) Let (Sn) be a stationary sequence of predictable times. Then
U An : An e fSn-,n > 1, AkAj = <D,k^ j\,
n=l J
n
6) Let S be a stopping time, and A G Tqq. If Sa is a predictable time,
then A G Js_.
7) If S is a predictable time, then so is Sa for all A G Ts- •
8) Let S be a predictable time, and £ be a real Ts--measurable r.v.
then the process £/[si00[ is predictable.
Proof. 1) We have
[VSn,00[=n[Sn,00[.

94
Chapter III Processes and Stopping Times
If (Sn) is stationary, then
lASn,oo[=U[S„,oo[.
n n
2) By Theorem 3.4, A[S < T] G Ft-,A[T = oo] G TT-, it suffices to
show A[S = T][T < oo] G .Ft-- Suppose that (Xt) is a predictable process
such that IAI[s<oo] = XsI[s<(X>] (Corollary 3.23.2)). Put Y = XI^,
Then Y is predictable, and
YtI[T«x>] = XTl[S=T]I[T«x>] = XsI[S=T«x>] = IaI[S=T<oo]-
Hence A[S = T < oo] G TT- (Corollary 3.23.2)).
3) It suffices to show TS- n TT- C ^(5aT)— Let ^ € Fs_ n Ft_.
Then
A[S<T] = A[S = SAT)eT{s„T)-,
A[T < S) = A[T = S A T] G F(5at)-.
Hence A = (A[S < T]) U [A[T < S]) G F(5AT)-.
4) We have
A[S < T) = A[S V T = T] G Ft-,
A[S <T) = (A[S < T))n[S <T)e FT-
Hence A[S = T] G Ft-. In view of the symmetry between S and T, we
obtain
A[S = T] G TS- n FT- = ^(saT)-
Let C G ^(svd— Put A = C[T < 5], B = C[S < T]. Then
C = AUB,AGf5-, BefT-,AB = Q.
5) The proof is completely similar to that of Corollary 3.5.6).
6) Put X = I[sAl00[' Then X is a predictable process. By Corollary
3.23.2),
Ia[s«x>] = xsl[s«x>]
is Fs_-measurable, i.e., A[S < 00] G Ps-- By Theorem 3.4.7), A[S =
00] G Ts-- Hence A G J*5_.
7) Let >1 G ^*5-. By Corollary 3.23.2), there exists a predictable
process X such that
W[S<oo] =*sJ[S<oo]-
Then [Sa] = [X = 1] f][S] is predictable. Since Sa is a stopping time,
Sa is a predictable time.
8) follows easily from 7). □

§3. Predictable and Accessible Times
95
3.30 Theorem. If A is a predictable set and contained in the union
of graphs of a sequence of predictable times, then A itself is the union of
graphs of a sequence of predictable times.
Proof. It is completely similar to that of Theorem 3.19. □
3.31 Theorem. Suppose that A is the union of graphs of a sequence
of stopping times (resp. predictable times). Then there exists a sequence
of stopping time (Tn) (resp. predictable times) such that A = \J[Tn] and
[Tn]n[Tm] = ^n^m.
Proof. We only show the predictable case. Let (Sn) be a sequence of
oo
predictable times such that A = (J [Sn]. Put T\ = Si, and for n > 2
n=l
Bn = nC)[Sk^Sn], Tn = (Sn)Bn.
Then Bn G .Fsn-, Tn is predictable, [Tn[n[rm] = 0 when n ^ m, and
00
A = U [Tnl n
Ti=l
3.32 Theorem. For any cadlag adapted process (Xt)t>o there exists
a sequence of strictly positive stopping times (Tn) such that
[AX ,* 0] = {(u>,t) :0<t<+oo, Xt(u) / X*-(w)} = \J[Tn). (32.1)
n
Proof In the proof of Theorem 3.17 take efc = -. We are going to
k
show [AX ? 0] C (J [Tn,kY Let 0 < t < +oo and |Xt(cj)--Yt-(cj)| > ^.
n,ifc>l *
Then for some n > 1 we have
r,J/*(w)<t<rift(w).
For all 5 €]T„1/fc(a;),T^{fc1(a;)[, from (17.1) we have
|-X,(w) - Xrv*(w)(w)| < -, |Xs_(a;) - XTi/k(u))(uj)
Kk'
2 i/t.
Thus \X3(uj) — Xs-(u)\ < -. This means that it must be t = 7V (cj).
A/
Now (32.1) follows immediately from Theorem 3.19. □
The following theorem provides a characterization of predictability for
cadlag adapted proceses.
3.33 Theorem. Let X = (Xt) be a cadlag adapted process. Then X
is predictable if and only if X satisfies the following conditions:

96
Chapter HI Processes and Stopping Times
i) There exists a sequence of strictly positive predictable times (Tn)
such that [AX ^ 0] C U[^n]-
n
ii) For each predictable time T, XtI[t<oo] € ?T- •
Proof. Necessity. Assume X to be predictable. Then in the proof of
Theorem 3.17 we are able to show that T„,n > 1, are predictable times
by induction on n. In fact, if T„ is predictable, then
A =]rnE,oo[n([|X^/]r5,oc[ - X-\ > e) U [\XnJmM -X\> e])
is a predictable set. We have already known that T*+1 is a stopping
time. Since [2J+1] C A, [T*+1] = AfllO, T£+1] is predictable, T*+1 is
predictable. Now the condition i) follows from the proof of Theorem 3.32,
and the condition ii) follows from Corollary 3.23.2).
Sufficiency. Assume that the conditions i) and ii) are satisfied. By
Theorem 3.31 we can assume that the graphs of (Tn) are disjoint. We
have
x = X-ir\[Tn]c + Y,XTni\Tn\
n Tl
Since XTnI[Tn<oo] e TTlx-, XTnI[Tn] = XTnI[Tn<oo](I[TniQol - /]Tn>oo[) is
predictable. Hence so is X. D
3.34 Definition. A stopping time T is called an accessible time, if
there exists a sequence (Tn) of predictable times such that [T] C UPn].
n
Obviously, predictable times are accessible.
3.35 Theorem. Let T be a stopping time, and (5n) be an increasing
sequence of wide-sense stopping times, dominated by T, i.e., Sn < T. Put
A[(Sn)} = {(Q[Sn < TDnyjm^n = T]}\J[T = 0] (35.1)
{i.e., (Sn) foretells T on A[(Sn)]). Then A[(Sn)] € TT-, and Ty4[(Sn)] is
accessible.
Proof Noting that lim Sn < T, we have [lim Sn = T} = [lim Sn < T]c €
TT- Hence A[(Sn)]) iTT- Put i?„ = {Sn\Sn<XxmSA A n. Then (K)
n
foretells R = lim(i2n), and R > 0. By Theorem 3.27, R is a predictable
n
time. Because [T[i4(sn)]] C [i?lU[0]»^4[(5n)] is accessible. □
3.36 Theorem. 1) If S and T are accessible times, so are SVT and
SAT.
2) IfT is an accessible time, so is Ta for all A G Tt-

§3. Predictable and Accessible Times
97
3) Suppose that (Tn) is a monotone sequence of accessible times, and
T = limTn. If(Tn) is increasing, thenT is accessible. If{Tn) is decreasing
n
and stationary, then T is accessible, too.
Proof. 1) and 2) are evident. Let us show 3). In the increasing case,
by Theorem 3.35 T^sn)] *s accessible, and by (35.1)
A[(Sn)]c = (\J[Tn = T])n[T>0}.
n
Hence [TyiKs^ye] C \J[Tn] and T^^ds accessible. At last, T = TA[(sn))
n
A^4[(5n)]c Is accessible. In the decreasing and stationary case, we have
[T] C Ul^nl- Hence T is accessible. □
n
Now that we have the concept of accessible time, we may define
accessible a-field according to Theorem 3.17.
3.37 Definition. Denote by A the collection of all accessible times.
The a-field on i2+ x ft, generated by {[5, oo[: 5 G A} is called accessible
a-field. The processes and sets, measurable w.r.t. the accessible a-field,
are called accessible processes and sets respectively.
If 5 is an accessible time, then [S] = [5, ooJ\]5, oo[ is an accessible
set.
Remark. If the stopping times and optional processes in Theorems
3.19, 3.20 and 3.31 are replaced by accessible times and accessible
processes respectively, the assertions remain true, their proofs are completely
similar.
The relation between accessible and predictable times or processes is
established in the following theorem.
3.38 Theorem. 1) Assume X to be an accessible process. Then X is
predictable if and only if for all predictable time T,XtI[t«x>] € ^T--
2) Assume S to be an accessible time. Then S is predictable if and
only if for all predictable time T [S = T] G Tt--
Proof. 1) The necessity comes from Corollary 3.23.2). Let us show
the suflBciency. According to the remark following Definition 3.37, there
exists a predictable process Y such that [X ^ Y] is the union of graphs of
a sequence of accessible times. Then by Definition 3.34 and Theorem 3.31,
there is a sequence of predictable times (5n) such that [X ^ Y] C U[Snl>
n

98
Chapter III Processes and Stopping Times
and [5n] fll^m] = 0 when n ^ m. We have
X = YIA + £*s„J[s„«x>]J[s„]>
n
where A = C\[Sn]c. Since A is predictable, so is YIa- On the other hand,
n
each Sn is predictable. By the assumption, XsnI[sn<oo] € ^5n-- Hence
^Sn^[Sn<oo]^[S„] is predictable. At last, X is predictable.
2) The necessity comes from Theorem 3.29.2). It suffices to show the
sufficiency. Put X = Jjsj. Then X is accessible, and by the assumption
for any predictable time T
[S = T<oo] = [S = T][T < oo] G TT-
(note that T is .^--measurable), i.e., XtI[t<oo] = J[S=T<oo] is
^immeasurable. By 1) X is predictable, namely, S is a predictable time.
□
3.39 Definition. A filtration F = (Tt)t>o is said to be quasi-left-
continuous if for any predictable time T, Tt = ^r- •
3.40 Theorem. 1) A filtration F = (Tt) is quasi-left-continuous if
and only if each F-asscessible time is F-predictable.
2) Suppose that F = (Tt) is quasi-left-continuous. Then for any
sequence of stopping times (Tn) we have
^Vr„=Y^rn. (40-1)
n
Proof 1) Necessity. Assume that F = (Tt) is quasi-left-continuous.
Let 5 be an F-accessible time, and T be an F-predictable time. Then
[S = T]eTT = TT— By Theorem 3.28.2), 5 is predictable.
Sufficiency. Assume that each F-accessible time is F-predictable. Let
T be an F-predictable time, and A G Tt- Then Ta is an accessible time.
By the assumption, Ta is predictable . Hence A G Tt- (Theorem 3.29.6)).
This means Tt = Tt-, i.e., F = (Tt) is quasi-left-continuous.
k
2) By Corollary 3.5.6), we have T k = V ^T„- In order to prove
n=l
(40.1), we may suppose that (Tn) is an increasing sequence. Put T =
00
V Tn. Let H = f][Tn < T]. Then H G TT— Let A G TT. We are going
n=l n

§4. Processes with Finite Variation
99
to show AH G \JFTn and AHC G V^Tn- We have
n n
oo
AHC= \J(A[T = Tn])eyTTn.
n=l n
Since T# > 0 and ((Tn)[Tn<Tj A n) foretells Th on whole ft, 7// is
predictable. Thus by the assumption we have Tth = Fth-, Hence AH G
Tt C TTh = Fth-. But T = TH on #. Then H = Hf][TH = T], and by
Theorem 3.29.2)
AH = AHn[TH = T]eTT-
fT_ = V TTn- C \IFTn always holds (Theorem 3.4.10)), we have AH G
VTTn• Finally, A = (AHC)\J(AH) e\lTTn- □
§4. Processes with Finite Variation
3.41 Definition. A process is called an increasing process, if its
trajectories all are non-negative right-continuous increasing real funtions. A
process is said to be with finite variation, if it is the difference of two
increasing processes.
Obviously, a process with finite variation is cadlag. Therefore, an
adapted process with finite variation is optional.
Let A = (At)t>o be a process with finite variation. For each w G
Q,A.(uj) is a real function on i2+ with finite variation, i.e., the variation
of A.(u) on every finite interval is finite, and it can be uniquely
decomposed as: A.(u) = A.c(u) + A.d(u), where A.c(uj) is a continuous function
with finite variation, Ad(u)) is a purely discontinuous function with finite
variation:
A*{w)= £ AA». (41.1)
0<s<t
Ac is called the continuous part of A, and Ad the purely discontinuous or
jump part of A.
A process with finite variation A is said to be purely discontinuous, if
AC = Q.
The following theorem describes the structure of adapted or
predictable processes with finite variation.

100
Chapter III Processes and Stopping Times
3.42 Theorem. // A is an adapted (resp. predictable) process with
finite variation, so is Ad {consequently, Ac is predictable). Furthermore,
there exists a sequence (Sn) of strictly positive stopping times (resp.
predictable times) with disjoint graphs such that
^ = EAi5n/[5n<t|. (42.1)
n
(By convention, AAqq = 0.)
Proof. We only give proof for the predictable case. By Theorems 3.31
and 3.33, there exists a sequence (Sn) of strictly positive predictable times
with disjoint graphs such that [AA ^ 0] C U[£n]- Since the series in (41.1)
n
is absolutely convergent, it does not depend on the order of summation,
so (42.1) holds. However, A A is predictable. Therefore, AAsn £ Fsn-,
and Ad is predictable by Theorem 3.29.8). □
Remark. Prom (41.1) and (42.1) we have
£ |AJ4s| = E|AA5n|/[sn<t].
0<s<t n
The following theorem is a consequence of the above theorem, and is
useful sometimes.
3.43 Theorem. Let A be an adapted (resp. predictable) purely
discontinuous process with finite variation. Then there exists a sequence (Tn)
of strictly positive stopping times (resp. predictable times) (here the graphs
of (Tn) need not to be disjoint in general), and a sequence (Xn) of reals
such that for allt>0
£ |An|/[r„<t] < oo, (43.1)
n
At = ZKI[Tn<t\- (43.2)
n
Moreover, if A is an increasing process, each Xn may be taken to be
positive.
Proof. We only give proof for the predictable case. Owing to (42.1),
it suffices to show the assertion for predictable increasing process A =
£i[s,oo[> where 5 is a strictly positive predictable time, and £ is a non-
negative real ^--measurable r.v.. Let (£n) be an increasing sequence of
non-negative ^--measurable simple r.v. such that lim £n = £. Then
n—►oo
oo
£ = X) (£n — £n-i), £o = 0. Hence there exits a sequence (An) of reals
n=l

§4. Processes with Finite Variation 101
oo
and a sequence (Hn) of ^--measurable sets such that £ = £ AnJ#n. For
Ti=l
each n define Tn = Snn, then Tn is a predictable time, and we have
At = £ An/[Tn<t]. □
n
3.44 Theorem. Le£ A = (At) 6e an adapted (resp. predictable)
process with finite variation. Then Bt = \dA3\, the variation process
J[o,t]
of A, is an adapted (resp. predictable) increasing process, and A can be
represented as the difference of two adapted (resp. predictable) increasing
processes.
Proof. By Theorem 3.42 we know that
i
•Ao,
M = £|AA5n|J[5n<t]
[0,t] n ~J
is an adapted (resp. predictable) increasing process, and
/ |iiA;| = |i4o|+Jfim £ \A%»t-Ac±t\
is an adapted continuous (consequently, predictable) increasing process.
Their sum Bt = /roti \dA3\, therefore, is an adapted (resp. predictable)
increasing process. Define
A+ = ±(B + A), A- = \{B-A).
Then A+ and A~ are adapted (resp. predictable) increasing processes,
and A = A+ - A~. □
Below we investigate the Lebesgue-Stieltjes integrals by paths of
measurable processes w.r.t. processes with finite variation.
3.45 Definition. Let H = (Ht) be a measurable process, and A =
(At) be a process with finite variation. If for all cj G Cl and t > 0 Lebesgue-
Stieltjes integral
/ Hs(uj)dAs(u)
J[o9t]
exists and is finite (i.e., / |i/a(u;)||cL4a(u;)| < oo), we say that H is
■/[CM]
integrable w.r.t. A. At this time we define
B = (Bt), Bt(uj) = I H3(uj)dA3(uj
J[ott]
)

102
Chapter III Processes and Stopping Times
as the stochastic integral of H w.r.t. A, and denote B = H.A. Trivially,
B is still a process with finite variation.
3.46 Theorem. Let H = (Ht) be a measurable process, and A = (At)
be a process with finite variation. Assume H is integrable w.r.t. A.
1) If H is progressive and A is adapted, then H.A is adapted.
2) If both H and A are predictable, so is H.A.
Proof Without loss of generality, we may assume that A is an
increasing process. It is easy to show by the monotone class argument that
for each measurable proess H with / \Hs\dAs < oo, / HsdAs is
./[0.oo[ «/[0,oo[
J*-measurable. Hence applying this assertion to #/[o,t] and At on (Vt,Tt)
leads to 1). In order to show 2), consider the following decomposition of
A:
At = Act+Z&ASnIlsn<t],
n
where (5n) is a sequence of predictable times with disjoint graphs
(Theorem 3.42). We have
/ HsdAs= ( HsdAcs + ZHsnAAsnI{sn<tV
J[Q,t] J[0yt] n ~
By 1) the first process on the right side is adapted and continuous.
Therefore, it is predictable. The second process on the right side is predictable
apparently. Since H is predictable, HsnI[sn<oo] € ^5n-, and by Theorem
3.29.8) HSn&AsnI[sn,ool is predictable . □
§5. Changes of Time
3.47 Definition. A family of stopping times r = (rt)t>0 is called a
change of time if r is an i2+-valued increasing process, i.e.,
i) for every t > 0, rt is a stopping time,
ii) for every u> G ft, t.(uj) is an ii+-valued righ-continuous increasing
function on i2+. A change of time r = (rt)t>o is said to be continuous, if
r is an R+ -valued continuous process.
The filtration changed by r is defined as:
G = (ft)t>o, Qt=TTi, t>0. (47.1)
If the original filtration F is right-continuous, so is G by Theorem 3.4.10).

§5. Changes of Time
103
For a change of time r = (rt)t>o its right-inverse process A = (At) :
At = inf {s > 0 : ra > t}
is an R+-valued adapted increasing process. In fact, for any t > 0
rt = inf{s > 0 : As > t}, rt- = inf{s > 0 : As > i)
i.e., r = (rt) is also the right-inverse process of A = (At) (Lemma 1.37).
Then for any t > 0 and s > 0
[rs- <t] = [At > s]. (47.2)
Because r5_ is a stopping time, so At G Tu i-e., A is adapted. Prom (47.2)
we also know that At is a G-stopping time in the wide sense.
3.48 Theorem. Let A = (At) be an R+-valued adapted (resp.
predictable) increasing process. Denote by r = (rt) the right-inverse process
of A:
rt = inf{s > 0 : As > t}.
Then for every t > 0,Tf_ is a stopping time (resp. predictable time), and
for every t > 0, rt is a wide-sense stopping time.
In particular, if the filtration F is right-continuous, then r = (rt) is
a change of time, and is said to be associated with R+-valued adapted
increasing process A = (At).
Proof. For every t > 0, again using (47.2), we see
[rt- < s] = [A3 >t]ef3, s> 0,
i.e., rt- is a stopping time. Then for every t > 0, r i [ rt,rt is a
wide-sense stopping time.
Furthermore, if A is predictable, noting that rt- = inf{s > 0 : As > t},
[rt-] = [0, Tf_] fl [A > t] is predictable, i.e., rt- is predictable.
The last assertion is trivial. □
As pointed out above, every change of time is associated with an
Revalued adapted process—its right-inverse process.
3.49 Theorem. Let r = (rt) be a change of timelK The filtration
G = (Gt) is defined in (47.1).
1) If S is a G-stopping time in the wide sense, then rs is an F-stopping
time in the wide sense, and Gs+ C FTS+-
2) If S is a non-negative r.v. such that rs is an F-stopping time and
S is FTs -measurable, then S is a G-stopping time, and Trs fl Goo C Gs-

104
Chapter III Processes and Stopping Times
Proof. 1) Let A G Gs+- F°r every t > 0 we have
A[rs <t] = (^[too < t, 5 = oo]) U ( (J A[S< r][rr < t]).
reQ+
Since A[S = oo] G 5oo C J>oo C ^oo, A[S < r] G QT = TTr , we obtain
A[rs <t]e Tt.
This means that rs is an F-stopping time in the wide sense (by taking
A = ft), and Gs+ C ^rs+-
2) Let A G 5oo n TTs. For every * > °
i4[5 < t] = (A[S < t])[rs < rt] G TTi = ft.
This means that S is a G-stopping time, and Goo H ^Vs C (7$. E
3.50 Corollary. Assume F is right-continuous. A non-negative r.v.
S is a G-stopping time if and only if r$ is an F-stopping time and S G
TTs. At this timej we have Gs = Goo H FTs.
Proof. Immediately it follows from Theorem 3.49, noting that G is also
right-continuous. □
3.51 Theorem. Let r = (rt) be a change of time, and A = (At) be
its right-inverse process: At = inf{s > 0 : rs > t},t > 0.
1) If T is an F-stopping time in the wide sense, then At is a G-
stopping time in the wide sense, and Ft+ H Goo C Gat+-
2) Assume that F is right-continuous, and TAt = t,t > 0 lK If T is a
non-negative r.v. such that At is a G-stopping time and T G Gat, then
T is an F-stopping time, and Gat C Tt-
Proof. 1) Let B G TT+ n ftx>. For every t > 0
B[AT <t} = (B[T = oo][i4oo < t]) U (\JB[T < 00][T < rt_1/n]).
n
Since (B[T < oo])[T < rt_1/n] € TTt_l/n C Tn = Qt, B[T = oo]^ <
t] = B[T = oo][Aoo < t][Tt = 00] € TTi = Qu we obtain
B[AT <t)€ Qt-
This means that At is a G-stopping time in the wide sense, and Tt+ l~l
Goo C <Mr+.
2) Let B € <mt. For every t > 0, At is a. G-stopping time, and
B[T <t} = (B[T < t])[AT < At] e gAl-
1' e.g. r is continuous, and tq = 0, t^ = oo.

§5. Changes of Time
105
By Corollary 3.50, Gax C TTAt = Tt- Hence T is an F-stopping time, and
Gat C Tt. a
3.52 Theorem. Let t = (Tt) be a change of time, and for each t>0,
rt < oo.
1) If X = (Xt)t>o is an F-optional process, then (XTt)t>o is a G-
optional process.
2) If X = (Xt)t>o is an F-predictable process, then (XTt_)t>o is a
G-predictable process.
Proof. It is easy to see that if (Xt)t>o is cadlag, so is (XTt)t>o, and
if (Xt)t>o is left-continuous, so is (XTt_)t>o. If (Xt)t>o is F-optional,
obviously (XTt)t>& and (XTt_)t>a are G-adapted.
Let T be an F-stopping time. If X = i[orr[> (^rt)t>o is cadlag and G-
adapted. Hence it is G-optional. If X = I\oyT\i (Xn- )t>o is left-continuous
and G-adapted. Hence it is G-predictable. Now If X = I[q]Ia, A G To, by
the same reason (XTt_)t>o is G-predictable. Now applying the monotone
class argument leads to the assertions. O
In the next theorem a useful and simple example of changes of time is
discussed in detail.
3.53 Theorem. Let T be a stopping time. Define
G = {Gt), Gt = TtM t > 0.
1) // S is an F-stopping time, then S AT and S[$<t] are G-stopping
times.
2) // S is an F-stopping time in the wide sense, then S[s<t\ is & G-
stopping time in the wide sense.
3) If S is a G-stopping time, then Gs = GsaT = TsaT-
4) If X is an F-optional (resp. F-predictable) process, then XT and
-XV[o,T] are G-optional (resp. G-predictable) processes.
5) // S is an F-predictable time, then S[$<T] is & G-predictable time.
Proof. Here we are concerned with the change of time t = (rt) :
rt = t AT. Its right-inverse process is A = (At) : At = £[t<r]- Because
Gt C Tt,t > 0, every G-stopping time is also an F-stopping time.
First of all, we show Goo = Tt- Goo C Tt is trivial. Let t > 0,A e
Tt. A[t <T] = (A[t < T])[t = tAT]e TtKT C Goo, SO A[T = OO] G Goo-
Therefore, if A G J*oo,then A[T = oo] G Goo- Now let A G Tt. A[T <

106
Chapter III Processes and Stopping Times
oo] = LMCT < n]), A[T <n]e TnKr C 5oo, n > 1, so A[T < oo] G £<*,
A = (A[T = oo]) \J(A[T < oo]) G Goo, TT C ^oc
1) Let 5 be a F-stopping time. For each t > 0
[5AT<t]€7sATAtCft,
[S[S<T] <t] = [S< T][S <t] = [S<TAt]e TiKT = Gt.
Hence S AT and S[$<t\ are G-stopping times.
2) Since S]s<t\ = As, the assertion follows from Theorem 3.51.1).
3) Let 5 be a G-stopping time. Immediately, we have Gs C Goo = ^r,
Gs C fs, Gs C TtH^s = Ft as- On the other hand, let A G Tsat- For
alU>0,
A[SAT <t)eTtnTSAT cGt.
This means A G GsaT and TsaT C Gsat C (?s. Thus (?s = GsaT = ^Sat-
4) The assertion concerned with XT comes directly from Theorem 3.52.
On the other hand, XJ[0,T] = XT — XtI]t}oo[- By 1) T is a G-stopping
time. Hence XtI[t<oo] = ^t^[t<oo] € £t, ^t^]t,oo[ *s G-predictable.
When XT is G-optional (resp. G-predictable), so is XI^j^.
5) Let S be an F-predictable time. Then X = I[s,oo[ *s -F-predictable,
XT = I[s[s<T]tool is G-predictable, i.e., S[s<t\ ls a G-predictable time.
D
Problems and Complements
3.1 Let (Xt)t>o be a right-continuous adapted process, 5 be a stopping
time in the wide sense, and B C R be an open set. Then
T = inf{* > 5 : Xt G B}( or inf{* > 5 : Xt G £})
is a wide-sense stopping time.
3.2 Let (Xt)t>o be a cadlag adapted process, 5 be a stopping time
and B C R be a closed set. Then
T = inf{* >S:XteBoTXt-eB}
is a stopping time. In particular, if (Xt) is a continuous adapted process,
S is a stopping time and B C iZ is a closed set, then
T = inf{* > 5 : Xt G 5}
is a stopping time.

Problems and Complements
107
3.3 Let (Xt)t>o be an adapted increasing process. For any a e R,
T = inf{* > 0 : Xt > a}
is a stopping time.
3.4 Let G = (Gn)n>o be a discrete time filtration. Define Tt = G[i\-
Then F = (Tt) is a right-continuous filtration.
1) If 5 is a G-stopping time, then S is also an F-stopping time, and
?s = Gs-
2) If T is an F-stopping time, then
(n, n<T<n + l,
| oo, T = oo,
is a G-stopping time and Gs = Ft-
3.5 Let X = (Xt) be an (^-adapted process. If for any e > 0, X is
(,Ft+<r)-progressive, then X is (^t)-progressive.
3.6 Let (Xt) be an adapted process, and D be a denumerable dense
set in R. Define
Yt+ = lim sup{Xa : s e Df)]t,t +-[},t > 0,
n—►oo 72
K" = lim va£{X3 : s 6 £>fl]M + -[},< ^ 0,
n—KX) fi
Z+ = lim sup{Xs : 5 6 Df)](t - -)+,t[},t > 0,
n—►oo 72
Zr = lim ini{Xs : s € Df)](t - -)+,t[},t > 0,
n—>oo n
Then (yt+), (Yp), (Z+) and (Zf) are all progressive.
3.7 Two filtrations F = (Tt) and G = (Gt) are given. Denote by
0(F),V(F),T(F) and 7>(F) the F-optional a-field, the F-predictable
cr-field, the collection of all F-stopping times and the collection of all
F-predictable times respectively. The notations 0(G), V(G),T(G) and
Tp(G) have the same meanings.
1) The following statements are equivalent:
(i) F = G, i.e., V* > 0 Tt = Gt-
(ii) 0(F) = 0(G).
(iii) T(F) = T(G).
2) The following statements are equivalent:
(i) ^o = Go, and Vi > 0 Ft+ = Qt+.

108
Chapter III Processes and Stopping Times
(ii) V(F) = V(G).
(iii) TP(F) = TP(G).
3.8 Let / G #(#+) x B(R), X = (Xt) be optional (resp. predictable).
Then Zt = f(t,Xt) is optional (resp. predictable).
3.9 Two filiations F = (Tt) and G = (Gt) satisfy the following
conditions:
^0 = 00, V*>0 7i+=ft+.
Then for any F-stopping time in the wide sense T
Ft- = Gt- •
3.10 Suppose that 5 is a stopping time, r.v. T > 5,T is
^-measurable, and T > 5 on [5 < oo]. Then T is a predictable time.
3.11 Let S be a predictable time and T be a stopping time. If S < T
and Ts- = Tt- , then T is a predictable time.
3.12 If S and T are predictable times, so is S + T.
3.13 Suppose that 5 is a predictable time, r.v. T > 5, T is T§-
measurable, and [T = S] G ^*5-. Then T is a predictable time.
3.14 Let T be a stopping time. Define
G = {Gt), Gt = Tt+u t > 0,
1) If 5 is an F-stopping time (resp. F-predictable time), then
~ f (S-T)+, T<oo,
\ 0, T = oo.
is a G-stopping time (resp. G-predictable time).
2) If (Xt) is an F-optional (resp. F-predictable) process, then (Xr+t
I[T<oo]) is a G-optional (resp. G-predictable) process.
3.15 Let X = (Xt) be a cadlag stochastic process, T be an F°(X)~
stopping time. If T takes only a denumerable number of values, then T is
measurable w.r.t. cr{XrAt,£ > 0}.
3.16 Let X = (Xt) be a cadlag stochastic process defined on (ft,.?7),
satisfying the following conditions: i) V* > 0 Xt(v) = Xt(u>') => uj = u/, ii)
Vcj G ft and £ > 0 there exists J G ft such that Vs > 0 Xa(u/) = XaAt(u;).
Then
1) A non-negative r.v. T is an F°(A")-stopping time if and only if
V*>0
T(u) <t,Vs<t Xs(u) = XS(J) => T(uj) = T(J).

Problems and Complements 109
2) If T is an F°(A>stopping time, then A G ?t(X) if and only if
ueA, T(u) = T(a/), Vs < T(u) X3(uj) = XS(J) => J G A.
3) For any F°(X)-predictable time T, T${X) = T^{XT), where XT =
(XTAt)-
3.17 In addition to the assumption in the above problem, we suppose
that Vu; G fi and t > 0 there exists uf e Q, such that Vs > 0 X3(uf) =
Xs(uj)3I[3<t]+Xt-(uj)I[3>t]. Then
1) A non-negative r.v. T is an F°(X)-predictable time if and only if
V*>0
T(uj) <t,Vs<t X3(uj) = Xa{J) =» T(u) = T(J).
2) If T is an F°(A>predictable time, then A G T^_{X) if and only if
w6A T(u) = T(u/), Vs < T(u) X3(uj) = XS(J) =► u/ G A.
3) For any F°(X)-predictable time T, ^£_(-Y) = ^(XT~), where
XT~ = XI[0T[ + XT-I[Tyool
4) An ^(-Y) x B(#+)-measurable process Z = (Zt) is
^(^-predictable if and only if Z is F°_(X) = (^_(X))-adapted.

Chapter IV
Section Theorems and Their Applications
In this chapter we mainly introduce section theorems and their
applications. We will show that predictable times are a.s foretellable. This
result will be used constantly later. By means of the concept of totally
inaccessible time we will investigate the jumps and quasi-left-continuity
of adapted cadlag processes.
§1. Section Theorems
In this paragraph, by using the theory of analytic sets and capacity
in Chapter I, we show the section lemma concerned with measurable sets.
Then, based on this lemma, we establish section theorems in general
theory of stochastic processes.
4.1 Definition. Let ft be a set, icftx .R+. Put
DA(u) = inf{t € R+ : (u,t) € A}, u € ft.
Da is called the debut of A. Recall that inf 0 = +oo.
4.2 Theorem. Let (ft, T) be a measurable space. For every A €
A{!F x B(R+)), Da is T-measurable.
Proof. Let r > 0. Then [DA < r] is the projection of A H (ft x [0, r[)
onto ft. Hence, by Theorem 1.32 [DA < r] G A(F). Since A(T) C T
(Theorem 1.36), DA is /"-measurable. □
Remark. Let (Xt) be a measurable process on (ft,^7). Then sup \Xt\
t
is .F-measurable. In fact, for all a > 0 put
Ta = inf{*>0:|Xt|>a},

§1. Section Theorems 111
then by the theorem we know that Ta is ^"-measurable. However,
[sup \Xt\ > a] = [Ta < oo] G T.
t
This means that sup \Xt\ is ^"-measurable.
t
The following lemma is usually called the section lemma. It is one of
the important applications of the theory of analytic sets and capacity in
probability theory.
4.3 Lemma. Let (ft,.F, P) be a probability space, A G A(PxB(R+)).
Then there exists a r.v. T G T* such that
T(u) < oo =» (uj,T(u)) G A, (3.1)
P([T < oo]) = P([DA < oo]). (3.2)
Proof. Put
P(A) = inf{P(B) :B eT,BDA}, A C ft.
Then P is a Choquet ^"-capacity on ft.
Let W be the paving consisting of all finite unions of sets in ,F<g)/C(i2+).
It is easy to see that 7i is closed under the formation of finite intersection.
By Theorem 1.32 we have
A(H) = A(T ® £(/*+)) = A(F ® £(#+)).
Denote by n the projection mapping from ft x i2+ onto ft. For any C G
>l(W) by Theorem 1.32 we have n(C) G >t(J"). Put
1(C) = P(tt(C)), Cc(lxfif
Prom the proof of Theorem 1.35 we see that J is a Choquet 7Y-capacity
on ft x ifc+.
Since A G A(T), by Theorem 1.35 for any given e > 0 there exists
B eH6, B CA such that 1(B) > 1(A) - e, i.e.,
P(jDb < oo) > P(DA < oo) - e.
Because D# is ^"^-measurable, there is a r.v. S£ G ^*"1' such that Se = Db
a.s.. Let C G ^*, C C [S£ = DB] such that P(C) = 1. Define
Te = 5e/c + (+oo)/Cc,
then Te G J7, Te = S£ = Db a.s., and Te = Db on [T£ < oo]. On the other
hand, for every u G ft, B(u) = {t > 0 : (v,t) G B} is a compact set in

112 Chapter IV Section Theorems and Their Applications
R+. Hence, DB{u) < +00 =*• (u>,DB(u)) € B CA We have
Te{u) < 00 =» (w,Te(w)) € A,
P(Te(u;) < 00) = P(DB < 00) > P(DA < 00) - e.
Putting To = +00, we define a sequence (Tn) of non-negative
^"-measurable r.v. satisfying (3.1) by induction as follows. If Tn has been defined,
put
An = A n {(w, t) : Tn(u>) = 00} = A n ([T„ = 00] x fl+).
Then An € .A(.F x /?(Jt+)). As shown above, there is a r.v. Sn € T+ such
that
5n(w) < 00 =* (w,Sn(u>)) € 4„,
P(Sn < 00) > \p{DAn < 00) = ^P([Tn = 00] n [DA < 00]).
Define Tn+i = Tn A 5n, then Tn+i satisfies (3.1) and
P(Tn+l < 00) = P(Tn < 00) + P(Sn < 00)
> P(Tn < 00) + \p{[Tn = 00] n [Dx < 00]). (3-3)
Set T = to^T* = /\Tn. Since Tn+1I[Tn<oo] = TnI[Tn<oo], we have
T/[Tn<oo] = Tn/[Tn<oo].nTherefore
OO OO
[T < 00] = U [Tn < 00], [T = 00] = fl [Tn = 00].
n=l n=l
Letting n —► 00 in (3.3) yields
P(T < 00) > P{T < 00) + £p([T = 00] n [Z^ < 00]).
Hence P([T = oo] n [DA < oo]) =, i.e., [D^ < oo] C [T < oo] a.s.. But
00
[T < 00] = (J [Tn < 00], so T satisfies (3.1), and [T < oo] C [DA < oo].
n=l
Therefore [T < oo] C [Da < oo] a.s., i.e., (3.2) is established. D
Remark. Later any non-negative r.v. T satisfying (3.1) is called a
section of A. (3.2) means that except for a P-null set, T is a full section
of A. Here comes the name of section lemma.
4.4 Lemma. Let (fi, T, P) be a probability space, C be a field
generating T. Then for any A £ T we have
P(A) = sup{P(B) :BeC6,BcA} = inf (P(C) :CeCa,CD A}.
(4.1)

§1. Section Theorems
113
Proof. Denote by Q the collection of all sets in T satisfying (4.1). Since
Ca = {A : Ac G C6}, we have AeG =» Ac G £.
Let An G G, An t A. We are going to show AeG- For any given
e > 0, take n large enough such that V(A\An) < -. Let B G C$, B C An
e 2
such that P(An\B) < -. Hence £ C A and P(A) < P(B) + e. On the
Li
other hand, for each n take Cn G Ca, Cn D An such that P(Cn\An) < —.
Li
OO
Put C = (J Cn. Then CGCff,CD4 and P(C\A) < e. This means
n=l
A G 5. We have shown that (/ is a monotone class. It is evident that
Cc5. Hence G = T. U
The following theorem is an important application of Lemma 4.3, and
will be used to prove section theorems.
4.5 Theorem. Let (fi,.F, P) be a probability space, G be a sub-o-field
of Tx B(R+), and C be a field generating G- Let AeG- Then for any
given e > 0, there exists B GCs such that
Be A, (5.1)
P(7r(A))<P(*(B)) + e. (5.2)
Proof Choose a r.v. T G T+ satisfying (3.1) and (3.2). Define a
measure /x on G as follows:
^(G) = P({u; : (cj, T(uj)) G G}), G G G.
Then A is the support of /i, and /i(A) = P([T < oo]) = P(7r(>l)). For any
G G G we have
{cj:(cj,T(u/))gG}Ctt(G).
Hence /i(G) < P(7r(G)). For any given e > 0, by Lemma 4.4, there exists
BGC^BcA such that p(A) < p(B) + e. Thus
P(n(A)) = fi(A) < fi(B) + e < P(n(B)) + e. D
Below we will make use of Theorem 4.5 to show section theorems in
general theory of stochastic processes. We suppose that a probability
space (fi,^*, P) and a filtration (Ft) are given.
4.6 Lemma. Let V be a family of wide-sense stopping times satisfying
the following conditions:
i) 0 G V, +oo G V,

114 Chapter IV Section Theorems and Their Applications
ii) S,T G V =► S A T G V, S V T G V,
iii) S,TeV=>S[s<T) € V,
iv) Sn G V, n = 1,2 • • •, Sn T 5 =► 5 G V.
Let C be the collection of all finite unions of sets in Co = {[£,T[: S <
T, S,T G V}. Then C is a field on Q x #+. For anj/ B € Cs we have
[Db] C £, w/iere Db is the debut of B (Definition 4.1), and there exists
a wide-sense stopping time T G V such that T = Db Q»s..
Proof The fact that C is a field follows from conditions i) and ii).
For each u e CI, B(u) = {t > 0 : (v,t) G S} is closed relative to the
right topology in ifc+. Hence [£>s] C B. Put
W = {5GV:S< Db}.
It is easy to see that Sn G H, n = 1,2, • • • => V Sn G 7i. Then, by Theorem
n
1.13 there exists T eH such that
T = ess sup H.
We are going to show T = Db a.s.. Let (Bn) be a decreasing sequence of
elements in C such that B = f]Bn. Put
n
C„ = BBn[T,oo[.
Then (Cn) is a decreasing sequence of elements in C, and
C\Cn = Bn[T,oo[=B.
n
Let G = [5,T[G C0. By condition iii) we have Dq = S[S<t] € V, where
Dg is the debut of G. Suppose Cn = Cn\ U Cn2 U • U Cnm, where
771
Cnfc € Co, fc = 1,2,■••,"». Then DCn = A DCnk € V (condition ii)),
fc=i
and DCn > T. Since C„ D 5, DCn < DB, i.e., I>cn € H. But T is the
essential supremum of 7i, it must be that Dcn = T a.s. for each n. Since
WcJ C C„, [T] is contained a.s. in f]Cn = 51). Thus T > DB a.s.. We
n
have already shown T < D^. Hence T = £># a.s.. □
The following two theorems are all called section theorems. They are
the most important results in general theory of stochastic processes.
4.7 Theorem. Let A be an optional (resp. accessible) set For any
given e > 0 there exists a stopping time (resp. accessible time) T such
that
l) Here the meaning of a.s. is that for almost all u> G [T < oo] we have (u;, T(lj)) e B.

§1. Section Theorems
115
i) IT] C A,
)i)P(T<oo)>P(n(A))-e,
where ir(A) is the projection of A onto Q.
Proof. We only give proof for the optional case. It is completely
similar to the accessible case. Let V be the collection of all stopping
times. Obviously, V satisfies the four conditions in Lemma 4.6. We will use
notations in Lemma 4.6. By Theorem 3.17 we know a(C) = O. According
to the assumption, A G <r(C). By Theorem 4.5 there exists B G Cs such
taht B C A and P(ir(B)) > P(*(A)) - e. By Lemma 4.6 there exists a
stopping time S eV such that 5 = Db a.s.. Put
L = {u:(uj,S(u))eB}.
Then IB(S)I[s<oo] = IL, and L € fs (Theorem 3.12). Since [DB] C J5,
we have P(L U [S = +oo]) = 1. Set T = SL. Thus T is a stopping time,
[T] C B C A, and T = 5 = jDb a.s.. Hence
P(T < oo) = P(jDb < oo) = P(n(B)) > P(n(A)) - e. U
4.8 Theorem. Let A be a predictable set. For any given e > 0 there
exists a predictable time T such that
i) IT] C A,
ii) P(T < oo) > P(n(A)) - e.
Proof. Let V be the collection of all predictable times. Then one may
copy the proof of Theorem 4.7, only noting that here L G Fs- (Corollary
3.23.2)), so SL is a predictable time (Theorem 3.29.7)). □
4.9 Definition. A subset A of fi x H+ is said to be evanescent,
(w.r.t. P) provided that the projection 7r(A) of A onto Q, is a P-null set
(it needn't be 7r(A) G T, but should be 7r(A) G FP)> A process X is said
to be evanescent, if {(u;,t) : Xt(u) ^0} is an evanescent set.
Two processes X = (Xt) and Y = (Yt) are said to be
P-indistinguishable (simply denoted by X = y), if {(u;,t) : Xf(u;) ^ ^t(^)} is an P-
evanescent set (refer to Definition 2.45). X is said to be not larger than
Y (denoted by X < Y), if {(uj,t) : Xt(uj) > Yt(uj)} is an P-evanescent
set. Henceforth, in a certain class of processes (e.g. the class of optional
processes or predictable processes) we identify two indistinguishable
processes.

116 Chapter IV Section Theorems and Their Applications
Below we give an application of section theorems to stochastic process
theory. We omit the accessible cases for brevity.
4.10 Theorem. Let X = (Xt) and Y = (Yt) be two optional (resp.
predictable) processes. If for every bounded stopping time (resp.
predictable time) T, we have Xt < Yt a.s., then X < Y.
Proof. We only discuss the optional case. Suppose A = {(u,t) :
Xt(uj) > Yt(uj)} is not evanescent. Since A is optional, by Theorem 4.7
there exists a stopping time S such that [5] C A and P(S < oo) > 0.
Choose a constant C > 0 such that P(S < C) > 0. Put T = S A C. Thus
T is a bounded stopping time, and Xt > Yt on [S < C\. It contradicts
the assumption. Hence A is evanescent. By definiton we have X < Y.
D
4.11 Corollary. Let X = (Xt) and Y = (Yt) be two optional (resp.
predictable) processes. If for every bounded stopping time (resp.
predictable time) T, we have Xt = Yt a.s., then X = Y.
In practical applications the following theorem is more effective than
Theorem 4.10 sometimes.
4.12 Theorem. Let X = (Xt) and Y = (Yt) be two optional (resp.
predictable) processes. If for every stopping time (resp. predictable time)
T, XTI[T«x>) and YTI[T<od\ are integrable, E[XtI[t<oo]] < E[>W[r<oo]],
then X < Y.
Proof. We only discuss the optional case. Suppose A = {(v,t) :
Xt(uj) > Yt(uj)} is not evanescent. By Theorem 4.7 there exists a
stopping time T such that [T] C A and P(T < oo) > 0. Thus we have
E[Xt/[t<oo]] > E[YtJ[t<oo]]- This contradicts the assumption. Hence A
is evanescent, i.e., X < Y. □
4.13 Corollary. // in Theorem 4.12 we have E[XtI[t<oo]] =
E[YTIlT<oo]], thenX = Y.
Remark. In Theorem 4.12 and Corollary 4.13, if only bounded
stopping times (resp. predictable times) are concerned, it is not sufficient to
arrive at the conclusion.

§2. a.s. Foretellability of Predictable Times
117
§2. a.s. Foretellability of Predictable Times
4.14 Definition. Let T be a wide-sense stopping time, (Tn)n>i be an
increasing sequence of wide-sense stopping times, and for all n, Tn < T.
Let A C fi. We say that the sequence (Tn) a.s. foretells T on A, if on
A\T > 0] we have Tn < T a.s. for all n and lim Tn = T a.s.. We say
briefly that (Tn) a.s. foretells T if (Tn) a.s. foretells T on whole ft.
A wide-sense topping time T is said to be a.s. foretellable, if there
exists an increasing sequence of wide-sense stopping times which a.s.
foretells T.
We will show every predictable time is a.s. foretellable.
4.15 Lemma. Let V be the family of all a.s. foretellable wide-sense
stopping times. Then V has the following properties:
i) 0 G V, +oo G V.
ii) S,TgV=> SAT, SvTgV.
iii) The limits of increasing sequences of elements in V belong to V.
iv) The limits of stationary decreasing sequences of elements in V
belong to V.
v) If S G V, T is a wide-sense stopping time, and T = S a.s., then
Tev.
vi) 5, TeV=> S[S<T) e V.
Proof i) and ii) are obvious.
iii) Let T be the limit of an increasing sequence (Tn) of elements in
V. For each n, let (5n)jfc)jt>i be a sequence of wide-sense stopping times,
which a.s. foretells Tn. Put
sk = s^k v s2,k v • • • v sKk.
Then (Sk)k>\ a.s. foretells T.
iv) Let T be the limit of a stationary decreasing sequence (Tn) of
elements in V. For each n let {Sn^)k>\ be a sequence of wide-sense stopping
times which a.s. foretells Tn. We may assume for all n and k
p(e~Sn>k - e"Tn > 2"fc) < 2~(n+fc)
(if necessary, by taking a subsequence). Put Sk = inf 5n^. Then (5n) is
an increasing sequence of wide-sense stopping times, and for all fc, Sk < T.
On [T > 0] we always have Tn > 0 for all n, and thus for all A;, Sn£ < Tn

118
Chapter IV Section Theorems and Their Applications
a.s.. Since (Tn) is stationary, on [T > 0] for all k we have Sk <T a.s.. Let
5 = Um Sk- For all k
k—>oo
P(e~s - e~T > 2~k) < P(e"s* - e~T > 2~k)
< P( (J [e-5"-* - e"T > 2"fc]) < £ P(e-5".* - e"T" > 2~k) < 2~k.
n=l n=l
Hence 5 = T a.s. and (5n) a.s. foretells T, i.e., T G V.
v) Suppose (5n) is a sequence of wide-sense stopping times which a.s.
foretells 5. Then (5n A T) a.s. foretells T. Hence T G V.
vi) Let (5n) and (Tn) be two sequences which a.s. foretell S and T
respectively. Put
Un = n A 5j5n<Tm].
For each fixed m, (C^l)n>i a.s. foretells the wide-sense stopping time
Um = 5[5<T-]fl[T->o]- (Since lS ^ Tml HP™ > 0] belongs to Ts+> but
not to Ts in general, Um is only a wide-sense stopping time, even if S
is a stopping time.) Hence Um G V. Evidently, (Um) is a stationary
decreasing sequence. Its limit U belongs to V by iv). But U = S\§<t] a-s..
Therefore, 5[5<T] G V by v). □
4.16 Theorem. All predictable times are a.s. foretellable.
Proof. Let V be the collection of all a.s. foretellable wide-sense
stopping times. We make use of notations in Lemma 4.6. By Corollary 3.28
we know V C cr(C). From the proof of Theorem 4.7 we can see that the
section theorem holds for sets in a(C) (because in this case Si = S a.s.,
thus SL G V).
Now let T be a predictable time. Then [T] G V, and [T] G a(C).
For any given e > 0 there exists S£ G V such that [Se] C [T] and
P(S€ < oo) > P(T < oo) - e. Define
Tn = 5^ A S* A • • • A Sn, n > 2.
It is easy to see that (Tn)n>2 is a stationary decreasing sequence of
elements in V and lim Tn = T a.s.. By properties iv) and v) of V we have
n—►oo
T G V, i.e., T is a.s. foretellable. □
Furthermore, we will show that all predictable times can be a.s.
foretellable by a sequence of predictable times.
4.17 Theorem. Let T > 0 be a predictable time, and A C [0,T[ be an
optional (resp. predictable) set such that for almost allu G [T < oo], T(u)

§2. a.s. Foretellability of Predictable Times 119
is a limit point of set A(u>) = {t > 0 : (v,t) G A}. Then there exists an
increasing sequence (Tn) of stopping times (resp. predictable times) such
that \J[Tn] C A a.s., Tn<T and lim Tn = T a.s..
n n—►oo
Proof. We only discuss the predictable case: A is predictable. Let
(V^) be a sequence of wide-sense stopping times which a.s. foretells T.
For each n the projection of predictable set A„ = Af)}Vn,T[ onto ft a.s.
contains [T < oo]. By Theorem 4.8 for each n there exists a predictable
time Un such that [Un] C A„ and
P(T < oo) < P{*{ATn)) < P(Un < oo) + ^.
Since lim Vn = T a.s., we have liminf Un = T a.s..
n—►oo n—►oo
For each n and k define
Snk = inf C/m, 5n = inf Sntk = lim Snjib.
n<ra<n+fc fc fc—^oo
It is obvious that (Sn,ife)ife>i is a decreasing sequence of predictable times.
oo
We will show that it is a.s. stationary. If cj G f| [^m = °o], then for all
77l=Tl
oo oo
* > 1,5»,fc(w) = oo. If w € ( U [Um < oo]) H( D [Vt < T]) fl[lim V, = T],
m=n J=i «—k»
then there exists mo > n such that Umo(uj) < oo. Since l/moO*') < ^(a;)
and lim Vi(uj) = T(u), there exists ko>mo~-n such that for fc > fco
I—►oo
0n+fc(") > Vn+fc(w) > 1^(4
This means for fc > fc0 Sn}k(uj) = 5n)jfc0(u;), i.e., (Sn,fcM)fc>i is stationary.
Noting that P( f| [VJ < T])H[ lim V/ = T]) = 1, we have shown that
(Sn,k)k>i is a-s. stationary. Put
oo
An,k = fl [Sn,k = Sn,k+i,], Rn,k = (Snyk)An,k A T.
i=l
Then An>jt G ^r5nfc_) #n,ifc is predictable and (-Rn,fc)ifc>i is a stationary
decreasing sequence (noting Ank C j4nfc+i). Hence .Rn = lim Rn^k is
' ' fc—►oo
predictable, and Rn = Sn a.s.. Put
Tn = Ri V #2 V • • • V Rn.
Since (5n) is an increasing sequence, we have Tn = Rn = Sn a.s.,
lim Tn = lim Sn = liminf Um = T a.s..
■ n—►oo n—►oo n—►oo
Finally, because all graphs of 5n>jt are contained in A and (Sn,ifc)fc>i is a.s.
stationary, the graph of Sn is a.s. contained in A, and so is the graph of
Tn. U

120 Chapter IV Section Theorems and Their Applications
4.18 Corollary. All predictable times are a.s. foretellable by a
sequence of predictable times.
Proof. Let U be a predictable time. Put T = £fy/>o]- Then T is pre-
dictable and T > 0. By Theorem 4.17 there exists an increasing sequence
(Tn) of predictable times such taht \J[Tn] C [0,T[ a.s. and lim Tn = T
n n—►oo
a.s.. Put
Un = TnAU[u=0].
Then (Un) is an increasing sequence of predictable times, and a.s. foretells
U. □
§3. Totally Inaccessible Times
4.19 Definition. Let T be a stopping time. T is called totally
inaccessible, if for every predictable time 5 P(T = 5 < oo) = 0.
Obviously, totally inaccessible times are a.s. positive. If a stopping
time is a.s. equal to a totally inaccessible time, then it is totally
inaccessible, too. If a stopping time is both accessible and totally inaccessible, it
must be a.s. equal to +oo.
4.20 Theorem. For each stopping time T there exists A £ Tt- such
that A C [T < oo], Ta is accessible and T\c is totally inaccessible. Such
set A is a.s. unique.
Proof. Put
H = {\J[Sn = T < oo] : (5n) is a sequence of predictable times}.
n
Obviously, H C Tt- and H is closed under the formation of countable
unions. By Remark 1.14, there exists A eH such that A = ess supH. It
is easy to see that T4 is accessible, and T^c is totally inaccessible. It is
not difficult to verify a.s. uniqueness of A. □
Remark. Usually, T4 and T^c are called the accessible part and totally
inaccessible part of T respectively, and denoted by Ta and Tl respectively.
They are determined a.s. uniquely.
Now we are ready to study the jumps of adapted cadlag processes.
4.21 Theorem. For any adapted cadlag process X = (Xt) there exists

§3. Totally Inaccessible Times
121
a sequence (Tn) of strictly positive stopping times satisfying the following
conditions:
i)[AX*0]C\J[Tn],
n
ii) each Tn is predictable or totally inaccessible^
iii) [Tn] n[Tm] = <bforn^m.
Such a sequence (Tn) of stopping times is called a standard sequence of
stopping times exhausting the jumps of X.
Proof. By Theorem 3.32 there exists a sequence (Un) of strictly positive
stopping times such that condition i) is satisfied. By Theorem 4.20 for
each n there exist an accessible time U% and a totally inaccessible time
\]xn such that
[Un] = IK] U fUn].
From the definition of accessible times we see that there exists a sequence
(5n) of strictly positive stopping times satisfying conditions i) and ii). Put
M\ = {n : Sn is predictable},^ = {n : Sn is totally inaccessible},
Ti = 5i,
{(I [Sk^Sn], neAfi,
k<n-l,kerfi
n>2,
( n [sk*sn])n( n [s**sn]), ^4
ifceA/i JKn-l.ifeeA/i
Tn = (Sn)Bn, n>2.
K Sn is predictable, then Bn G ^*5n-, and Tn is predictable. If Sn is totally
inaccessible, then Bn € Tsn^Tn is totally inaccessible. Apparently, (Tn)
satisfies all conditions i), ii) and iii). □
Remark. If in condition ii) predictable times are replaced by
accessible times, then it is desirable that in condition i) equality holds.
4.22 Definition. Let X = (Xt) be an adapted cadlag process, and
T > 0 be a stopping time. T is called a jump time of X, if on [T < oo]
we have XT ^ XT- a.s., i.e., P(T < oo,XT ^ XT-) = P(T < oo). We
say that X has only accessible jumps, provided that each jump time of
X is a.s. equal to a certain accessible time. We say that X has only
totally inaccessible jumps, provided that each jump time of X is totally
inaccessible.

122
Chapter IV Section Theorems and Their Applications
An adapted cadlag process having only totally inaccessible jumps is
said to be quasi-left-continuous.
4.23 Theorem. Let X = (Xt) be an adapted cadlag process. Then
the following statements are equivalent:
1) X is quasi-left-continuous;
2) For every predictable time T > 0, Xt = Xt- a.s. on [T < oo];
3) If(Tn) is an increasing sequence of stopping times and T = lim Tn
then lim Xtu = Xt cl-s. on [T < oo].
Tl—►OO
Proof. 3) => 2) follows from Theorem 4.16. 2) => 1) follows from
Theorem 4.20. We are going to show 1) ^ 2) => 3).
Assume that X is quasi-left-continuous, and T > 0 is a predictable
time. Put
B = [T < oo] n [XT ? XT-].
Then Tb is a jump time of X. By the assumption Tb is totally inaccessible,
P(TB = T < oo) = 0.
This means Xt = Xt- a.s. on [T < oo]. 1) =*► 2) is established.
Assume 2) holds. Let (Tn) be an increasing sequence of stopping times,
and T = lim Tn. Put
n—►oo
A = f][Tn<T}.
n
Then Ta > 0 and (Tn)[Tn<x\ A n foretells T^. Hence Ta is predictable.
By 2) XTa = XTa- a.s. on [T < oo], i.e., XT = XT- a.s. on Af\[T
< oo]. Therefore, lim XTn = XT a.s. on Af][T < oo]. But on Acf][T <
n—►oo
oo], lim Xpn = Xt holds always. Thus
n—►oo
lim Xr = Xt a.s. on [T < oo].
n->oo In L J
2) => 3) is established. □
Remark. A thin set is said to be totally inaccessible if A = U[Tn],
n
where each Tn is a totally inaccessible time. Then an adapted cadlag
process X is quasi-left-continuous if and only if [AX ^ 0] is totally
inaccessible.
4.24 Theorem. Let X = (Xt) be an adapted cadlag process. In order
that X have only accessible jumps it is necessary and sufficient that for

§3. Totally Inaccessible Times
123
any totally inaccessible time T we have
Xt = Xt- a.s. on [T < oo].
Proof. It is analogous to proof 1) -<=>► 2) in Theorem 4.23. □
The following theorem gives a useful decomposition for adapted
processes with finite variation.
4.25 Theorem. Let A = (At) be an adapted process with finite
variation. Then A has a unique decomposition as follows:
A = AC + Ada + Adi,
where Ac is an adapted continuous process with finite variation, A^ is
an adapted purely discontinuous process with finite variation having only
accessible jumps, and Adx is an adapted quasi-left-continuous purely
discontinuous process with finite variation.
Proof. Let (Tn) be a standard sequence of stopping times exhausting
the jumps of X. Let M\ = {n : Tn is predictable}, A/2 = {n : Tn is totally
inaccessible}. Put
A? = £ AATn/[Tn<t], Af= £ AATJ[Tn<t], t>0.
By Theorem 3.42 A = Ac + A**0, + Adl is a decomposition satisfying the
requirement of the theorem. Suppose A has another decomposition
satisfying the requirement of the theorem: A = Ac + A + A . Denote
B = A^ - IT = Tl - Adi. Then B is an adapted purely discontinuous
process with finite variation. Let T > 0 be a stopping time, Ta and
1* be its accessible and totally inaccessible parts respectively: [T] =
[T1] \J[T% We have ABT« = AA$a - AA#a = 0 a.s. on [Ta < 00], and
ABTi = AAfc - AA^i = 0 a.s. on [T* < 00]. Hence ABT = 0 a.s. on
[T < 00]. By Corollary 4.11, AB is indistinguishable from a null-process,
i.e., B is indistinguishable from a continuous process. But B is a purely
discontinuous process with finite variation, so B is indistinguishable from
a null-process, i.e., A**" and A are indistinguishable, Adx and A are
indistinguishable. The uniqueness of the decomposition has been proved.
□

124 Chapter IV Section Theorems and Their Applications
§4. Complete Filtrations and the Usual Conditions
4.26 Theorem. Assume (Tt) is complete (see Definition 2.63).Then
all evanescent measurable processes are predictable.
Proof. Let X be an evanescent measurable process, and A = {u :3t e
JR+ such that Xt(u) ^ 0}. Then P(A) = 0, and A e T0. Put
C = {C x [t, oo[: C G T,t G R+}.
It is easy to see that C is a 7r-class, and a(C) = T x B(R+). On the other
hand, put
H = [Y : Y is T x B(fl+)-measurable and YIaxR+is predictable}.
Then for all H G C, /# G W. By the monotone class theorem (Theorem
1.4) 7Y is just the collection of all T x B(i2+)-measurable processes. In
particular, X = XIaxR+ is predictable. □
4.27 Corollary. Assume that (Tt) is complete. Let A G T and
P(A) = 0. Then for each non-negative r.v. £, £a = £Ia + (+°o)Iac is a
predictable time.
Proof. Since [f^+o0! is an evanescent measurable set, it is
predictable. By Definition 3.25 £a is a predictable time. □
4.28 Corollary. Assume that (Tt) is complete. Let X and Y be two
indistinguishable measurable processes. If X is optional (resp. accessible,
resp. predictable), soisY.
Proof. Note that Y = XI[x=y] + YI[x?Y]i hx^Y\ an(* ^[*=>1 are
predictable. Then the assertion follows immediately. □
4.29 Theorem. Assume that (Tt) is complete.
1) Let T be a stopping time (resp. accessible time, resp. predictable
time), and S = T a.s.. Then S is a stopping time (resp. accessible time,
resp. predictable time) and Tt = Ts,Tt- = Ts--
2) Let S > 0 be a r.v.. In order that S be a stopping time (resp.
accessible time, resp. predictable time) it is necessary and sufficient that
[5] be an optional (resp. accessible, resp. predictable) set.
Proof 1) follows from Corollary 4.28 since 5 is a r.v. and measurable
process. i[s)0o[ and ^[T,oo[ are indistinguishable. The equalities Tt = Ts
and Tt- = Ts- follow directly from the definitions of Tt and Tt--

§4. Complete Filtrations and the Usual Conditions
125
2) Only the sufficiency needs to be proved. We discuss only the
predictable case, the others are similar. Let [S] be a predictable set. By the
section theorem, for any given e > 0 there exists a predictable time Te
such that [T£] C [5] and P (Te < oo) > P(S < oo) - e. Put
Sn = T^ A---ATn,n>2.
Then (Sn)n>2 is a stationary decreasing sequence of predictable times,
and lim Sn = S a.s.. By 1) 5 is a predictable time. □
n—►oo
4.30 Theorem. Assume that (Tt) is complete. The debut Da of any
progressive set A is a wide-sinse stopping time. Furthermore, if [Da] C
A, Da is a stopping time.
Proof. For any t > 0 put
At = {(u>,s) :s<t,(u;,s)e A}.
Then At = A^Q, x [0,t[) G Tt x B(R+), and [Da < t] = 7r(At), where
7r(Af) is the projection of At onto fi. Since Tt is complete w.r.t. P, by
Theorems 1.36 and 1.40 [DA < t] = 7r(At) G A(Tt) = ^i- Hence Z^ is a
wide-sense stopping time.
If [DA] C A, then [Da < *] = *r(At) where
At = {(cj, a) : s < t, (a;, a) G A}.
Similarly, [D^ < t] = n(At) G ^(^t) = Tt. Hence D^ is a stopping time.
□
4.31 Theorem. Assume that (Tt) is complete. Let H be a predictable
set, and [Dh] C H, where Dfj is the debut of H. Then Dh is a predictable
time.
Proof. First, by Theorem 4.30 Dh is a stopping time. Thus [Dh] =
Hf)[0,Dn] is a predictable set. Then by Theorem 4.29.3) Dh is
predictable. □
4.32 Theorem. Assume that (Tt) is complete. All adapted right-
continuous processes are optional.
Proof. Let X be an adapted right-continuous process. For any given
e > 0 denote by A the collection of all stopping times 5 satisfying the
following condition: there exists an optional process Y^ such that
{(w, t):te [0,5(w)[, \Xt(u) - y/5)(u,)| > e}

126
Chapter IV Section Theorems and Their Applications
is an evanescent set. It is easy to see that A is not empty (because 0^4)
and has the following properties:
i) s, T e A =► S v T e A
ii) Sn G A n = 1,2, • •., Sn | 5 =► 5 G A
iii) 5 G A T is a stopping time, T = S a.s. => T G A
By Theorem 1.13 there exists T G A such that T = ess sup A We are
going to show T = +oo a.s.. Let
A = {(W, t) : t > T(w), |Xt(w) - XTM(u>)\ > e}.
A is a progressive set. Denote by U the debut of A. In view of right-
continuity of X, we have [£/] C A. Hence U is a stopping time. Put
Then U € .4 and U > T. Therefore U = T a.s.. On the other hand, again
by the right-continuity of X, we have T < U on [U < oo]. This means
T = U = +00 a.s. Thus by the property iii) of A we know +00 6 A. Put
X£ = y (+*>). Then Xe is an optional process and
{(W,t):|XtM-Xf(a;)|>e}
It.
is an evanescent set. Take en = —. Put
n
F = liminf X£n, Y = YIUyUoo].
So y is an optional process, X and Y are indistinguishable. Hence X is
optional by Corollary 4.28. □
4.33 Theorem. Assume that (Tt) is complete. Let X be an adapted
cadlag process. In order for X to be predictable it is necessary and
sufficient to satisfy the following conditions:
i) For each totally inaccessible time S Xs = Xs- a.s. on [S < oo];
ii) For each predictable time T,XtI\t<oo] is Ft--measurable.
Proof. The necessity follows from Theorem 3.33. We are to show the
sufficiency. Suppose conditions i) and ii) hold. By Theorem 3.32 there
exists a sequence (Un) of strictly positive stopping times such that
[AX^0] = [JlUnl
n
For each n, let U% and Uxn be the accessible and totally inaccessible parts
of Un respectively:

§4. Complete Filtrations and the Usual Conditions
127
By condition i) U\ = +00 a.s.. Hence U%n is a predictable time (Theorem
4.29.2)). Then Un = U% A JJ%n is accessible, and there exists a sequence
(Tn) of strictly positive predictable times such that
[ax ^ 0] c upy.
n
Combining with condition ii), from Theorem 3.33 we know that X is a
predictable process. □
Remark. Prom the above proof one can see that condition i) is
necessary and sufficient for X to be accessible ( see the Remark of Definition
3.37). Theorem 4.33 is more convenient than Theorem 3.33 in
applications.
4.34 Theorem. Assume that (Tt) is complete. All predictable times
are foretellable.
Proof. Let T be a predictable time, and (Sn) be an increasing sequence
of stopping times which a.s. foretells T. Put
A = {(f][Sn < T]) n[T= lim Sn}} U [T = 0],
n n-*oo
Tn = (Sn)AA(T-±)+c.
n
Since P{A) = 1, (Tn) is a sequence of stopping times by Theorem 4.29.
Obviously, (Tn) foretells T. □
4.35 Theorem. Assume that (Tt) is complete. The following
statements are equivalent:
1) (Ft) is quasi-left-continuous, i.e., for every predictable timeT,Tr —
?T-,
2) All accessible times are predictable,
3) If(Tn) is an increasing sequence of stopping times andT = lim Tn,
then
rT = \IFTn.
n
Proof 1) <=> 2) and 1) => 3) have already been shown in Theorem
3.40. It remains to show 3) => 1). Let T be a predictable time, and (Tn) be
a sequence of stopping times foretelling T (Theorem 4.34). By Theorem
3.4.11) we have
n
By 3) we obtain Tt = Tt~- Hence [Tt) is quasi-left-continuous. □

128
Chapter IV Section Theorems and Their Applications
4.36 Theorem. Assume that F = (Tt) is the usual augmentation of
a filtration F° = (T?), i.e.,
Tt = T«+\/N, t>0,
where M is the a-field generated by all P-null sets (see Definition 2.63).
1) For each F-stopping time T there exists an F° -stopping time in the
wide sense U such that T = U a.s.. In addition, we have
TT = T%+ V Af, TT- = T%_ V M.
2) For each F-optional process X there exists an F\-optional process
Y such that X and Y are indistiguishable.
Proof. 1) Since
T = n!fe>l {^l^TKfr + (+00)7[r<^]u[T>£]}'
we may assume that T has the following form:
T = aIA + {+oo)IAc, a G i*+, A G
Tain this case, take B G .7^+ such that P(BAA) = 0. Then
U = aIB + (+oo)IBc
is an ir>0-stopping time in the wide sense, and U = T a.s..
By Theorem 4.29.1) we have TT = Tu D ^+ V M. On the other
hand, for any L G Tt we may take V G T^ such that P(LAL') = 0, and
an F°-stopping time in the wide sense V such that V = Tl a.s.. Put
M = (V H [U = oo]) U [V = U < oo].
Then M G T%+, and P(MAL) = 0. This means L G T%+ V Af. Hence
Tt = T$+ V M. The equality for Tt- is trivial.
2) Suppose X = /pr,oo[> where T is an F-stopping time. By 1) take
an F+-stopping time U such that U = T a.s.. Then Y = I[u,oo[ls an &+-
optional process, and indistinguishable from X. The general assertion
follows by the monotone class argument. □
4.37 Theorem. Assume that F = (Tt) is the completion of a
filtration F° = (T?), i.e.,
Ji=^VAT, t>0.
1) For each F°-stopping time T we have

§4. Complete Filtrations and the Usual Conditions
129
2) For each F'-predictable time T there exists an F°-predictable time
S such that T = S a.s. and Tt- = T%- v M.
3) For each F-predictable process X there exists an F°-predictable
process Y such that X and Y are indistiguishable.
Proof. 1) is trivial.
2) Let (T71) be a sequence of F-stopping times foretelling predictable
time T[r>0]. For each n let Rn be an F+-stopping time such that Rn = Tn
a.s. (Theorem 4.36.1)). Replacing Rn by R1 V • • • V Rn (if necessary), we
may suppose (Rn) is increasing. Denote R = lim Rn. Put An = [Rn <
n—KX)
R]. Since (An) is decreasing, (Un = i?^n A n) is an increasing sequence of
F^_-stopping times. (Un) foretells U and U > 0, so U is an ^-predictable
time (see the proof of Theorem 3.27). Because P(An) = 1, Un = Rn A n
a.s., U = R = T[T>0] a.s.. Take H G Jj such that P(HA[T = 0]) = 0.
Put
S = UA0H-
Then 5 is an F°-predictable time, and T = S a.s..
By 1) and Theorem 4.29.1) we have TT- = FS- = ?$- v-^-
Proof 3) is similar to that of Theorem 4.36.2). □
4.38 Theorem. Assume that F = (Tt) is the usual augmentation of
a filtration F° = (/?).
1) For each F-accessible time T there exists an F^.-accessible time U
such that T = U a.s..
2) For each F-accessible process X there exists an F+-accessible
process Y such that X and Y are indistinguishable.
Proof. 1) Let 5 be an ^-predictable time such that S = T a.s..
Assume [T] C UF^nL where (Tn) is a sequence of F-predictable times.
n
For each n there exists an ^-predictable time Sn such that Sn = Tn a.s.
(Theorem 4.37.2)). Obviously, [5] C \J[Sn] a.s.. Put
n
u = /\(s)[s=Sn].
n
Then U is an J^ -accessible time: [U] C (J[SnJ, amdU = S = T a.s..
n
2) follows from 1) by the monotone class argument. □

130 Chapter IV Section Theorems and Their Applications
§5. Applications to Martingales
In this paragraph we assume that (fi,^*, P) is a complete probability
space and the filtration F = (Ft) satisfies the usual conditions.
4.39 Theorem. Let X = (Xt) be a right-continous supermartingale
(resp. martingale) and T be a stopping time. Then the stopped process
XT = (XtAr) is also a supermartingale (resp. martingale).
Proof. For each t > 0, XtAT is ^tAT-measurable. Hence XT is adapted.
Let a G i2+, 0 < s < a. Applying Theorem 2.59 to Xa = (XtAa) yields
E[Xa/KT\Fs] < X3AT (resp. = XsAT) a.s..
Therefore, XT is a supermartingale (resp. martingale). □
Remark. Put Qt = TtKT- Filtration (Gt) satisfies the usual conditions,
and XT is a (^t)-supermartingale (resp. martingale).
The following theorem is very useful sometimes, though it is simple.
4.40 Theorem. Let X = (Xt)t>o be an adapted measurable process,
and Xoo be an integrable Too-measurable r.v.. If for every stopping time
T, Xt is integrable, and E[Xt] is independent of T, then X is a
uniformly integrable martingale. Moreover, if X is optional, then almost all
its trajectories are right-continuous.
Proof. For any stopping time T and A G Tt we have
/ XTdP = E[XTa] - I XoodP = E[Xoo] - I XoodP = / X^dP.
J A JAC JAc J A
(40.1)
Take T = t G R+ in (40.1). Since Xt G Tu we obtain
E[Xoo\ft]=Xt a.s..
Hence X is a uniformly integrable martingale. Moreover, if X is optional,
for any stopping time T, Xt is ^r-naeasurable. From (40.1) we obtain
E[Xoo\Tt)=Xt a.s..
Let Y = (Yt) be the right-continuous adapted modification of martingale
(E[Xoo\Ft\) ( with the convention Y^ = A^). By Theorem 2.58 we have
YT = E[Xoo\rT] = XT a.s..
Since Y is optional, by Corollary 4.11 X and Y are indistinguishable.
Therefore, almost all trajectories of X are right-continuous. □

§5. Applications to Martingales
131
Remark. Suppose X = (Xt) is an optional process. If for every
bounded stopping time T Xt is integrable and 15[AV] is independent of
T, then applying the theorem to Xn = (XtAn),n > 1, we know that X is
a martingale, and almost all its trajectories are right-continuous.
The following theorem is the predictable form of the stopping theorem.
It is the basis of defining predictable projections of processes in the next
chapter.
4.41 Theorem. Let (Xt)te-]s be a right-continuous supermartingale
(resp. martingale). Then for every predictable time T and stopping time
U >T we have
E[Xu\Tt-\ < XT- (resp. = XT-) a.s., (41.1)
where Xt- is a.s. well-defined (Theorem 2.43) and integrable.
Proof. Let (Tn) be a sequence of stopping times foretelling T. By
Theorem 3.4.11) we have
n
For brevity we discuss only the supermartingale case. By the stopping
theorem 2.58 we have
E[Xu\fTn} < XTn a.s..
Applying Corollary 2.19, we obtain
E[Xu\?t-] = lim E[Xu\TTn] < lim XTn = XT- a.s..
n—►oo ™J n—►oo
The integrability of Xt- follows from Corollary 2.61. □
4.42 Theorem. Let £ be an integrable r.v., S and T be two predictable
times. Then
E[E[t\Fs-}\FT-} = E[Z\FiSAT)_] a.*.. (42.1)
Proof Let (Xt) be the right-continuous adapted modification of mar-

132 Chapter IV Section Theorems and Their Applications
tingale (E[£\Ft])' Then we have
E[E[Z\rS-]\rT-] = E[XS-\Ft-\ = E[I[s>t\X{Svt)- + I[s<t\Xs-\Ft~]
= I[s>T\E[E[XsvT\F(SvT)-]\fT-] + I[S<T]XS-
= I[s>t]E[Xsvt\Ft-] + I[s<T]Xs-
= I[S>T\Xt- + I[S<T]Xs- = ^(SaT)-
= «KI^(SAT)-] as"
D
Problems and Complements
4.1 Let 5 be a wide-sense stopping time and A c]S, oo[ be an optional
(resp. predictable) set such that for all lj G [S < oo] S(lj) is a limit point
of set A(u) = {t > 0 : (u;, t) G A}. Then there exists a decreasing sequence
(5n) of stopping times (resp. predictable times) such that (Jl^n] C A and
n
lim Sn = S a.s..
n—►oo
4.2 Suppose T = Ti A T2,7i V T2 = +oo, where Tx is an accessible
time and T2 is a totally inaccessible time. Then T\ = Ta, T2 = Tl a.s.
4.3 Assume that (^) is complete. Let H be a predictable set and
Dfj be the debut of H. Then there exists an increasing sequence (Tn)
of stopping times such that lim Tn = Dh and for all n Tn < Dh on
n—►oo
{u : Dh(w) > 0 and (lj,Dh(uj)) G H}.
4.4 Assume that {Ft) is complete. Then every predictable time can be
foretold by a sequence of stopping times taking values in the set of dyadic
numbers.
4.5 Let F° = (Tt) be a right-continuous filtration and F = (Tt) be
its completion. If F° is quasi-left-continuous, so is F.
4.6 Suppose F = (!Ft) is the usual augmentation of a filtration F° =
(ff) and X = (Xt) is a right-continuous i^-supermartingale (resp.
martingale).Then X is also an F-supermartingale (resp. martingale).
4.7 Assume that (Tt) is complete. Let S and T be predictable times,
S < T. Then for any e > 0 there exists a sequence (i?n) of predictable
times such that Rq = 5, 0 < Rn+i — Rn < e> ™ > 0, and limn Rn=T.
4.8 Assume that (^t) is complete. Let T > 0 be a predictable time.
Then there exists an adapted continuous strictly increasing process A such

Problems and Complements
133
that Ao = 0, At = 1.
In the following problems we always assume that the filtration F =
(Tt) satisfies the usual conditions.
4.9 If X = (Xt) is a cadlag supermartingale (resp. martingale) and T
Is a predictable time, then XT~ is a supermartingale (resp. martingale),
too.
4.10 Suppose X = (Xt) is a predictable process, Xqq = lim Xt exists
and is finite, for each predictable time T Xt is integrable and £[Xt] is
independent of T. Let Y be the cadlag adapted modification of martingale
(J5[Xoo|7i]). ThenX = y_.
4.11 Suppose (Xt) is a cadlag supermartingale (resp. martingale). If
S and T are predictable times, and S <T, then
X5__ > E[XT-\Fs-\ > E[XT\Fs-]
(resp. Xs- = £?[Xr-|Js-] = «[*r|Js-])-
4.12 Let 5 C f be a sub-a-field such that Fs- C (7 C Js for a certain
stopping time S. Then for every stopping time T and integrable r.v. £
Moreover, if T is predictable, then £[£|J*t-|0] = Kl^l^r-].
4.13 Let X = (Xt) be an optional (resp. predictable) porcess such
that for every bounded stopping time (resp. predictable time) T^Xt is
integrable.
1) If for every decreasing sequence (Tn) of bounded stopping times
(resp. predictable times) lim £7[Xrn] exists (may be ±00), then almost
n—*oo
all trajectories of X have right limits on R+ (may be ±00).
2) If for every uniformly bounded increasing sequence (Tn) of
stopping times (resp. predictable times) lim i£[Xrn] exists, then almost all
trajectories of X have left limits on ]0,00[.
Moreover, if for every bounded stopping time (resp. predictable time)
T,E\Xt\ < K, where K is a constant, then all right or left Umits above
are finite.
4.14 Let X = (Xt) be a bounded optional process. If for every
increasing sequence (Tn) of finite stopping times tending to +00, lim E[XTn]
exists, then lim Xt a.s. exists.
t—*oo
4.15 Let X = (Xt) be an optional process and
sup{i£|Xr| : T is a bounded stopping time} < 00 .

134
Chapter IV Section Theorems and Their Applications
If for every decreasing sequence (Tn) of bounded stopping times, (Xrn)
is uniformly integrable and
KmE[XTn) = E[XuraTn),
n->oo n_+00
then almost all trajectories of X are right continuous.
4.16 Let X = (Xt) be an optional process. In order that almost all
trajectories of X be right-continuous it is necessary and sufficient that for
every decreasing sequence (Tn) of bounded stopping times we have
Xrn —* X um Tn •
n—»oo
4.17 Let X = (Xt) be a predictable process and
sup{^|Xt| : T is a bounded predictable time} < oo.
If for every uniformly bounded increasing sequence (Tn) of predictable
times, (Xrn) is uniformly integrable and lim £?[Xrn] = E[X um tJ> then
Tl—*OQ n—>oo
almost all trajectories of X are left-continuous.
4.18 Let X = (Xt) be a predictable process. In order that almost all
trajectories of X be left-continuous it is necessary and sufficient that for
every uniformly bounded increasing sequence (Tn) of predictable times we
have
Xrn —* X um Tn-
n—»oo
4.19 Let (XW) n>i be a sequence of right-continuous supermartingales
such that for all n and t € R+, Xt(n) < Xt(n+1) a.s.. Put
Xt(uj) = supXln)(uj), t>0.
n
If Xo is integrable, then (Xt) is a supermartingale, and almost all its
trajectories are right-continuous.
4.20 Let (X(n))n>i be a sequence of right-continuous submartingales
such that for all n and t 6 R+, Xt(n) < Xt(n+1) a.s., and for all n, A"(n+1> -
is a submartingale. Put
Xt(uj) = supXln)(uj), t>0.
n
If for every t G i2+,Aj is integrable, then (Xt) is a submartingale, and
almost all its trajectories are right-continuous.
In other words, (Xt) is bounded in L1.

Chapter V
Projections of Processes
In this chapter we mainly present optional and predictable projections
of measurable processes, dual optional and predictable projection of
processes with finite variation. As an application of projection theory, we will
show the extremely important Doob-Meyer decomposition theorem for su-
permartingales. At last, we will give a detailed discussion about filtrations
of discrete type to conclude the general theory of stochastic processes.
In this chapter we suppose (fi,^*, P) is a complete probability space,
and the filtration F = (Tt) satisfies the usual conditions, unless otherwise
stated. Usually, (fi, T, F, P) is called a filtered probability space.
§1. Projections of Measurable Processes
5.1 Theorem. Let X = (Xt) be a measurable process such that for
every stopping time T XtI[t<oo] i>s 0-integrable w.r.t. Tt1^• Then there
exists a unique optional process, denoted by °X, such that for every
stopping time T we have
E[XtI[t<oo]\Ft]=0XtI[t<oo} as- (1-1)
In this case, we say that the optional projection of X exists2^, or X has
optional projection, and refer to °X as the optional projection of X.
Proof. The uniqueness comes from Corollary 4.11. It suffices to
show the existence. We proceed in four steps.
1J It is easy from Theorem 1.16 to see that this condition is equivalent to the
seemingly weaker one: for every bounded stopping time T, Xt is a-integrable w.r.t. Tt-
2) Rigorously speaking, we should say that the optional projection of X exists and
is finite. Because from the proof of the theorem one can see that if the value +oo is
allowed for °X, then the optional projections of all non-negative measurable processes
exist. But it is stipulated that processes take only finite values. The existence of °X
means °X is finite-valued.

136
Chapter V Projections of Processes
a) Assume X = f/[r,a[, where f is a bounded (or integrable) r.v.,
0 < r < s < +00. Let °X = YI[ts[, where Y = (Yt) is the cadlag
modification of martingale (E[£\Tt]) (see Corollary 2.48). Obviously, °X
is optional. It is easy from Theorem 2.58 to see that °X satisfies (1.1),
i.e., °X is the optional projection of X.
b) Let X and Y be two measurable processes. If X and Y have optional
projections °X and °Y respectively, then for any real A and /?, XX + (3Y
has optional projection X°X + (3°Y. Moreover, if X < Y, then °X <
°Y (Theorem 4.10). If (X^) is an increasing sequence of non-negative
measurable processes, for each n the optional projection of X^ exists, and
the limit process X = lim X^ is bounded, then the optional projection
of X exists, and °X = lim °(X^). Hence by a) and the monotone
n—►oo v
class theorem we conclude that the optional projections of all bounded
measurable processes exist.
c) Suppose X is a non-negative measurable process satisfying the
assumption of the theorem. Let X^ = X/\n. By b) the optional projection
of X^ exists, and °(X^) is increasing (up to an evanescent set). For
every stopping time T, (°X^ I\t<oo]) ls increasing (up to a null set), and
by Theorem 1.19 we have
E[XTI[T<oo]\FT] = ^E[4n)/[r<oo]|JFT] = ^<>4%<oo].
Let Y = limsup0**") and °X = YI[Y<oo]. Then °X is an optional process,
n—►oo
and for every stopping time T
°XtI[t<oo] = YtI[t<oo] = nlimj0x£l /pr«x>] = e[XtI[t<oo]\Pt] a.s..
Thus °X satisfies (1.1), i.e., °X is the optional projection of X.
d) Suppose measurable process X satisfies the assumption of the
theorem. Then sodoX+=XV0 and X~ = -(X A 0). By c) °(X+) and
°(X~) exist. Now it is easy to see that °X = °(X+) — °(X~) is the optional
projection of X. □
Remark. If X is a progressive process, then the optional projection
of X exists, and for every finite stopping time T we have Xt = °Xt a.s..
In particular, °X is an optional modification of X.
5.2 Theorem. Let X = (Xt) be a measurable process such that for
every predictable time T, XtI[t<oo] 25 <r-integrable w.r.t. T?-- Then
there exists a unique predictable process^ denoted by PX, such that for
every predictable time T we have
E[XtI[T<oo]\Ft-] = PXTI[T<oo] <*•*•• (2.1)

§1. Projections of Measurable Processes
137
In this case, we say that the predictable projection of X exists, or X has
predictable projection, and refer to PX as the predictable projection of X.
Proof Assume X = f/[r,a[, where £ is a bounded (or integrable) r.v.,
0 < r < s < +oo. Let Y = (Yt) be the cadlag modification of martingale
(E[Z\Ft]). Put PX = Y-I[rs[. Then PX is predictable, and satisfies (2.1)
(Theorem 4.41), i.e., PX is the predictable projection of X. The other
parts of the proof are completely similar to that of Theorem 5.1. □
5.3 Remarks. 1) If X is a uniformly integrable cadlag martingale,
then X- is the predictable projection of X.
2) Suppose the optional projection of a measurable process X exists.
In order to verify that an optional process Y is the optional projection of
X it suffices by Corollary 4.11 to justify the following equality
E[Xt\Ft] = Yt a.s.
for any bounded stopping time T. Moreover, if for every stopping time T
XtI[t«x>] is integrable, it suffices by Corollary 4.13 to justify the following
equality
E [XTI[T< oo]] = E[YTI[T<oo}]
for any stopping time T. For the predictable case we have similar
assertions.
The following theorems illustrate that projections have the smoothing
property, analogous to conditional expectations.
5.4 Theorem. Let X be a measurable process and Y be an optional
(resp. predictable) process. • // the optional (resp. predictable) projection
of X exists, then the optional (resp. predictable) projection of XY exists,
and °(XY) = (°X)Y (resp. P(XY) = (PX)Y).
Proof. It follows from Theorem 1.21 easily. □
5.5 Theorem. Let X be a measurable process. If the optional and
predictable projections of X exist, then the predictable projection of °X
also exists, and P(°X) = PX. In addition, [°X ^ PX] is indistinguishable
from a thin set.
Proof. The first assertion comes from Theorem 1.22 easily. We are
going to show the second assertion. Let X = €I[ryS[, where £ is a bounded
(or integrable) r.v., 0 < r < s < +oo. Let Y be the cadlag modification
of the martingale (E^Tt]). Then
[°X^pX] = [Y^YJi.
We have already known that [Y ^ YL] is a thin set. Thus, [°X ^ PX] is
indistinguishable from a thin set. The assertion for general measurable

138
Chapter V Projections of Processes
processes follows by the monotone class argument as usual. □
5.6 Theorem. Let T be a stopping time, and £ be a real r.v.. Put
X = ^[T,oo[i y = £J]T,oo[> Z = ^[T]-
1) The optional projection of X exists if and only if £^[t<oo] is a-
integrable w.r.t. Tt>
2) Assume that T is predictable. Then the predictable projection of X
exists if and only if £/[t<oo] is a-integrable w.r.t. Tt->
3) // £/[r<oo] is cr-integrable w.r.t. Ft, then the predictable projection
of Y exists, and Z has optional projection
°Z = E[^I[T<OQ]\JrT]I[T]'
4) IfT is predictable and £/[t<oo] is a-integrable w.r.t. Tt-> then the
predictable projection of Z exists, and
VZ = E[£I[T<00]\FT-]I[T]-
Proof. We only give proof for 1). The others are left to readers. Let
5 be a stopping time. Obviously,
XsI[S«x>] =£J[T<S«x>]-
If the optional projection of X exists, then XtI[t<oo] = £^[T<oo] is a-
integrable w.r.t. Tt- Conversely, assume that £/[t<oo] is ^-integrable
w.r.t. Ft- Let An G Tt such that An | ft and £/[t<oo]^4„ is integrable
for each n. Put
nn = (An[T<S))U[S<T).
Then fin G Fs, ^n T ft, and XsI[S<oo]Inn = £*>!» J[T<S«x>] is integrable.
Hence XsI[s<oo] is cr-integrable w.r.t. Fs-> i-e-> the optional projection of
X exists. □
5.7 Theorem. Let X be a measurable process. If the optional (resp.
predictable) projection of X exists, then for any stopping time {resp.
predictable time) T the optional {resp. predictable) projection of XT exists,
and
(°*)'[o,n = °(xT)il0,n (7
Proof We have
X = ^/[0,T[ + ^T/[T<oo] J[T,oo[-
By Theorems 5.4 and 5.6 we know immediately that the optional (resp.
predictable) projection of XT exists. (7.1) follows from Theorem 5.4 easily
since XT/[0,r] = XI{oTi. □

§1. Projections of Measurable Processes 139
Remark. If the optional and predictable projections of X exist, then
for any stopping time T the predictable projection of XT exists, too. In
fact,
XT = XI^r\ + xtI[t<oo]I]t,oo[,
the existence of the predictable projection of XT follows from Theorems
5.4 and 5.6.
5.8 Theorem. 1) Let X be a measurable process, and (Tn) be a
sequence of stopping times (resp. predictable times) such that supTn =
n
+oo a.s., and for each n the optional (resp. predictable) projection of
XIipTnl exists. Then the optional (resp. predictable) projection of X
exists.
2) Let X be a measurable process, and (Tn) be a sequence of
stopping times such that supTn = +oo a.s., and for each n the predictable
n
projection of -X"/[o,T„] exists. Then the predictable projection of X exists.
Proof. We only show 2). The proof of 1) is left to readers. Let 5 be a
predictable time. Put fin = [5 < Tn] U [5 = oo]. Then ftn G TS-,\J^n =
n
Q a.s., and
xsI[s«x>]Inn = xsI[s<Tn]I[s«x>i
In view of the assumption, XsI[s<Tn]I[s«x>] ls cr-integrable w.r.t. Ts--
By Theorem 1.23 we know that XsI[s<oo] ls cr-integrable w.r.t. Ts-, ie.,
the predictable projection of X exists. □
5.9 Theorem. Let T be a stopping time, and £ be an integrable r.v..
1) X = £/[o,r[ and Y = E[^\TT-]I[oyT[ have the same optional
projections.
2) X = C^[o,T] and y = E[£\Tt-]I[q,t] have the same predictable
projections.
Proof. 1) Let 5 be a stopping time. We have
E[XsIls<oo]} = E[ZI[s<t]} = E[E[t\FT-]Ils<Tl] = E[YsI[s<oo]}.
By Remark 5.3.2) X and Y have the same optional projection.
2) Let 5 be a predictable time. We have
^[*s/[S<oo]] = E[£I[S<T]I[S<oo]]
= E[E[^T-)I[S<T]I[S<oo]} = ^[F5/[5<oc]].
Hence X and Y have the same predictable projection. □

140 Chapter V Projections of Processes
Remark. Obviously, £/[o,r] and E[^\JrT]I[o,T\ have the same optional
projection.
§2. Dual Projections of Increasing Processes
At first, we study the measures on T x B(R+) generated by increasing
processes.
5.10 Definition. Let A be an increasing process. Define a set function
/jla on T x B(.R+) as follows:
fiA(H) = E\f I„(;s)dAs(-)V HeTxB(R+). (10.1)
Then ha is a measure1). Put
T„ = inf {t > 0 : At > n}.
Then Tn is a r.v., [0, Tn[€ T x B(R+), U(0, Tn[= Q x fl+ and /m([0, Tn[)
n
< n. Thus, /jla is a cr-finite measure on T x B(i2+) and is said to be
generated by A.
If A is optional (resp. predictable), then Tn, n > 1, are stopping times
(resp. predictable times). Consequently, [0, Tn[, n > 1, are optional (resp.
predictable) sets. Hence restricted on the optional (resp. predictable) a-
field, fiA is also cr-finite.
It is easy from (10.1) to see that ha does not charge any evanescent
set, i.e., for any evanescent set i/, ha(H) = 0, and for any t > 0, F € T
fjLA(Fx[0,t]) = E[IFAt). (10.2)
5.11 Theorem. A measure /i on T x B(iZ_|_) is generated by an
increasing process if and only if for each t > 0 the set function Qt(F),
defined on (fi, F) as follows:
Qt(F) = »(Fx[0,t}),Fef, (11.1)
is a a-finite measure and absolutely continuous w.r.t. P. In this case, the
increasing process generating fi is uniquely determined.
Proof. The necessity is trivial (see (10.2)). We are going to show the
sufficiency. Let A't be the Radon-Nikodym derivative -—. When s < t,
we have A's < A't a.s.. Assume £* j t. For each n put Fn = [A'tl < n].
Then
Urn E[IFn(A'tk - At)] = lim p(Fnx]t,tk]) = 0.
1 * The term "measure" always means non-negative measure.

§2. Dual Projections of Increasing Processes 141
Since \JFn = fi, we obtain lim A't = At a.s.. Define
n fc—►oo *
At = M{A'r :r>t,r€ Q+}, t > 0.
Then for all t > 0 we have At+ = At and At = A\ a.s.. Modifying the
trajectories of A = (Aj) on a P-null set (if necessary), we may consider A
as an increasing process, and for all F € T we have
»(Fx[0,t}) = Qt(F) = j AtdP
= E[IFAt] = E[f IFx[0tt](;s)dAa(-)}.
This means that /i is generated by A.
If B = (Bt) is another increasing process generating /i, then Bt =
—— a.s.. In consequence, B is a modification of A. Owing to the right-
dP
continuity of A and B, they are indistinguishable. □
5.12 Definition. Let /i be a measure on T x S(i2+) not charging
any evanescent set. /jl is said to be optional (resp. predictable), if for any
non-negative bounded measurable process X1^
ti(X) = »(°X) {resp.»(X) = ix(rX)),
where fi(X) = JXdfi = E^X}.
Remark. Let X be a bounded measurable process. By Theorem 5.4
for any bounded optional (resp. predictable) measure /x we have °X =
E^[X\0) (resp. "X = E^V)).
5.13 Theorem. Let A be an increasing process, and \la be the measure
on T x B(iZ-i-), generated by A. /jla is optional (resp. predictable) if and
only if A is adapted (resp. predictable).
Proof. Sufficiency. Let A be adapted, and C = (Ct) be the change
of time associated with A (see Theorem 3.48). Let X be a non-negative
bounded measurable process. By Lemma 1.38, Theorem 5.1 and Fubini's
theorem we have
E[Jooo XsdA3] = E[J"Xc.Ilc.<oo]d8] = Iq°°E[XcJ[C9<oo}}ds
= f°° E[0XcsI[Cs<oo])ds = E\[ °X3dA3].
JO lJ[0,oo[ J
This is just iia(X) = /ia{°X). Hence ha is optional.
2* In fact, it is enough that the requirement is satisfied for the processes of the form
X = /h, where H € T x B(R+).

142
Chapter V Projections of Processes
Let A be predictable. For every t > 0, Ct- is a predictable time
(Theorem 3.48). By using Lemma 1.38, Theorem 5.2 and Fubini's theorem
in the similar way we have /jla(X) = /xa(J*X), i.e., /jla is predictable.
Necessity. Let /jla be optional. Take X = IFI[o}t], where F G T. X
and E[IF\Tt]I[o,t] have the same optional projection. Hence
E[AtIF) = fjLA(X) = fjLA(°X) = E[AtE[IF\Ft))
= E[E[At\Tt]E[IF\Tt]] = E[E[At\Ft]IF].
Therefore At = E[At\Pt] a.s., i.e., A is adapted.
Let /jla be predictable. For any non-negative bounded measurable
process X, X and °X have the same predictable projection. Hence /jla(X) =
Ha(pX) = /jla(°X), i.e., /jla is optional. In consequence, A is adapted. We
are to show that A satisfies the requirements in Theorem 4.33. Let S be
a totally inaccessible time. Obviously, the predictable projection of J[sj
is 0. By predictability of /ia,
E[AAS] = /ia(I[s]) = /M(0) = 0.
Since AAs > 0, we obtain A As = 0 a.s., i.e., P([As ^ As-, S < oo]) = 0.
Let T be a predictable time. Take X = IfI[o,t\i where F G T. By
Theorem 5.9.2) X and Y = ^[/^l^r-l^o.T] have the same predictable
projection. By predictability of /ia,
E[IFAT] = i*a(X) = /jla(Y) = E[E[If\FT-]At]
= E[E[If\Tt-]E[At\TT-]] = E[IFE[AT\FT-]].
Hence At = E[At\Tt-] a.s., i.e., At is ^--measurable. Since T G Tt—>
we have AtI[t<oo] € Ft-- By Theorem 4.33 A is predictable. D
As a simple application of Theorem 5.13, we obtain Radon-Nikodym
theorem concerning processes with finite variation.
5.14 Theorem. Let A and B be two adapted {resp! predictable)
increasing processes. Then the following statements are equivalent:
1) For almost all u),dB.(u) <C dA.(u),
2) /iB < VA on T x B(jR+ ),
3) /ib < HA on O(resp.V),
4) There exists a non-negative optional (resp. predictable) process H
such that for almost all u we have
Bt(u>) = ( Hs(u)dA3(cj). (14.1)
J[o,t)
Proof. 4) =>• 1) is trivial. 1) => 2) is easy by the definition of absolute
continuity. 2) =*• 3) is trivial. At last, we show 3) =*• 4). Let H be the

§2. Dual Projections of Increasing Processes 143
Radon-Nikodym derivative —— on O (resp. V). Then H is optional
(resp. predictable) and non-negative. Put
Tn = inf{* >0: Bt > n}.
Then
E[ f H3(u)dA3(uj)] =/iB([0,TnD < n.l)
«/[0,Tn[
Hence (/ Hs{w)dAs{w)) is an adapted increasing process, and generates
the same measure as (Bt(uj)). Therefore, it is indistinguishable from (Bt),
i.e., (14.1) holds. D
5.15 Theorem. Let A and B be two adapted (resp. predictable)
increasing processes. Then the following statements are equivalent:
1) For almost all v dA.(u) _L dB.(u),
2) /ia -L Hb on O (resp. V),
3) [lA^-iis on J7 x B(R+),
4) There exists D G O (resp. V) such that for almost all uj
/ Id(u, s)dA3(uj) = 0 and J Idc(u, s)dBs(u) = 0.
J[0,oo[ -M0,oo[
Proof. 2) <£=> 4) => 3) is obvious.
3) => 1). There exists J G T x B(R+) such that
fjLA(J) = E\ f Ij(uj,s)dA3(uj)\ = 0,
tiB(Jc) = E\ f IJC(u;,s)dB3(uj)\ = 0.
Hence for almost all uj
f Ij(u, s)dAs(u) = 0, / IJC(u, s)dB3(uj) = 0,
J[0yoo[ «/[0,oo[
i.e., dA.(u) _L dB.(v).
1) => 2). Put C = A + B. By Theorem 5.14 there exist non-negative
optional (resp. predictable) processes H and K such that for almost all u
At(uj) = f Hs(u)dC3(u), Bt(uj) = f Ks(uj)dCs(uj)
J[o,t] J[oyt]
and H + K = 1, HK = 0. Take J = [H = 0], then J G O (resp. V),
Jc = [K = 0],
/iA^)=0, /iB(JC) = 0,
1 * Henceforth, under the symbol of integral we use, for instance, [5, T[ to denote the
stochastic interval [5, T[.

144 Chapter V Projections of Processes
i.e., [Xa 1 /ifl on O (resp. V). D
Remark. Let A and B be two adapted (resp. predictable) processes
with finite variation. If for almost all cj, \dB.(u)\ «C |dA(u;)|, then there
exists an optional (resp. predictable) process H such that for almost all
Bt{u)) = I Hs{u))dAs{uj), t > 0.
J[o,t)
In fact, applying Theorems 5.14 and 5.15 to A"1", A~ and C = ( / |cL43|),
■/[o,t]
we know that there exists an optional (resp. predictable) process L such
that A = L.C and \L\ = 1. Then L.A = L2.C = C. Applying Theorem
5.14 to B+, B~ and C, there exists an optional (resp. predictable) process
K such that B = K.C. Finally, B = H.A, where H = KL.
5.16 Theorem. 1) Let A be an adapted increasing process and S <
T be two stopping times. For every non-negative measurable process X
having optional projection
E\ J XsdAs\Fs] =E\ f °XadAB\Ts}. (16.1)
lJ[S,T[ ' J lJ[S,T[ ' J
2) Let A be a predictable increasing process, and S <T be two
predictable times. For every non-negative measurable process X having
predictable projection
E\ j XsdAs\FS-\ =E\ f *XadAa\TsX (16.2)
lJ[S,T[ ' J lJ[S,T[ ' J
Proof. We only show 2). The proof of 1) is analogous. Let F £
Fs-' Then F G Tt~, Sf and 7> are predictable times, and [Sf,Tf[ is
predictable. We have (by Theorems 5.4 and 5.13)
EHs,n ^dA3)IF]=E[j[0oo[IlSFm(.,s)X3dA3}
= EU[0MIlSF'TFli',S)PXsdAs\ = EHs,T[PX*dA3)lF}-
Hence (16.2) follows. D
Remarks. 1) If in (16.1) and (16.2) [S,T[ is replaced by ]S,T[, or
]5,T], or [5, T], the equalities remain true.
2) If in (16.2) S and T are two stopping times (resp. S is predictable,
resp. T is predictable), then, replacing [S,T[ by ]S,T] (resp. [S,T], resp.
]S,T[), (16.2) still holds.

§2. Dual Projections of Increasing Processes
145
Below we define the projections of measures. They are the basis of
studying dual projections of increasing processes.
5.17 Definition. Let /i be a a-finite measure on T x B(R+) not
charging any evanescent set. For every non-negative bounded measurable
process X put
lx°{X) = tx{°X), ^(X) = ^X).
Then /jl° and [jiP are optional and predictable measures on T x B(i2+)
respectively, not charging any evanescent set (but need not to be cr-finite).
We call /i° and /jlp the optional and predictable projection of /jl respectively.
Obviously, /jl is identified with /x° on optional a-field O, and with /jlp
on predictable cr-field V. Besides, in order for /jl to be an optional (resp.
predictable) measure on T x B(R+) it is necessary and sufficient that
/i = fi° (resp. /i = /jlp).
5.18 Definition. Let A be an increasing process. A is said to be
integrable, if A^ = Mm At is an integrable r.v.. A is said to be locally
*Too
integrable, if Aq is a-integrable w.r.t. To, and there exist stopping times
Tn | +oo a.s. such that for each n Arn — Ao is integrable. A is said to be
prelocally integrable, if there exist stopping times Tn | +oo a.s. such that
for each n ATn-I[rn>o] ls integrable.
Obviously, locally integrable increasing processes are prelocally
integrable. In fact, it needs only to deal with the case of At = Ao, t > 0,
where Ao is a-integrable w.r.t. ^b- Let En G To such that En | ft and
each AolEn is integrable. Put Tn = 0 • /^c + (+oo)/£n. Then Tn | oo and
i4Tn-/[Tn>o] = AolEn' Hence A is prelocally integrable.
Let A be a process with finite variation. Put Vt = I \dAs\. If
V = (Vt) is an integrable (resp. locally integrable, resp. prelocally
integrable) increasing process, we call A a process with integrable (resp. locally
integrable, resp. prelocally integrable) variation.
Obviously, in order that A be a process with integrable (resp. locally
integrable, resp. prelocally integrable) variation it is necessary and
sufficient that A be the difference of two integrable (resp. locally integrable,
resp. prelocally integrable) increasing processes.
5.19 Theorem. Adapted processes with finite variation are processes
with prelocally integrable variation. Predictable processes with finite
variation are processes with locally integrable variation.
Proof. We only deal with increasing processes. Let A be an adapted

146
Chapter V Projections of Processes
increasing process. Put
Tn = inf{t > 0 : At > n}.
Then Tn,n > 1, are stopping times, Tn | +oo, and ATn-I[rn>o] < n.
Hence A is prelocally integrable. If A is predictable, so are Tn,n > 1.
Without loss of generality, we may assume Aq = 0. In this case, Tn > 0.
For each n let (SU}k)k>i be the sequence of stopping times foretelling Tn.
n
Put Sn = V Siyn- Then Sn < Tn and Sn | +oo, ASn < n. Hence A is a
locally integrable increasing process. D
5.20 Theorem. Let /jl be the measure on T x B(R+) generated by an
increasing process A, /jl° and /jlp be its optional and predictable projection
respectively.
1) In order that /jl° be generated by an (adapted ) increasing process it
is necessary and sufficient that A be prelocally integrable.
2) In order that jjP be generated by an (predictable ) increasing process
it is necessary and sufficient that A be locally integrable.
Proof. 1) Necessity. Assume that /jl° is generated by an increasing
process A°. By Theorem 5.13 A° is adapted. Put
Tn = ini{t > 0 : A°t > n}.
Then Tn, n > 1, are stopping times, Tn | +oo, and A^n_I^Tn>0^ < n.
Thus
E[ATn.I[Tn>0]] = /i([0,Tn[) = M°([0,Tn[) = E[A°Tn_I[Tn>0]] < n.
This means A is prelocally integrable.
Sufficiency. Assume that A is prelocally integrable, i.e., there exist
stopping times Tn | +oo and for each n E[ATn-I[Tn>o]] < °°- Denote
Qt(F) = fi°(F x [0,*]), F e T. Let Fn = [Tn > t]. Then \JFn = fi,
Fn x [0,^] = [0,Tn[, Qt(Fn) < /i°([0,Tn[) = f7[Arn_/[Tn>0^] < +oo.
Therefore, Qt is a a-finite measure on (Q,T). Since for any P-null set
F, F x [0,t] is a predictable set, Qt(F) = fi°(F x [0,t]) = /i(F x [(),£]) =
jB^^] = 0. This means Qt is absolutely continuous w.r.t. P. By
Theorem 5.11 we know that fi° is generated by an increasing process. By
Theorem 5.13 this increasing process is adapted.
2) Necessity. Assume that /ip is generated by an increasing process Ap.
By Theorem 5.13 Ap is predictable. Put Fn = [A% < n]. Then Fn G T0,
Fn T fy and
E[AoIFn] = »(Fn x {0}) = fip(Fn x {0}) = E[Ap0IFn) < n.
Hence Aq is a-integrable w.r.t. Jb- Since AP is locally integrable (Theorem
5.19), take stopping times Tn | +oo such that for each n A1^ - A^ is

§2. Dual Projections of Increasing Processes
147
integrable. Then
E[ATn - Ao] = Ai(10,rn]) = /ip(]0,Tn]) = E[ApTn - AP0] < +00.
This means A is locally integrable.
Sufficiency. Assume that A is locally integrable. Put
Bt = Ao, Bf = E[Ao\T0), *>0.
Obviously, [i?B is generated by predictable increasing process Bp. So we
may assume Ao = 0 (otherwise we deal with A — Ao). Similar to the
proof of the sufficiency in 1), one can show that Qt(F) = /jP(F x [0, t]) is
a a-finite measure on (fi,^*), and absolutely continuous w.r.t. P. Then
by Theorems 5.11 and 5.13 jjP is generated by a predictable increasing
process. □
Theorem 5.20 hints us to define dual projections of increasing processes
as follows.
5.21 Definition. Let A be a prelocally integrable increasing process,
fi be the measure on T x B(i2+) generated by A, and /i°be the optional
projection of/i. By Theorem 5.20 there exists a unique adapted increasing
process A° such that /i° is generated by A°. A° is called the dual optional
projection of A (note that by Theorem 5.9 the optional projection °A of A
also exists, but °A is no longer an increasing process in general). Let A be
a locally integrable increasing process, /jl be the measure on T x B(R+)
generated by A, and /ip be the predictable projection of /i. By Theorem
5.20 there exists a unique predictable increasing process Ap such that /jP is
generated by AP-AP is called the dual predictable projection of A (similarly,
by Theorem 5.9 the predictable projection PA of A also exists, but PA is
no longer an increasing process in general).
Let A be a process with prelocally (resp. locally) integrable variation.
A can be decomposed as the difference of two prelocally (resp. locally)
integrable increasing processes A\ and A2. Define A° = A\ — A^ (resp.
AP = APX — A%). It is easy to see that A° (resp. Ap) does not depend
on the concrete decomposition. A° (resp. Ap) is called the dual optional
(resp. predictable) projection of A. Dual predictable projections are also
called compensators.
5.22 Theorem. 1) Let A be a process with prelocally integrable
variation. For every optional process H we have
Elf \Hs\\dA°s\] < E[ f \Hs\\dA3\]. (22.1)
L^[0,oo[ J LV[0,oo[ J

148 Chapter V Projections of Processes
2) Let A be a process with locally integrable variation. For every
predictable process H we have
E\ I \Hs\\dA*\] <e\( \Hs\\dAs\\. (22.2)
L./[0,oo[ J lJ[0,oo[ J
Proof. We only show 1), the proof of 2) is similar. Put
Then A = A+ — A~, A+ and A~ are prelocally integrable increasing
processes. Since A° = (A+)° — (A~)°, we have
Elf \Hs\\dA°\} < e[ [ \Hs\\d(A+)i\ + E[ f \Hs\d(A-)°]
= e[[ \H.\dAt]+E[[ \H.\dAj]
= e[[ \Hs\\dAs\}. D
5.23 Theorem. 1) Let A be a process with prelocally integrable
variation, and H be an optional process such that H.A is a process with
prelocally integrable variation. Then H.A° is an adapted process with finite
variation, and (H.A)° = H.A°.
2) Let A be a process with locally integrable variation, and H be a
predictable process such that H.A is a process with locally integrable variation.
Then H.AP is a predictable process with finite variation, and (H.A)P =
H.A*>.
Proof. We only show 1). There exists a sequence (Tn) of stopping
times with Tn | +oo such that
Elf \H3\Il0,Tnl(.,s)\dAs\} =E[ f \H3\\dAs\]
lJ[0yoo[ J lJ[0,Tn[ J
< +00.
By (22.1) we know that H is integrable w.r.t. A°. Then H.A° is adapted
by Theorem 3.46.1). Without loss of generality, we may assume that A is
an increasing process and H is non-negative. Then for every non-negative
bounded measurable process X we have
HhMx) = »°h.a(x) = mA°x) = ha(h°x)
= fiA(°(HX)) = n°A{HX) = tiA°(HX) = n„MX).
This implies (H.A)° = H.A°. □
5.24 Corollary. 1) Let A be a process with prelocally (resp. locally)
integrable variation. Then for every stopping time T we have (AT)° =
(A°)T (resp. {ATY = {AP)T).

§2. Dual Projections of Increasing Processes 149
2) Let A be a process with locally integrable variation. Then for every
predictable time T we have (AT~)P = (AV)T~, where AT~ = AZ[0)t[ +
at-I[t,ooI
Proof. Putting H = /[o,rj (resp. H = Z[o,t[) m Theorem 5.23 gives 1)
(resp. 2)). D
5.25 Theorem. 1) Let A be an adapted process with finite variation,
H be a measurable process having optional projection such that H.A is a
process with prelocally integrable variation. Then °H is integrable w.r.t.
A, and (H.A)° = (°H).A.
2) Let A be a predictable process with finite variation, H be a
measurable process having predictable projection such that H.A is a process
with locally integrable variation. Then PH is integrable w.r.t. A, and
{H.A)* = (W).A.
Proof. We only show 1). There exists a sequence (Tn) of stopping times
with Tn T +oo such that E[ f \Hs\\dAs\] < +oo. Since \°H\ < °{\H|),
J[o,Tn[
we have
Consequently, °H is integrable w.r.t. A. We may assume that A is an
increasing process and H is non-negative. Then for every non-negative
bounded measurable process X we have (noting that fiA = /*% °(°HX) =
°{H°X) = °H°X)
V>(H.A)°{X) = iih.a(°X) = fiA(H°X) = fiA(°HX) = fioHA(X).
This implies (H.A)° = °H.A. □
Remark. In fact, Theorems 5.23 and 5.25 can be unified in the
following more general assertion. Let A be a process with prelocally (resp.
locally) integrable variation and H be a measurable process such that H.A
is a process with prelocally (resp. locally) integrable variation. Then there
exists an optional (resp. predictable) process K such that (H.A)° = K.A°
(resp. {H.Af = K.A?). In addition, we have K = E^A[H\0] (resp.
K = E»A[H\V\).
We deal with the prelocally integrable case only. Observe that fiH.A is
absolutely continuous w.r.t. /jla on J*xB(iZ+) and = H. Restricted
dfiA
on the optional a-field 0, [lu.A is a-finite. Hence H is a-integrable w.r.t.

150 Chapter V Projections of Processes
O and \pa\- The Radon-Nikodym derivative of /jlh.a w.r.t. /jla on O is
dfjLH.A
Denote K = E^A[H\0). Then (H.A)° = K.A°. Moreover, if His optional
or A is adapted and H has optional projection, it is easy to see K = °Hn
l/i^l-a.e.. Theorems 5.23 and 5.25 follow immediately.
5.26 Theorem. 1) Let A be a prelocally integrable increasing process,
S < T be two stopping times. Then for every non-negative measurable
process X having optional projection we have
E\ I °XsdAs\Fs\=E\l XadA°\fs\=E\ f °XadA°^s].
lJ[S,T[ ' J lJ[S,T[ ' J lJ[S,T[ ' J
(26.1)
2) Let A be a locally integrable increasing process, S <T be two
predictable times. Then for every non-negative measurable X having predictable
projection we have
E\ f pXsdAs\FS-] =E\ I XadA*\TS-\ = E\ I *XBdA*\F8X
VJ[S,T[ ' J lJ[S,T[ ' J lJ[S,T[ I J
(26.2)
Proof. It is completely similar to that of Theorem 5.16. We also have
the remarks, similar to that after Theorem 5.16. □
The following theorem provides the computing method for jumps of
dual projections.
5.27 Theorem 1) Let A be a process with prelocally integrable
variation. Then AA has optional projection: °(AA) = AA°, i.e., for every
stopping time T
AA°TI[T<oo] = E[AATI[T<oo]\TT} a.s.. (27.1)
2) Let A be a process with locally integrable variation. Then AA has
predictable projection: P(AA) = AAP, i.e., for every predictable time T
A^/[T<oo] = E[AATI[T<oo]\TT-] a.s.. (27.2)
Proof. We only show 1) and assume that A is an increasing process. By
Theorem 5.8 we know that A has optional projection. Since A- < A, A-
has optional projection, too. Hence so does AA. Then for every stopping
time T, AAtI[t<oo] *s tf-integrable w.r.t. T?, and for every F G Tt we
have
E[AA$I[T<oo]IF] = E[[ IlTF]dA°] =E[[ IlTF]dAs]
L./[0,oo[ J L./[0,oo[ J
= E[AATI[T<oo]IF],

§2. Dual Projections of Increasing Processes
151
i.e., (27.1) holds. □
5.28 Corollary. 1) Let A be a process with prelocally integrable
variation. If A is continuous, so is A°. If AA is bounded, so is AA°.
2) Let A be a process with locally integrable variation. If A is
continuous, so is A?. If AA is bounded, so is AAP.
3) Let A be an adapted locally integrable increasing process. Ap is
continuous if and only if A is quasi-left-continuous (see Definition 4.22).
We give two simple examples of dual projections in the next theorem.
5.29 Theorem. 1) Let T be a stopping time, and £ be a real r.v..
A = £Jpr,oo[ *5 a process with prelocally integrable variation if and only if
£I[T<oo] *5 (T-integrable w.r.t. Tt- In this case, the dual optional projection
of A is
^ = ^[^[T<oc]|^r]/[T,oc[.
2) Let T be a predictable time, and £ be a real r.v.. A = f/[r,oo[
is a process with locally integrable variation if and only if £I[t<oo] *5 °-
integrable w.r.t. Tt-- In this case, the dual predictable projection of A
is
A*> = E[tI[T<oo)\rT-]IiTM-
Proof. We only show 1). Necessity. There is a sequence (Tn) of
stopping times with Tn | +oo such that for each n, E[ATn-.I[rn>o]] < +°°.
Note that ATn_/[Tn>0] = £/[Tn>r]- Take Fn = [T = oo] U [Tn> T] e TT.
Then Fn | fi and for each n, £/[r<oo]^F„ = ^Tn-I[Tn>o] ls integrable.
Hence £I[t<oo] is a-integrable w.r.t. Tt-
Sufficiency. We may assume that £ is non-negative. Let /jla be the
measure generated by A. For every non-negative bounded measurable
process X we have
HA(°X) = E[°Xt^[t<oo)} = EI°XtI[t<oo]E[SIt<oo)\Ft}}
= E[XTI[T<oo]E[£I[T<oo]\fT}} = Mflp0,
where B = E[^I[T<oo]\^T]I[Tyoc[' Since B is adapted, by Theorem 5.23
and Definition 5.21 we know that A is prelocally integrable and B — A°.
D
Finally, we prove a very useful result.
5.30 Theorem. Let A and B be two processes with integrable
variation.

152
Chapter V Projections of Processes
1) A and B have the same dual optional projection if and only if for
every stopping time T
^[Aoc - AT-I[T>o]} = £[#oo - BT-I[T>0]] (30.1)
(i.e., (Aoo — j4t_J[t>0]) and (B^ — Bt-I[t>0]) have the same optional
projection). In particular, if A and B are adapted, then A and B are
indistinguishable if and only if for every stopping time T (30.1) holds.
2) A and B have the same dual predictable projection if and only if
E[A0\T0] = E[B0\T0] a.s., (30.2)
and for every stopping time T
ElAn - AT] = £[Boo - Br] (30.3)
(i.e., (Aqq — At) and (Boo — Bt) have the same optional projection), or
equivalently
E[A0\T0] = E[B0\T0] <*•«•>
(30.4)
ElAco - At\Ft] = Efta - Bt\rt] a.s., t>0.
In particular, if A and B are predictable, then A and B are
indistinguishable if and only if Aq = Bo and for every stopping time T (30.3) holds, or
equivalently, (30.4) holds.
Proof. 1) Let /jla and /jlb be the measures generated by A and B
respectively. Then (30.1) reads: ^aUT, oo[) = /i#([T, oo[) for every stopping
time T. Thus the necessity is trivial. We are going to show the sufficiency.
Put C = {[T, oo[: T is a stopping time}. Then C is a 7r-class and generates
the optional a-field (Theorem 3.17). In addition, fi x R+ = [0, oo[e C.
By (30.1) /ia and /i# are identical on C. Then /jla and /jlb are identical on
O by the monotone class argument. Therefore, /jla and /jlb have the same
optional projection, i.e., A and B have the same dual optional projection.
2) (30.3) reads: /jla(]T, oo[) = /jlb(]T, oo[) for every stopping time T,
and (30.2) is equivalent to that /i^dO^]) = /^([Of]) for every F G To.
Put C = {{0a} : A e To) U {]T, oo[: T is a stopping time }. Then C is a
7r-class and generates the predictable a-field (Theorem 3.21). In the same
way, if /jla and /i# are identical on C, so are on V. Then A and B have
the same dual predictable projection if and only if (30.2) and (30.3) hold.
If we consider
C = {[0A] :AeT0}U {Bx]t, oo[: B G Tut > 0}
instead of C, it is easy to see that (30.4) is also a necessary and sufficient
condition. □
5.31 Corollary. 1) Let A be a (resp. adapted) process with integrable
variation, and B be a predictable process with integrable variation. In

§3. Applications to Stopping Times and Processes
153
order that B be the dual predictable projection of A it is necessary and
sufficient that Bo = jE[j4o|.Fo] wn>d °A — B (resp. A — B) be a uniformly
integrable martingale, where °A is the optional projection of A.
2) Let A be a process with integrable variation. Then °A — A° is a
uniformly integrable martingale.
Proof. 1) For all t > 0, °At = E[At\Tt] a.s.. Thus, (30.4) holds if and
only if Bo = E[A0\To] and for all t > 0
°At -Bt = E[Aoo - Bool^i] a.s..
Hence 1) is established.
2) By 1) °A — Ap is a uniformly integrable martingale. On the other
hand, A° — AP = A° — (A°)p is also a uniformly integrable martingale.
Hence so is °A - A°. □
§3. Applications to Stopping Times and Processes
5.32 Theorem. Let A be an adapted increasing process^ and M be
a non-negative uniformly integrable cadlag martingale. Then for every
stopping time T we have
E\ f MtdAt\ = E[MTAT). (32.1)
L J[0,T] J
Proof. Put X = MT/[0)Tj. Since E[MT\Tt) = MtAT, t > 0, MT is the
optional projection of Mr- Hence °X = MtI\o,t\ = ^7[o,t]- Thus
E[MTAT] = E\ f XtdAt\ =E\ j °XtdAt\ = e\ f MtdAt].U
5.33 Theorem. 1) Let A be an adapted increasing process. Suppose
for allt > 0 At is integrable. Then A is predictable if and only if for every
non-negative bounded cadlag martingale M and t > 0
E\ f MsdAs] =E\ f Ms-dAs]. (33.1)
lJ[oyt] J lJ[o,t] J
2) Let A be an adapted integrable increasing process. Then A is
predictable if and only if for every non-negative bounded cadlag martingale M
E\ [ M3dAs] =E\ f Ms-dAs]. (33.2)
L./[0,oo[ J lJ[0,oo[ J
Proof. 1) The necessity comes from Theorem 5.13, because the
predictable projection of MI[0}t] is M_/[0,t]- We are going to show the
sufficiency. At first, suppose A is integrable. Put
C = {Fx [0,t]: <>0,F6f}.

154
Chapter V Projections of Processes
Then C is a 7r-class and a(C) = T x B(-R+). Let C = F x [0,t] G C,
and M be the cadlag modification of (E[IF\Tt)). Then °/c = ^f J[o,*J>
p/c = -W-^[o,t]- By adaptedness of A and (33.1) we obtain
»a{Ic) = »a(°Ic) = /m(p/c),
where /jla is the measure generated by A. Put
G = {CeTxB(E+): fiA(Ic) = HA(pIc)}.
Then Q is a A-class and C dQ. Hence (? = <r(C) = T x B(i*+), i.e., /i^ is
predictable. Consequently, by Theorem 5.13 A is predictable. For general
cases we consider An = (j4tAn). Since An is integrable, it is shown above
that An is predictable. At last, A is predictable.
2) If M is a non-negative bounded cadlag martingale, so is M' =
MI[o9t] + MtI¥M. By (33.2) we have
E\ f MsdAs]+E[Mt(A00-At)}
= E\ I M'sdAs] =E\ f M's_dAs] (33.3)
= E\[ Ms-dA3} + ElMtiAoo - At)\.
Substracting ElM^A^ - At)] from the two sides of (33.3), we obtain
(33.1). By 1) A is predictable. □
As an application of Theorem 5.33, we obtain a characterization for
predictable times.
5.34 Theorem. Let T be a stopping time. Then T is predictable if
and only if for every non-negative bounded cadlag martingale M
E[MT-] = E[MT].
Proof. The necessity follows from Theorem 4.41. We are to show the
sufficiency. Put A = /[t,oo[- Then A is an adapted integrable increasing
process, and for every non-negative bounded cadlag martingale M
E\ I MsdAs] = E[MTI[T<oo}} = E[MT-IlT<oo]] = e\ f M3-dA3]
L./[0,oo[ J L^[0,oo[ J
(note that M^ = Moo_). By Theorem 5.33 A is predictable, and so is T.
D
The next theorem provides a useful characterization for totally
inaccessible times.
5.35 Theorem. Let T > 0 be a stopping time. Then T is totally
inaccessible if and only if there exists a uniformly integrable martingale

§3. Applications to Stopping Times and Processes
155
M with Mo = 0 such that M is continuous outside [T] and AMt = I on
[T<oo\.
Proof. Necessity. Let T be a totally inaccessible time, and A = J[t,oo[-
Then A is quasi-left-continuous, and the dual predictable projection Ap
of A is continuous (Corollary 5.28.3)). Put M = A - Ap. Then M is
a uniformly integrable martingale and Mo = 0 (Corollary 5.31.1)), and
satisfies the requirements.
Sufficiency. Suppose there exists a uniformly integrable martingale M
satisfying the requirements. Then for any predictable time S we have
AMS = AMsI[S<oo] = AMt/[t=s<oo] = /[t=s«x>]-
By Theorem 4.41 P[T = S < oo] = E[AMS] = 0. Hence T is totally
inaccessible. □
The following theorem provides a characterization for
quasi-left-continuous filtrations (see Definition 3.39).
5.36 Theorem. A filtration F = (Tt) is quasi-left-continuous if
and only if all uniformly integrable cadlag F-martingales are quasi-left-
continuous.
Proof. Necessity. Assume (Tt) is quasi-left-continuous. Let M be a
uniformly integrable cadlag martingale. Then for any predictable time
T>0
E[MTI[T<oo] \Tt-\ = MT-I[r<ao] as~
But Tt = Ft- and MtI[t<oo] € ^r, so AMtI[t<oo] = 0 as> le•> M is
quasi-left-continuous.
Sufficiency. Assume that (Tt) is not quasi-left-cotinuous. Then there
exists a predictable time T such that P(T < oo) > 0 and Tt ^ Ft—
Take a set H G TT \ Tt— Put M = (IH - jE?[/h|Jt-])/[t,oo[- M is a
uniformly integrable cadlag martingale (see Problem 5.3), but not quasi-
left-continuous. □
5.37 Definition. A filtration F = (Tt) is said to be completely
continuous if for any stopping time T Tt = Tt-- Obviously, the complete
continuity of a filtration implies the quasi-left-continuity.
5.38 Theorem. The following two statements are equivalent:
1) All stopping times are predictable.
2) All uniformly integrable cadlag martingales are continuous.
And in this case, (Tt) is completely continuous.
Proof. 1) => 2). At first, we know by Theorem 3.40 that (Tt) is
quasi-left-continuous. For any stopping time T and uniformly integrable

156
Chapter V Projections of Processes
cadlag martingale M, we have by Theorem 5.36 Mt = Mt- a.s. since T is
predictable. Hence M is indistinguishable from M_, i.e., M is continuous.
The complete continuity of {Ft) is trivial in this case.
2) => 1) comes from Theorem 5.34 immediately. □
5.39 Theorem. Let A G T x B(i*+). Then A' = [P(IA) > 0)] is the
only predictable set (up to evanescence) such that for any predictable time
T A[T] is evanescent if and only if A'lT} is evanescent. A' is called the
predictable support of A.
Proof. Let T be a predictable time. Since
p(Ia)tI[t<oo) = E[(Ia)tI[t<oo]\Ft-) a.s.,
it is not hard to see
p(Ia)tI[t«x>) = 0 a.s. <=> (Ia)tI[t<c6\ = ° as-
Now,
A{T] is evanescent <£=>► (Ia)tI[t<oo] = 0 a.s.
A'[T\ is evanescent <=> (Ia')tI[t<oo] = 0 as-
<=> v{Ia)tI\t<oo] = 0 a.s..
Hence A[T] is evanescent -<=>• i4'[T] is evanescent. The uniqueness comes
from the predictable section theorem as usual. □
5.40 Lemma. Let An E T x B(i2+) and A!n be the predictable support
of An,n > 1. Then the predictable support of\JAn is \JA'n.
n n
Proof Immediately follows from the definition of predictable support.
□
5.41 Lemma. Let A be a locally integrable increasing process. Then
the predictable support of [AA ^ 0] is [AAP ^ 0], where Ap is the dual
predictable projection of A.
Proof. For any predictable time T AA^I]iT<OQ\ = E[AAtI[t<oo]\Ft-]i
[AA ^ 0][T] is evanescent <*=> AAtI[t<oo] = ° a.s. <*=> AA^I[T<0o] = 0
a.s. <=> [AAP ^ 0][T] is evanescent. Hence predictable set [AAP ^ 0] is
the predictable support of [AA ^0]. □
5.42 Theorem. The predictable support of any thin set is the union
of graphs of a sequence of predictable times.
Proof. On account of Lemma 5.40, we need only to show the assertion
for set [T], where T > 0 is a stopping time. Put A = /pr,oo[- Then A is
an integrable increasing process. By Lemma 5.41 the predictable support

§4. Doob- Meyer Decomposition Theorem
157
of [71 = [AA ^ 0] is [AA? ^ 0]. Obviously, [AA? ^ 0] is the union of
graphs of a sequence of predictable times. □
5.43 Corollary. Let A be a thin set and A' be its predictable support.
1) A is totally inaccessible «£=>• A' is evanescent.
2) A is predictable «<=> A = A!. In this case, A can be represented as
the union of graphs of a sequence of predictable times.
3) A is accessible «<=> A C A!. In this case, A can be represented as
the union of graphs of a sequence of accessible times.
§4. Doob-Meyer Decomposition Theorem
5.44 Definition. Let T be the collection of all stopping times. A
measurable process X is said to be of class (D), if {XtI[t<oo] : T € T} is
a uniformly integrable family of r.v..
By Doob's stopping theorem it is not hard to see that all uniformly
integrable cadlag martingales or non-negative right-closed cadlag submartin-
gales are of class (D).
5.45 Theorem. Let A = (At) be a predictable integrable increasing
process with Aq = 0, and Z = (Zt) be the optional projection of(A00 — At).
Then Z is a potential of class (D), and A is uniquely determined by Z. Z
is called the potential generated by A.
Proof. We have already known that the optional projection of A^ is
the cadlag modification of the martingale (jE[j4oo|Tt])- Hence Z is cadlag
and Zt = ElAoolFt] - At a.s.. For 5 < t
E[Zt\Ts] = EiA^Ts) - E[At\Fs] < ElA^Fs] -AS = ZS a.s.,
i.e., Z is a non-negative supermartingale. On the other hand,
lim E[Zt] = lim ElA^ - At] = 0.
t—>oo t—►oo
Therefore, Z is a potential. Finally, Zt < ElA^^t] a.s., then Z is of class
(D). Let /ia be the measure generated by A. fiA is finite, /mIM) = 0
and for every stopping time S
I*a(]S, oo[) = £7[^oo - As] = E[ZS). (45.1)
It is clear that, being restricted on P, ha is uniquely determined by Z.
Since A is predictable, so A is uniquely determined by Z. □
From Theorem 5.45 it is natural to ask if any potential of class (D)
is generated by a predictable integrable increasing process. The answer is
affirmative. In fact, (45.1) is the key to solving this problem.

158
Chapter V Projections of Processes
Let C be the field generated by
{[0F] :F6f0} U{]S, T]:S <T are stopping times}.
Then C generates V and each element H of C has the form [0^]u
771
( U ]Ui, Vi\), where F e To and [/*, V{,i = 1, • • • ,ra, are stopping times
Let Si be the debut of i/n]0, oo[, TX be the debut of Hcn]Si, oo[, S2 be
the debut of i/nJTi, oo[, T2 be the debut of Hcn}S2, oo[, and so on. Then
i/ can be uniquely represented as
ff=[OFlUlSi,T1]U"-UlS„,T„],
where F G To, Si,Ti,i = l,-,n, are stopping times, and Si < T{ on
[5i < oo], i = l, — ,n; Ti < Si+i on [Ti < oo], i = l,---,n - 1. This
representation of H is said to be canonical. Denote
H=lOF]\JlSuTi]{J.--{JlSn,Tnl
5.46 Lemma. Let Z = (Zt) be a potential of class (D), and Z^ = 0.
Let H G C with canonical representation
H = [0F] \J\SuTi] U • • • \J]Sn, Tnj. (46.1)
Define
l*(H) = E[ZSl - ZTl] + • • • + E[ZSn - ZTn\. (46.2)
Then /i is a finite measure on C.
Proof. First of all, we show the following fact: for any given e > 0
and H G C there exists K e C such that JC C #, iiTfl [0] = 0 and
/i(Zf) 5* M^O + £• To this end, we may suppose H has the form ]S,T],
5 < T and 5 < T on [5 < oo]. Put
5n=(5+n)[S+i<rf Tn = TP+i<T]-
We have Sn > S, S = limSn, and Sn > S on [5 < oo]. At the same
time, Tn > T, limTn = T, and T = Tn on [5n < oo]. Thus for each n,
n
[5n,Tn] C]5,T]. Since Z is right-continuous and of class (D),
ZSn-^Zs, ZTn^ZT and Umf;[Z5n - ZTn] = E[ZS - ZT\
Take n large enough such that E[Zsn — Zrn] > E[Zs — Zt] — e, and
K =]5n,Tn]. Then If satisfies the requirements.
The finiteness and finite additivity of /i are evident. What remains is to
show the a-additivity of /i, or equivalently, Hn G C, Hn | 0 => /i(^n) j 0.
For any given e > 0, take ifn^ such that /T n [0] = 0, ~Kn c tfn and

§4. Doob-Meyer Decomposition Theorem
159
tAHn) < V>{Kn) + e2~n. Put Ln = Kx n • • • n Kn. Then for all n, Ln G C,
Ln C H and
/i(/fn) < /i(Ln) + e. (46.3)
On the other hand, Ln j 0. Denote by Dn the debut of Ln. Then [Dn] C
Ln and Z?n | +oo. Because Ln c]Dn,oo[,
fi(Ln) < ii(]Dn, oo[) = E[ZDn - Zoo] = £[ZdJ.
Noting that Zr>n —^ 0 (Z is a potential of class (£>)), we have lim/i(Ln) =
0. In view of (46.3), lim fi(Hn) < e. Letting e | 0 gives limfi(Hn) = 0.
n n
D
5.47. Theorem. Let Z be a potential of class (D). Then there
exists a unique predictable integrable increasing process A such that Z is
generated by A.
Proof. The uniqueness has been implied in Theorem 5.45. Only the
existence needs to be proved. The finite measure /i on C defined by (46.2)
can be uniquely extended onto the predictable a-field V, and we still
denote it by fi. fi does not charge any evanescent set. In fact, for any
evanescent set H its debut Dh = oo a.s. is a predictable time, and
#C[0F]U]|0F,oo[,
where F = [Dh < oo] G Tq. Since P(F) = 0, Op = oo a.s., we have
fi(H) = 0.
For every non-negative bounded measurable process X define
H(X) = fi^X). (47.1)
Then ~p is a finite measure on T x B(i2+), and does not charge any
evanescent set. Because fi is the extension of/i, by (47.1) we know Ji(X) = /Z(PX),
i.e., /Z is predictable. By Theorems 5.11 and 5.13 there exists a unique
predictable integrable increasing process A such that fi is the measure
generated by A. E[A0] = /l([0]) = /i([0]) = 0, thus A0 = 0 a.s..
Furthermore, by (46.2) for any stopping time S
E[A00-As}=iJi(]S,oo[) = E[Zs}.
This means Z is the optional projection of (Aqq — At), i.e., the potential
generated by A. □
As an important application of Theorem 5.47, we obtain Doob-Meyer
decomposition theorem for supermartingales of class (D).
5.48 Theorem. Let X be a right-continuous supermartingale of class
(D). Then X can be uniquely decomposed as:
X = M -A, (48.1)

160 Chapter V Projections of Processes
where M is a uniformly integrable martingale, and A is a predictable
integrable increasing process with Aq = 0. (48.1) is called the Doob-Meyer
decomposition of X.
Proof. Existence. Put
Zt = Xt-E[X00\ft).
Then Z = (Zt) is a potential of class (D). By Theorem 5.47 there exists
a predictable integrable increasing process A such that
Zt = E[A00\Ft}-At.
Put Mt = ElXoo + Aoo|5i]. Then X = M - A is the Doob-Meyer
decomposition of X.
Uniqueness. Let X = M — A be another Doob-Meyer decomposition
of X. Then A — A = M — Misa predictable martingale with integrable
variation. By Corollary 5.31 A — A = 0, A = A and M = M. □
5.49 Definition. Let X be a uniformly integrable cadlag supermar-
tingale. X is said to be regular, if for any predictable time T > 0
jE/fXT1-] = jE7[-Xj,J,
or equivalent^, XT_ = E[XT\TT-) (since XT- > E[Xt\Ft-\ by
Theorem 4.41).
It is easy to see that quasi-left-continuous uniformly integrable cadlag
supermartingales and uniformly integrable cadlag martingales are regular,
and regular predictable uniformly integrable cadlag supermartingales are
continuous. The following theorem is also evident.
5.50 Theorem. Let X be a cadlag supermartingale of class (D), and
X = M — A be its Doob-Meyer decomposition. Then A is continuous if
and only if X is regular.
§5. Filt rat ions of Discrete Type
5.51 Definition. Suppose
i) (Gn)n>o is a filtration with discrete parameter,
ii) (Tn)n>o is a strictly increasing sequence of r.v., i.e., Vn > 0 Tn <
oo =» Tn < Tn+i, with T0 = 0 and Tn | oo,
iii) Vn > 1, Tn is ^-measurable.

§5. Filiations of Discrete Type 161
Define
oo
\J(Gnn[Tn<t<Tn+i\)
n=0
oo
= { U An[Tn < t < Tn+i] :AneGn,n> 0}, t > 0, (51.1)
n=0
and denote it by Tt. It is easy to see that for each t > 0, Tt is a a-field,
and To = Go- We will show that F = (^i) is a right-continuous filtration,
and is called a filtration of discrete type. It is the object we deal with in
this paragraph.
5.52 Theorem. 1) F = (Tt) is a right-continuous filtration.
2) Vn > 1, Tn is an F-stopping time.
oo
3) IfG'n = Gn V A/\ T[ = (J (G'n n [Tn < t < Tn+i]), w/iere ^ is the
n=0
o-field generated by all P-null sets, F' = (F't) is the usual augmentation
ofF.
Proof. Let 5 < t, A G J*5 and A = \Jf=0Ak[Tk < s < Tk+x],Ak G Gk.
Then
A[Tn < t < Tn+i]
= {(nU AfclTib < s < Tk+{\) U (An[Tn < *])} fl [Tn < t < Tn+i]
€ Cn 0 [Tn < t < Tn+i].
Hence A G ^. Consequently, Ts C ^i- We are going to show Ji =
fln^t+i- Let ^ ^e (rin^i+i J-measurable. Then for each n > 1
h = Ex 4n)/[Tn<t+i<Tfc+1]^ 4n) € &.
Put /ijt = limn_oo hr^K For each cj G [Tfc < £ < 7fc+i] there exists an
integer nw such that
n > n„ => uj G [Tfc < t + - < Tfc+i] =► /i(cj) = 4n)(w) = JijbM-
Evidently, hk eGk- Hence
k=o
This implies ^ = Tt+.
For all n > 1, t > 0
oo
/i = £ /lfc/[Tt<t<rit+1] e Tt.
[Tn <t}= \J[Tk<t< Tk+1] € ^i.
fc=n
Therefore, Tn is a stopping time.

162 Chapter V Projections of Processes
The third claim is apparent. □
5.53 Theorem. Tqq = Goo if and only if for each n > 0
Gn n [rn+i = oo] = Goo n [rn+i = oo]. (53.1)
Proof As usual, F^ = V Ft and Goo = V 0n- Obviously, J^ C Goo-
t>0 n>0
Sufficiency. Let A e Gn- For each £ > 0
A[Tn < t] = U ^[7* < t < Tfc+i] G Ji C J*oo.
fc=n
(By the way, we obtain Gn C ^t„) Hence A[Tn < oo] G J*oo. On the other
hand,
A[Tn = oo] = (J A[Tk-i < oo, Tk = oo]. (53.2)
By (53.1), A[Tk = oo] = Ak-i[Tk = oo], Afc-i G Gk-v We have shown
^[T^ < oo] G Too. Thus
^[Tfc.! < oo,Tfc = oo] = A^Tk-x < oo)[Tk = oo] G J^.
By (53.2) we have A[Tn = oo] G J*oo- In consequence, A G J*oo, Gn C ^
and ^oo C ^"oo-
Necessity. For £ > 0 and A G Ji
A[Tn+1 = OO] = {(Tj^fclTfc < t < Tfc+i]) U(^n[Tn < tDjlTn+i = Oo],
A* G &.
Hence A[Tn+i = oo] G 0n n [Tn+i = oo], i.e., Tt n [Tn+i = w]cfi„n
[Tn+i = oo]. Then
0oo H [Tn+i = oo] = J*, fl [Tn+i = oo]c5nn [Tn+i = oo],
(53.1) holds. D
The theorem illustrates that in order to guarantee T^ = Goo the
increase of Gn cannot go behind Tn. In order to satisfy (53.1) it suflBces to
replace Gn by {Gn n [Tn+i < oo]) U {Goo n [Tn+i = oo]), i.e., to amplify
Gn properly. On the other hand, it is desirable to have Gn = Frn for
each n. To this end (53.2) is necessary, as we will see later. Hence in the
remainders of this paragraph we always suppose ^oo = Goo-
5.54 Theorem. T > 0 is a stopping time if and only if for each n > 0
there exists Rn G Gn such that
T[T<Tn+l] = (#n)[Rn<Tn+1], (54.1)

§5. Filtrations of Discrete Type 163
or equivalently, any one of the following conditions holds:
T < Tn+! =>Rn = T and T>Tn+i=*Rn> Tn+1. (54.2)
Rn<Tn+i=*T = RnandRn> Tn+i =► T > Tn+1. (54.3)
T < Tn+! «=► Rn < Tn+! =*T = Rn. (54.4)
TATn+^/inATn+i. (54.5)
Proof. The equivalence of (54.1)-(54.5) can be proved directly, and its
proof is left to readers.
Sufficiency. For all t > 0
oo oo
[T < t) = U ([T < t][Tn < t < Tn+1)) = U ([Rn < t][Tn < t < Tn+l}),
n=0 n=0
since on [Tn < t < Tn+i] T < t => T < Tn+i => T = Rn, and Rn <
t=> Rn< Tn+l => Rn = T. Because Rn G Qn, we have [T < t] G Ji,
and hence T is a stopping time.
Necessity. For all n > 0, £ > 0 and A G Ji we have
4[* < Tn+1] = (J Ak[Tk < t < Tfc+i] G5nn[t< Tn+1], (54.6)
where ^ G <?*, k = 0, • • •, n. Let Fr = [r < Tn+i], r G Q+. Due to (54.6),
there exists GT G Gn such that
[T < r]Fr = GrFr. (54.7)
We may suppose (Gr,r G Q+) is monotone increasing. In fact, when
r' < r, we have Fr/ D Fr, and
Gr/Fr = Gr/Fr/Fr = [T < r']Fr/Fr = [T < r']Fr C [T < r]Fr = GrFr.
Thus
[T < r)Fr = ( U Gr0^r.
r'<r
If necessary, GT may be replaced by (J Gr/. Now define
r'<r
Rn(uj) = inf{r G Q+ : a; G Gr}.
Obviously, #n > 0. For * > 0, [Rn < t] = (J GT G 0n, hence #n G 0n.
We show that (54.4) holds. If (54.4) does not hold, it must be that one
of T{uj) and Rn{uj) is smaller than Tn+i(u;), and T(u) ^ Rn(uj). At
this time, we may take t < Tn+i(uj) such that T(u) > t > Rn(uj) or
Rn(u) > t > T(u). In the first case, take r G Q+ such that r < t and
u; G Gr. Then u & [T < r]Fr, but cj G GrFr, contradicting (54.7). In
the second case, take r G Q+ such that T(u) < r < t and u & GT. Then

164 Chapter V Projections of Processes
u) & GrFT, but u G [T < r]FT, contradicting (54.7) similarly. In a word,
(54.4) must hold. □
5.55 Theorem. 1) X = (Xt) is optional if and only if for each n > 0
there exists a process X^ G Gn x B(i2+) such that
oo , .
X = £ *("V..r..+1|. (55-1)
n=0
2) X is predictable if and only if for each n > 0 there exists a process
XW eGn* B(-R+) such that
X = XoI[0] + £ X^I]Tn^l}. (55.2)
n=0
Proof Since C/n c ^t„? the sufficiency is apparent. Only the necessity
needs to be verified.
1) By the monotone class argument it suffices to verify (55.1) for X =
^[T,oo[> where T is a stopping time. Suppose Rn G Gn, n > 0, satisfy
(54.2). Put
Then X<n) G^x B(R+), and on [Tn < t < Tn+i] T<t*=>Rn<t, i.e.,
-XJ[T„fTn+i[ = ^(n)^[Tn,Tn+l[-
(55.1) follows immediately.
2) By the monotone class argument it suffices to verify (55.2) for X =
I[o,t\ > where T is a stopping time. Now put
Similarly we have X^ € Gn x B(R+), on [T„ < * < Tn+i] T < t <$=►
iZ„ < t (e.g. T < t => T < t < Tn+i =► T = Rn < t), and
XI]Tn,Tn+1) = X(n)l]Tn,Tn+l]-
(55.2) follows immediately. □
5.56 Theorem. Let T be a stopping time. Then for each n > 0
tt n [T„ < r < r„+i] = e„ n [rn < r < rn+1]. (56.i)
Proof. Because Qn C TTn, we have Qn n [T„ < T] C Tt and
an n [Tn < T < Tn+1] cfTn[Tn<T< rn+i].
On the other hand, if A € .Tt, there is an optional process X =
g *(«) 7[rn)Tn+l[ such that IA[T<oo] = XTI[T<oo], XW eGnx B(R+).
n=0
On [Tn<T< Tn+1], IA = X^ = X<£, thus
A[Tn <T< Tn+1] = [X<£ = l][Tn < T < TB+1], (56.2)

§5. Filtrations of Discrete Type 165
where Rn G Gn is determined as in Theorem 5.54. Since [XJg* = 1] € 0n,
by (56.2) we know
TTn[Tn<T< Tn+l) CGnn[Tn<T< Tn+1].
Hence (56.1) holds. □
5.57 Corollary.
?Tn = Gn, n> 0,
TTn-=Gn-lV<T{Tn}, n>l.
Proof. For each n > 0
;Fn n [r„ < oo] = rTn n [rn < Tn < Tn+1)
= Qnr\ [Tn <Tn< Tn+1] = gnn [rn < oo].
By (53.1)
TTn n [T„ = oo] = ^oo n [r„ = oo] = (?«, n [r„ = oo] = an n [rn = oo]
(here we use the assumption ?<*, = Goo)- Since [Tn < oo] € T^n C\Qn, we
obtain Trn = Gn-
For each n > 1, Tn-\ < oo ^ Tn_i < Tn. Hence ^r„_i C ^r„-, and
(?n_i V <r{Tn} C ^r„— On the other hand, let A € ^i, then
A[*<rn] = TMfc[rJk<t<rfc+1], AkeGk, o<*<n-i.
Therefore A[t < T„] € £„_i V o~{Tn} and Jtb_ C 0n_i V o~{Tn}. D
5.58 Theorem. Suppose Qq is complete, i.e., F satisfies the usual
conditions. A stopping time T is predictable if and only if for each n > 0
there exists Rn € Gn such that
T[T<Tn+1] = (#n)[ftn<Tn+i] (58.1)
or equivalently, any one of the following conditions holds:
T < Tn+i =>Rn = T andT> Tn+1 =^Rn> Tn+1. (58.2)
Rn < Tn+i =► Rn = T and R* > Tn+1 => T > Tn+1. (58.3)
Rn < Tn+i «=»> T < Tn+1 ^T = Rn. (58.4)
Proof. The equivalence of (58.1)-(58.4) can be proved directly and its
proof is left to readers.
Sufficiency. Put X = 7[r,oo[. *(n) = '[Rn,ao[> n > 1. Then (55.2)
holds, where Xq = 7[r=o]> because on [Tn < t < Tn+i],T < t <=> Rn <t.
By Theorem 5.55.2) process X is predictable, and so is T.

166
Chapter V Projections of Processes
Necessity. Let T be a predictable time. There is a sequence (Sk) of
stopping times foretelling T, i.e., Sk T T and on [T > 0] Sk < T for all k.
Let Uk,n € Qn such that Uk,n A Tn+1 = Sk A Tn+1. Put
' 1<7<« «—►OO
oo
k=l
Rn = Un + IFn[T>0].
It is easy to see Rn G Qn. We show that (58.2) holds.
Assume T < Tn+i, n > 0. If T = 0, then for all fc,Sfc = 0.
Consequently, Ukin = 0, C^fn = 0, Un = 0, Rn = 0, and i^ = T. If T > 0,
then for all fc, Sfc < Tn+i. It must be Sk = Uk,n- Since Sk T r, we have
U'kn = Sk, Un = T. Noting that Sk <T,IF = 0, we obtain Rn = Vn = T.
Assume T > Tn+i (then T > 0). For sufficiently large fc, Sk > Tn+i,
then Ukin > Tn+i, C^n > Tn+1. If JF = 0, Un > U'kn > Tn+i, Rn = Un>
Tn+i. If JF = 1, C/n > tfjU > Tn+i, /J* = C7n + 1 >'Tn+i. □
Remark. In the theorem the sufficiency does not need the
completeness of the filtration, and the necessity holds for any foretellable stopping
time.
5.59 Corollary. Suppose for some k > 0, T G Frk, T > Tk and
Tfc < oo => Tjb < T. Then T is a predictable time.
Proof. Put
Rn
{oo, n < fc,
T, n>fc.
Clearly, fln G Gn. It suffices to show (58.2). When n > fc, (58.2) is
trivial, since Rn = T. When n < k, Rn = oo, T > Tn+i => Tn+i <
oo => Rn > Tn+i. It remains to verify T < Tn+i =^ T = Rn. When
T = oo, it is trivial. But [T < oo,T < Tn+i] = 0. In fact, if n + 1 < k,
then T < oo and T < Tn+i are impossible, since T > Tk. If n + 1 = A:,
T < Tn+i =► T = Tfc, but Tk < oo => Tk < T. U
5.60 Corollary. Suppose Qq is complete. For any n > 1, Tn is
predictable if and only ifTn G Gn-i-
Proof. Applying Theorem 5.58 to Tn yields the necessity: there is
Rn-i € Gn-i such that Tn = Rn-i' The sufficiency comes from Corollary
5.59 by taking T = Tn and k = n - 1. □
5.61 Theorem. Let T be a foretellable stopping time. Then for each

§5. Filtrations of Discrete Type
167
n>0
Tr- n[Tn<T< Tn+i, T < oo] = Qn n [Tn < T < Tn+1, T < oo]. (61.1)
Proof. Since Qn = TTn, Qn n [T„ < T] € ^T-,
a„ n [r„ < t < t„+i] c J-T- n [rn < r < r„+i].
Let A 6 Jr-- There is a predictable process X = £ ^n^]r„,r„+i] +
n=0
X0/[o] such that XT/[r<oo] = IaI[t<°o}, *(n) € £„ x B(R+). Hence
A[Tn <T< Tn+1,T < oo] = [x£Vn<0o] = 1][T„ < T < Tn+1,T < oo],
^ir- n [T„ < T < Tn+UT < oo] C Qn n [Tn < T < T„+i,T < oo],
where i^ € ^„ is determined in Theorem 5.58. Then (61.1) follows. □
5.62 Theorem. A stopping time T is totally inaccessible if and only
if
[T\ C U \Kl (62.1)
n=l
where T£ is the totally inaccessible part ofTn.
Proof. The sufficiency is easy: for any predictable time S
oo
P(T = S < oo) < Z P(K = S < oo) = 0.
n=l
It remains to show the necessity. First, for each n > 0, T is foretellable
on [Tn<T < Tn+i]. In fact, there exists Rn € TTn satisfying (54.2). Put
Sk = Tn V (l^ - -), k > 1. Then Sk > Tn and Sk € ^r„, so Sk is a
stopping time. On [Tn < T < Tn+i], Sk < T = Rn, k > 1, and Sk T T.
Hence if T is totally inaccessible, it must be P(Tn < T < Tn+i) = 0,
n > 0. On the other hand, P(T = T£ < oo) = 0, n > 1. (62.1) follows. □
5.63 Theorem. .4 stopping time T is totally inaccessible if and only
if
i) for all n > 0, P(Tn <T< Tn+i) = 0,
ii) for alln>0 and r.v. R € rTn,P(T = Tn+i = R<oo) = 0.
Proof. Necessity, i) has been proved in the previous theorem, similarly,
it can be shown that T is foretellable on [T = Tn+i = R < oo]. To this
end, it suffices to take 5* = Tn V (r - -). Then (5jk)fc>i foretells T on
[T = Tn+i = R < oo]. Hence P(T = Tn+i =R<oo) = 0.
Sufficiency. Suppose there exists a sequence (Sk)k>\ of stopping times
such that P(A) > 0, where A = (f)kLi[Sk < T]j n [Sk | T < oo]. By
i) P(A) = ££L0P(A[T = Tn+1 < oo]), there is some n > 0 such that

168 Chapter V Projections of Processes
P(A[T = Tn+i < oo]) > 0. For each k > 1 there exists Rk G TTn such
that Sk < Tn+i ^ Sfc = Rk. Put R = fiE^oo Rk G TTn> On A[T =
Tn+i < oo], i? = Umifc^oo Rk = lim^oo 5fc = T, i.e., A[T = Tn+i < oo] c
A[T = Tn+i = R<oo]. Hence P(A[T = Tn+i = R < oo]) > 0 contradicts
ii). So T is totally inaccessible. □
5.64 Theorem. Suppose Qo is complete. (Ft) is quasi-left-continuous
if and only if
i) for each n> 1, T% is predictable, where T£ is the accessible part of
T„,
ii) for each n > 1, Tt* = ?T*--
Proof. The necessity is obvious. We will show the sufficiency. Let T
be a predictable time, and A € Tt- Then
A = (A[T = oo])U ( U MTn = T < oo]) U ( U A[Tn < T < Tn+1]) a.s..
Vn=0 ' Vn=0 '
Evidently, A[T = oo] € Jt_, A[T = 0] € :FT_. For n > 1
yt[T£ =T<oo]€Jrn[7^ = T<oo]= -FTl? n [T% = T < oo]
= :Fr«- n [T° = T < oo] = TT- n [T» = T < oo].
Noting that [T£ = T < oo] € :FT-, we have A[T° = T < oo] € TT—
By Theorem 5.56, 4[Tn < T < Tn+1] = ^'[Tn < T < Tn+i], where
A' € TTn- Thus A'[rn < T] € ^r— In order to show A € ^r- it remains
to verify [T < Tn+1] € Jr_:
[T < T„+i] = [T < oo] \ [T„+i < T < oo],
[Tn+1 < T < oo] = [2j+1 < T < oo] U [T°+1 < T < oo]
= PS+i < T < oo] U [T°+1 < T < oo] a.s.
€ ^r— D
5.65 Theorem. Suppose Qo is complete. (Tt) is completely
continuous if and only if
i) for each n>l,T£ is predictable,
ii) for each n>l,Qn = Q0\/ a{Tu • • •, Tn}.
Proof. Necessity, i) is evident. By Corollary 5.57 for each n > 1
Qn = TTn = TTn- = a„-i V a{Tn}.
ii) follows by induction.
Sufficiency. By ii) and Corollary 5.57 we have Trn = ?Tn- immedi-

§5. Filtrations of Discrete Type 169
ately.
TT« H [Tna < oo] = ?t* H [Tna = Tn < oo] = TTn H [Tna = Tn < oo]
= FTn- n [1% = Tn < oo] = Jts- n [T- < oo].
Clearly, TT% n [T£ = oo] = FT]?- n [2£ = oo] and [2% < oo] G TT%-, so we
obtain Fr« = Tt*-> Similarly, TTi = TTi_.
Let T be a totally inaccessible time and A G Tt fl [T < oo]. By
oo
Theorem 5.62 A = \J (A[T = Tn < oo]) a.s.. For n > 1, A[T = Tn <
71=1
oo] G TTix fl [I* < oo] = TTit_ fl [T = T^ < oo] = FT- n [T = 2* < oo]. In
order to obtain A G Ft- it suffices to check [T = T£ < oo] G .Ft-:
[T = Tn < oo] = pt-i < T < l£][T < oo] G FT-
Hence it follows that Tt = Tt- •
By the previous theorem we have known that (Tt) is
quasi-left-continuous. For any stopping time T, Ta is predictable, and
TT n [Ta < oo] = FTa- fl [Ta < oo].
In consequence,
TT n [T < oo] = (FTa fl [Ta < oo]) U (TTi n [T* < oo])
= {TT-- n [Ta < oo]) n (FT,_ n pr < oo])
= FT-n[T< oo].
Finally, we have Tt = Tt-- a
5.66 Lemma. Le£ H be a sub-a-field, and £ be an integrable r.v..
Then for all A G T and H eH
J AH J AH *j[lA\rl\
Proof Put G = [E[IA\H] ^ 0]. Then G G H and
P(,4GC) = / E[IA\H]dP = 0
Jgc
Hence the integrand on the right-hand side of (66.1) makes sense. Now
Jah E[IA\H\ Jahg E[IA\H] Jhg E[IA\H] iAar
= f E[ZIA\H]dP = f £IAdP = f £dP. □
Jhg Jhg Jah

170 Chapter V Projections of Processes
5.67 Theorem. Let W = (Wt) be a bounded measurable process.
Then
our ^ E\WtI[Tn+i>t]\Gn] T ( .
Wt = 2s -FriT ^T7[Tn<KTn+1]. (67.1)
n=0 ^[Tn+iMll^nJ
Moreover, suppose Go is complete. Then
>Wt = E[VW]/|t,„, + t ELT'T,"lct'^<'^'l- («■»)
n=0 il/i[T„+i>t]l^nJ
Proof. For each n > 0 it is not difficult to choose Qn x
/^immeasurable versions of (E[WtI[Tn+l>t]\Gn]) and (^[/[Tn+1>i]l^n])- By
Theorem 5.55 °W defined by (67.1) is optional. It suffices to justify for any
stopping time T
E[WTI[T<oo]] = E[°WTI[T<oo]}.
Let Rn G Qn such that T < Tn+1 <=» R„, < Tn+1 ==$► T = Rn. Using
Lemma 5.66, we have
oo oo
E[WTI[T<oo]] = E E[WTI[Tn<T<Tn+l]} = E ^[W*»J[T»<ft.<rn+1]]
n=0 n=0
~ £oEi E[i[Tn+1>Rn]\gn] hr^T^)
~ So^ E[I[Tn+l>t]\Qn} Ln>S*»<r"«l/
oo oo
= E E^W^t^^t^] = £ ^[°^T/[Tn<T<TB+l]]
n=0 n=0
= £?[°Wt/[t<oo]]-
(67.2) can be proved in a similar way. □
5.68 Corollary. Let £ be an integrable r.v. and T be a stopping time.
Moreover, if T is predictable and Go is complete, on [T < oo]
E[t\rT-)=E[t\fo]I[T=0]+ g ^^^^^KTKT^] O.S..
n=0 ^[Tn+1>T]\yn\
5.69 Theorem. Let A be an integrable increasing process. Then for
each n > 0 there exist a modification B^ o/(J5[j4tn+1~"|(/n]) and a modi-
fication C^ of (E[Ajn+1 \Gn]) such that fl<n> and C^ all are integrable

§5. Filtrations of Discrete Type 171
increasing processes. And we have
A°* = I ,S pit \ .10 M<*<Tn+i]> t > 0, (69.1)
•/[0,t]n=0 P[Tn+l > s\Qn\ L " n+1J
rt oo dC^
Apt=E[A0\G}+ £__L^/[r ,, *> 0.(69.2)
Proof. We only deal with the predictable case. For every r G Q+ take
a version CT of jE[j4rn+1|(/n] such that (Cr)reQ+ is increasing. Put
Ct(n) = inf{Cr :r>t,re Q+}, * > 0.
It is easy to see that C^ satisfies the requirement, and for any non-
negative Qn x B(.R+)-measurable process H
E\ f HsdAjn+1] =E\ f HsdC^A. (69.3)
L^[0,oo[ J L-/[0,oo[ J
At the same time, for every non-negative measurable process H we can
take a Qn x S(i2+)-measurable version H = (Ht) of (E[Ht\Gn\), and
Ef / HsdCinA =E\ f HsdC^]. (69.4)
(In fact, if we take a filtration F(n) = (jf°) such that J*t(n) = Qn,t > 0,
then the predictable a-field of F(n) is just Qn x B(R+), and C(n) is the
dual predictable projection of ATn+1 w.r.t. F^n\)
Denote by Gn(ds) the conditional distribution of Tn+i w.r.t. Qn:
Gn([«,Oo]) = ^[/[T„+i>a]|0n]. Put 5n = inf{* > 0 : Gn([t, oo]) = 0}.
Then 5n G 5n and
P(Tn+1 > 5n) = E[Gn(]Sn, oo])] = 0,
ie, Tn+i < 5n a.s.. Since Gn([Sn, oo]) = Gn([Sn]) = £[J[7Wl=5n]|en],
Gn([5n,oo]) > 0 a.s. on [Tn+i = 5n]. Hence Ap defined by (69.2) makes
sense, and is a predictable increasing process.
Let if be a non-negative predictable process:
H = H0Im + g tf (n)/]Tn,rn+1j, #(n) € «?„ x B(R+).

172
Chapter V Projections of Processes
Then
E\ I HadAs] = E[H0A0] + § E f H<pdA*»»
VJ[0M J n=0 J]Tn,Tn+i)
= E[HQE[A0\F0}} +EE[f flj«) I[Tn<s)dCinA (by(69.3))
n=0 lJ]0,oo[ J
= e[h0a*)+ze[[ HV/[?T<3-n+ll^n)]
n=0 L^]0,oo[ Gn([s,OOj) J
(by (69.4) and / dC<n) = 0)
./{5:Gn([5,oo])=0}
= £?[/ HadA*\.
Therefore, Ap is the dual predictable projection of A. □
5.70 Example. Let T > 0 be a r.v. and Q = a{T) V AT. Let
Tt = (Afn [t < T]) U(Gn[T< i]), t > 0.
Then (Ji) is one of the simplest complete filtrations of discrete type. We
claim
1) S > 0 is a stopping time if and only if there is a constant C (may
be +oo) such that 5[5<^j = C[c<T] a-s--
2) A stopping time S is predictable if and only if there is a constant
C (may be +oo) such that 5[5<^j = C[q<t] as- In particular, T is
predictable if and only if T is a.s. a constant.
3) T{ = T[TeBc], Ta = T[TeB], where B = {b > 0 : P{T = b) > 0}.
4) A stopping time S is totally inaccessible if and only if 5 = T[^€i4]
a.s., where A G B(R+) and A C Bc. In particular, T is totally inaccessible
if and only if the distribution of T is continuous on ]0, oo[.
5) (Tt) is quasi-left-continuous if and only if there is a constant C (may
be +oo) such that P(T > C) = 0 and the distribution of T is continuous
on ]0, C[. In this case (Tt) is also completely continuous.
1) and 2) follow directly from Theorems 5.54 and 5.58 respectively. The
total inaccessibility of Tl follows from Theorem 5.63. The accessibility of
Ta is obvious. Then 4) follows from Theorem 5.62 and 3). In order to
deduce 5) from Theorems 5.64 and 5.65 it suffices to show that Ta is
predictable if and only if B = 0 or B = {C} and P(T > C) = 0. If
B = 0,Ta = oo is predictable trivially. If B = {C} and P(T > C) = 0,
Ta = T[T=C] satisfies the requirement in 2), and is predictable. Conversely,
if Ta is predictable, there is a constant C satisfying the requirement in
2). If C = oo, then T < oo => T < Ta a.s., it must be B = 0. If
C < oo, and b G B, then P(T = 6)>0, T = b=*Ta = T>C

Problems and Complements
173
(since T < C => T < Ta) and Ta = C consequently. Hence b = C, i.e.,
B = {C}. On the other hand, if T > C, Ta = T[T=C\ = oo ^ C. Hence
P(T > C) = 0.
Problems and Complements
5.1 Let T be a stopping time, and £ be an integrable r.v.. Put X =
£I[T] and Y = -B[£|^r]/[^]. Then X and Y have the same optional
projection.
5.2 Let T be a stopping time, and £ be an integrable r.v..
1) The optional projection of X = £/[o,t[ ls 0 ^=^ -E[£|-^t-]J[t>o] = 0.
2) The predictable projection of Y = £/[o,n is 0 <=> E[^\TTJ\ = 0.
5.3 Let T be a stopping time, and £ be an integrable .^--measurable
r.v.. X = £/[x,oo[ is a martingale if and only if
E[Z\TT-]I[o<t<oo] = 0.
5.4 Let X be a uniformly integrable cadlag martingale. Then for any
predictable time T, AXtI[t,oo[ ls a uniformly integrable martingale.
5.5 Let N = (Nt) be a Poisson process with parameter A. Then
N? = \t, pNt = Nt-, t>0.
p.6 Let X be a bounded measurable process such that both X and °X
are cadlag. Then
P(X.) = (°X)-.
5.7 Let X be a bounded measurable process and a > 0. Put Y* =
l°°°Xae-°t(a-t)ds, t > 0. Then Mt = °Yt - / (a°Ks - °Xs)ds, t > 0, is a
martingale.
5.8 Let X be a non-negative accessible process. If the predictable
projection of X is evanescent, then so is X.
5.9 Let A be an adapted process with finite variation, and H be a
progressive process, integrable w.r.t. A. Then °H is integrable w.r.t. A,
and (°H).A = H.A.
5.10 Let H be an adapted measurable process. Then H has an
optional modification.
5.11 Let T be a totally inaccessible time, 0 < T < oo, A = Jpr,oo[-
Then A1^ is exponentially distributed with rate 1.
5.12 Let X be a cadlag supermartingale. Then there exists a sequence
(Tn) of stopping times with Tn] oo such that for each n, XTn is of class
(D).

174
Chapter V Projections of Processes
5.13 Let X be a non-negative cadlag supermartingale, and
Rn = inf{* > 0 : Xt > n}, n > 1.
Then X is of class (D) if and only if
hmoE[XpnIlRn<oo]] = 0.
5.14 Let A be a predictable integrable increasing process with Aq = 0,
and Z be the potential generated by A. Then
roo
E[A200] = E[ (Z8 + Za-)dA,].
Jo
(^ElAlc] is called the energy of A or Z.)
5.15 Let A and B be two predictable increasing processes nidi at 0,
Y and Z be the potentials generated by A and B respectively. If Y < Z,
then
E[Al\ < 4E[Bl].
5.16 Let X be a potential, and T > 0 be a stopping time. Then
(jEfAV+jl^]) is a potential of class (D).
5.17 Let T > 0 be a r.v. with distribution F, and A = Jpr,oo[- Put
Tt = (ATfl [t < T}) U ((AT V a{T}) H[T< *]), t > 0.
Then the dual predictable projection of A is
F(ds)
i
/l0,tAT] F([5,00])' •
5.18 Let (/> be a non-negative Borel function on i2-|_. Then the following
statements are equivalent:
1) For any stopping time T defined on any probability space with a
filtration, (f>(T) is still a stopping time,
2) There exists C G ii+ such that
\ >t, if
*(')•! -• r, ;^:
5.19 Let Tn be the n-th jump time of Poisson process X. Then for
n ^ 1^ Tn-i = esssup{5 : 5 is an F(X)-stopping time and S < Tn}. In
particular, if 5 is an F(X)-stopping time and S < Ti, then 5 = 0 a.s..
5.20 Let T be an accessible time, and D be the debut of the predictable
support of [TJ. Put
A = {S : S is a stopping time, 5 < T and 5 < T on [T > 0]}.
Then D = ess sup A

Chapter VI
Martingales with Integrable Variation
and Square Integrable Martingales
Beginning with this chapter, we start to develop modern theory of
martingales and stochastic integrals. Unless otherwise stated, our starting
point is always a complete probability space (fi,^7, P) with a filtration
F = (Tt)t>o satisfying the usual conditions.
Hereinafter we use the following notations:
A—the collection of all adapted processes with integrable variation,
A*—the collection of all adapted integrable increasing processes,
V—the collection of all adapted processes with finite variation,
V+—the collection of all adapted increasing processes,
M—the collection of all uniformly integrable martingales.
We stress that all the elements of M are supposed to be cadlag.
Moreover, martingales always mean cadlag martingales.
For any class V of processes we denote by Vq the sub-class of V
consisting of all elements of V with null initial values, e.g., Mq is the collection
of all uniformly integrable martingales which are null at zero.
For any adapted process with locally integrable variation A we also
denote by A the dual predictable projection or compensator of A. The
process A — A is called the compensation of A.
§1. Martingales with Integrable Variation
6.1 Definition. X = (Xt) is called a martingale with integrable
variation, if it is both a martingale and a process with integrable variation.
Obviously, a martingale with integrable variation is a uniformly
integrable martingale. Denote by W the collection of all martingales with
integrable variation. Hence, W = M n A.

176 Chapter VI Martingales with Integrable Variation and Others
Prom Corollary 5.31 we know that for every A € A, A — A € Wq. The
next theorem illustrates that this is the general form of all elements of
6.2 Theorem. Let M be a martingale with integrable variation. Put
At= £ AM3, t>0.
0<s<t
Then A is an adapted process with integrable variation having continuous
compensator A, and
M = M0 + i4-A (2.1)
Moreover, for any predictable process H = (Ht) we have
E\ I \H3\\dM3\] < 2E\ Z \Hs\\AM3\l (2.2)
Proof. Put Dt = Mt- M0, t > 0. Then D = (Dt) G W0 and
D = Dc + Dd = Dc + A.
Because D G Mo, by Corollary 5.31 we know that A = —Dc. Hence, A
is continuous, and (2.1) holds. (2.2) can be easily deduced from Theorem
5.22.2). □
6.3 Theorem. 1) All predictable uniformly integrable martingales are
continuous.
2) IfM is a predictable martingale with integrable variation, then Mt =
M0.
Proof. 1) Let M G M. If M is predictable, then for any predictable
time T, Mt G ^t-? and by Theorem 4.41,
MT = E[Mt\Tt-] = MT-,
i.e., M and M_ axe indistinguishable. So M is continuous.
2) If M is a predictable martingale with integrable variation, then by
Corollary 5.31, M = M, Mt = M0. □
The following theorem is the key result concerning martingales with
integrable variation.
6.4 Theorem. Let M be a martingale with integrable variation. Then
for any bounded martingale N we have
ElMooN^} = E[M0No] + E\Z AM3AAtJ. (4.1)
L3>0 J
In addition, (Lt) = (MtNt — £ AMSANS) is a uniformly integrable mar-
8<t
tingale.

§2. Square Integrable Martingales 177
Proof. Since N is the optional projection of JVqo,
ElM^N^] = E\ f N^dMs] =E\ I NsdMs}.
On the other hand, Mt = Mo,
E\ [ Ns.dMs] =E\ [ N3-dM3] = E[M0No}.
L./[0,oc[ J L«/[0,oc[ J
Hence
ElM^N^] - E[MQNo] = E\ f ANsdM3] = E\ £ AM3AN3].
LJ[0,oo[ J La>0 J
Let T be a stopping time. Applying (4.1) to NT yields
E[MtNt} = E[M00Nt} = E[M0No} + e\ £ AM3AN3],
l0<s<T J
i.e., E[LT] = E[L0]. By Theorem 4.40 we know L G M. □
The next theorem illuminates the special role of predictable processes
in the theory of stochastic integrals.
6.5 Theorem. Let M be a martingale with integrable variation, and
H be a predictable process such that
E\ I \Hs\\dM3\\
L./[0,oc[ J
< OO.
Then H.M is a martingale with integrable variation.
Proof. By Theorem 5.23.2) we have
(HM) = H.M = H0M0.
Hence H.M - (HM) = H.M - H0M0 e W0, H.M e W.
§2. Square Integrable Martingales
6.6 Definition. A martingale M is called a square integrable
martingale, if supE[M2] < oo. Denote by M2 the collection of all square
t
integrable martingales. By Theorem 1.7.2), M2 C M.
6.7 Theorem. Let M be a square integrable martingale. Then for
any A > 0 we have
p(sup|Mt| > A) < ±suPE[M?}. (7.1)

178 Chapter VI Martingales with Integrable Variation and Others
Proof. Let {*i, *2» * * •} be a countable dense set of il+. By Corollary
2.13 we have
p(sup|Mti| > A) < ± supE[M?\
Since [sup|M*| > Al = U [sup|M*.| > Al and f[sup|Mt.| > Al) is
1 t J n=l lj<n J VLj<n J/n>l
monotone increasing,
P(sup|Mt| > A) < ± supE[M?\. (7.2)
Replacing A by A - e in (7.2) and letting e j 0 give (7.1). □
(7.1) is called Kolmogorov inequality as usual.
6.8 Theorem. 1) Let M G M. Then M G M2 if and only if
JE7[M^] < oo. In this case, we have
E[Ml\= sup E[M?\. (8.1)
t
2) Al2, endowed with inner product (M,N) = EIMoqNqq], is a Hilbert
space, isomorphic to L2(Q,JrOQ,P) by the mapping: M i—► M^.
Proof. 1) Let M £ M2. By Fatou's lemma we have
^[M^sup^M-2]. (8.2)
t
Conversely, let M G M and £[M£J < oo. Since Mt = jB[Moo|Ji], by
Jensen's inequality, we have M2 < ElM^Tt]. Then
sup E[M2]<E[Ml\. (8.3)
t
This implies M G M2. (8.1) follows from (8.2) and (8.3).
2) is evident. □
6.9 Theorem. Let (Mn)n>i C M2, M G M2. If
lim \\MZ - Moolb = Km {«[(AC - M^)2]}1/2 = 0,
n—*oo n—►oo
Mere existe a subsequence (Mn*, fc > 1) suc/i £/ta£ /or almost all a;,
(M^ '(a;))^ converges to Mt(uj) uniformly in t.
Proof Take a subsequence {Mnk)k>i such that £ ||M£* - Mooh <
oo. By Doob's inequality we have

§2. Square Integrable Martingales 179
E\ f sup|Mtn* - Mt\] = g E\sup\M?* - Mt\]
lk=l t J Jfc=l L t J
< g{£[sup(Mtnfc-Mt)2]}1/2
K—1 *
< 2 g {e[(M£ - M^)2] }V2 < oo.
In particular, ]T sup \M?k — M*| < oo a.s.. This implies that for almost
k=i t
all u;, (Aftn*(a;))jk>i converges to M^(a;) uniformly in t. □
6.10 Corollary. Denote by M2jC the collection of all continuous
square integrable matingales. Then M2'c is a closed subspace of M2.
6.11 Theorem. 1) Let M e M2. Then for any stopping time T,
MT eM2. Furth ermoref if (Tn)n>i is a sequence of stopping times with
L2
Tn T oo a.s., then Mrn —► M^.
2) Le*(Mn)n>i CM2, MG.M2. // ||M£ - M^ — 0, then for any
stopping time T, ||Mj; - Mt||2 —> 0.
Proof 1) At first, by Theorem 4.39 we know MT e M. On the other
hand
E[M$\ < f;[(sup|Mf|)2] < 4supE[M?} < oo,
by Theorem 6.8.1) we obtain MT G M2.
Let (Tn) be a sequence of stopping times with Tn | oo a.s.. By
Doob's stopping theorem, (M^n)n>i is a square integrable martingale
w.r.t. (^rjn>i. By Corollary 2.20, M^^M^.
2) Since (Mn — M)2 is a submartingale, we have
E[(M$ - Mr)2] < E[(M^ - M^)2}.
Then 2) follows immediately. □
6.12 Definition. Let M and TV be two square integrable martingales.
We say that M and TV axe mutually orthogonal, if MqNo = 0 a.s. and for
every stopping time T, E[MtNt] = 0, and denote it by MAIN. The
notation M _L N is reserved for orthogonality between M and N as two
elements in the Hilbert space M2, i.e., EIMoqNoo] = 0. In order to
distinguish the two kinds of orthogonality, we call the latter weak orthogonality.
6.13 Theorem, Let M,N € M2 and M0No = 0. Then M and N are
mutually orthogonal if and only if MN is a martingale.

180 Chapter VI Martingales with Integrable Variation and Others
Proof. The necessity can be deduced directly from Theorem 4.40. Now
we show the suflSciency. Let MN be a martingale. Because sup \Mt\ G L2
t
and sup \Nt\ G L2, we have sup \MtNt\ G L1. Hence MN is a uniformly in-
t t
tegrable martingale. For every stopping time T, E[MtNt] = E[MqNo] =
0, i.e., MAIN. D
6.14 Theorem. Let X C M2 and C(X) be the closed linear subspace
generated by X. Let N G M2. IfNALX, then NALC(X).
Proof. It is obvious that NALC(X), where C(X) is the linear subspace
generated by X. Let M G £(#). Take (Mn) C C(X) such that ||M£ -
L2
-Moolb —> 0- By Theorem 6.11, for any stopping time T, M£^>Mt, and
M%NT^MTNT. Hence M0AT0 = 0 a.s. and £[MT7VT] = 0, i.e., M1LJV.
Therefore, NALC(X). U
6.15 Definition. A family of measurable processes X is said to be
stable, if
i) for every stopping time T, M e X => MT G X,
ii) for every A G T0, M G X =* /^M G A'.
Stable closed linear subspaces of M2 axe simply called s£a&/e subspaces
ofM2.
6.16 Theorem. £e* X C A42. //A' zs stable, then XL and C(X) are
stable subspaces, and X±ALC(X), where
XL = {M e M2 :VNeX,M _L N}.
Proof. Let N G XL, M G X. For every stopping time T, MT G X.
Thus
B[MooA£] = EIM^Nt] = E[MTNT] = ElM^N^} = 0.
This means NT G XL. On the other hand, if A G ^o? then T^M € <*\ and
EIIaNoqMoo] = 0. Consequently, /^TV G A'1. Therefore, Af1 is a stabe
subspace. So is C(X) = (X1)1.
Now let M G £(#) and AT G A'-1. For every stopping time T we
have MT G £(#) and 7VT G XL. Then £[MT7VT] = EfM^AT^] = 0.
Furthermore, for every A e To we have IaMt G £(*), E[IaMtNt] = 0.
Especially, taking T = 0, we obtain MoTVo = 0 a.s.. This means MALN.
□
6.17 Corollary. Let A42'd = (.M2'0)1. Then M2>c and X2^ are
stable subspaces, and M2,cALM2,d.

§3. The Structure of Martingales in M2>d 181
6.18 Definition. The elements of M2'd axe called purely
discontinuous square integrable martingales.
Let M e M24. It is evident that M0 = 0 a.s..1)
Let M e M2. By Corollary 6.17, M can be uniquely decomposed as
follows:
M = M0 + Mc + Md,
where Mc 6 A10'c, Md 6 M?'d. Mc is called the continuous martingale
part of M, and Md is called the purely discontinuous martingale part of
M.
Let M € M2 and T be a stopping time. Obviously, we have
(MT)C = (MC)T, (MT)d = (Md)T.
§3. The Structure of Purely Discontinuous Square
Integrable Martingales
6.19 Lemma. Let AeA+. Then
E[Al] < 4E[Al\.
Proof. Let N = (Nt) be the cadlag modification of the martingale
(f^ool^t]), N^ = sup\Nt\. By Doob's inequality,
EKN^)2} < 4E[Nl] = 4E[Al).
Because A — A 6 M, we have
As-As = £[^00 - A^Ts] = Nt- -E[ioo|^s]-
Then by Theorem 5.13,
ElAU = E\ f AoodAs] =E\ f EiAoolfsjdAs]
L./[0,oo[ J LV°°[ J
= E\ f (As-- As. + Ns-)dAs]
L-/[0,oo[ J
= E\ j (As-- As-+ Ns-)dAs]
L -/[0,oo[ J
< ElAooAao +
J) It should be stressed that in many literatures M2yC is used for the space of all
continuous square integrable martingales with initial values zero, and the initial values
of purely discontinuous square integrable martingales need not be null.

182 Chapter VI Martingales with Integrable Variation and Others
But
£[ioo4x>] = E[J AoodAs] =E[J Ns-dAa} < EiN^Aoo].
Hence we have
E[Al] < 2E[N*00A00) < 2{E[(N*O0)2}E[Al]y/2 < 4E[Al\. D
6.20 Lemma. Let M G M2. Then for every stopping time T,
E[(AMT)2} < 16E[M^].
Proof. Put M£c = sup | Mt |. By Doob's inequality
t
2SKAC)2] < AE[Ml] < ex,
Since |AMT| < 2M^, E[(AMT)2} < 16£[M£]. □
Now we start to study the structure of purely discontinuous square
integrable martingales.
Let T > 0 be a stopping time. Put
M2[T] = {Me M24 : [AM ^ 0] C [T]}.
Obviously, A/J2[T] is a stable subspace.
6.21 Theorem. Let T > 0 be a totally inaccessible time or predictable
time.
1) Me M2[T] <=>M = A-A, where A = £/[T,oo[, £ € L2(FT).
2) Let M e M2[T\. For all N e M2 we have
ElMooNn] = E[AMTANT\. (21.1)
3) Let M £ M2. The (orthogonal) projection of M onto M2\T] is
N = A — A, where A = AMx/[Too[.
Proof. 1) Sufficiency. Let £ G L2(fT), A = £ipr,oo[ By Lemma 6.19,
A — A € M2. If T is a totally inaccessible time, then A is continuous
(Corollary 5.28.3)). If T is a predictable time, then A = #[f |^r-Kpr,oo[
(Theorem 5.29.2)). In both cases, A — A is continuous outside [T]. What
remains is to show A - A G M2'd. Let TV e M2'c. Put
Tn = inf{t > 0 : \Nt\ > n}.
Then Tn,n > 1, axe stopping times, and Tn | +oo. For each n, NTn is a
bounded continuous martingale. By Theorem 6.4,
E[(A - AUNtJ = E[(A - i)ooA£n] = 0.
By Theorem 6.11.1), E[(A - A^N^} = 0. This means A - A € M2'd.
Hence A-AeM2[T].

§3. The Structure of Martingales in M2*d 183
Necessity. Let M G A42[T]. Put A = AMt/[t,oo[- As shown above,
A-A G M2[T}. Then MZ(A-A) G M2[T}. On the other hand, if T is a
totally inaccessible time, A is continuous, and A{A-A)t = A At = AMy.
If T is a predictable time, A = jB[AMt|^t-]^[t,oo[ = 0 (Theorem 4.41),
and A{A - A)T = AMT. This means M - (A - A) G X2'c. Hence
M - (A - A) = 0, i.e., M = A - A.
2) Let N^ be the cadlag modification of bounded martingale (E[Noo
%oo|<n]l^i])- % Theorem 6.4,
^[MoeA^] = JB[AMTAi4n)]. (21.2)
Since N& = A^ V^n] -^ N^ as n -+ oo, by Lemma 6.20,AA$° -^
AiVr as n —> oo. Letting n —* oo in (21.2) yields (21.1).
3) Prom 1) we know that N = A-A e M2[T] and M-N has no jump
on [T]. By 2), M - N _L M2[T]. This implies that N is the projection of
M onto M2[T]. U
The following theorem describes the structure of purely discontinuous
square integrable martingales.
6.22 Theorem. 1) Let M eM2. Then
E[M2} + e[u±Ms)2] < E[Ml\. (22.1)
Equality holds in (22.1) if and only if M — Mo € M2'd.
2) Let M € M2'd and (Tn)n>i be a standard sequence of stopping times
exhausting the jumps of M {see Theorem 4.21). Denote by Mn the
projection of M onto A42[Tn] {i.e., Mn is the compensation of AMrn/[Tn,oo[)-
oo
Then the orthogonal series £ Mn converges to M in M2 .
71=1
3) Let M G M2'd. M has a unique decomposition as follows:
M = Mdu + Mdi,
where M^ G M2'd has only accessible jumps, and Mdl G M2ld has only
totally inaccessible jumps.
Proof. 1) Let (Tn)n>i be a standard sequence of stopping times
exhausting the jumps of M. Put
An = AMrn/[Tft)00[, Mn = An - An, Hk = £ Mn.
71=1
ThenM-M0-^fchasnojumpon[Ti]U-U[rfcl. By (21.1), M-Mq-
Hk is weakly orthogonal to Hk. But M1, • • • ,Mfc are mutually weakly

184 Chapter VI Martingales with Integrable Variation and Others
orthogonal,
E[Ml] = E[M$] + E[(H^) + ^[(Meo - M0 - tf *,)2]
= E[Mg] + £ E[(M^)2] + EWoo - Mo - H^)2).
71=1
By (21.1) E[(M£)2] = E[(AMTn)2}, hence
E[Ml] = E[M2} + £ £[(AMTJ2] + ^[(Moc - M0 - H*,)2]. (22.2)
n=l
oo
In particular, the orthogonal series £ Mn converges to an element H in
l
M2, and H € M2,d. Moreover, by Theorem 6.9, H and M have the same
jumps. Thus M — H is a continuous square integrable martingale, and
Md = H, MC = M -M0-H. By (22.2) we have
E[MU = E[M2} + g E[(AMTn)2} + £[(M^)2]
n=l
= E[Mg) + E £(AM3)2 + E[(M^)2).
S
Hence (22.1) holds, and equality holds in (22.1) if and only if Mc = 0, i.e.,
M - M0 = Md e M24.
Proof 2) has been included in the above proof 1).
3) Let (Tn)n>i be a standard sequence of stopping times exhausting
the jumps of M. Put N\ = {n : Tn is predictable}, N2 = {n : Tn is
totally inaccessible}, and
Mda = £ Mn, Mdi= D Mn,
n€N\ neN2
where Mn is the compensation of AMTnI[Tn}oo[- By Theorem 6.9 we know
that M^ has only accessible jumps and Md% has only totally inaccessible
jumps. By 2) we have M = M^ + Mdl. The uniqueness of the decompo
sition is easy to prove (see the proof of Theorem 4.25). □
6.23 Theorem. 1) Let M, N e M2. Then
E\M0N0\ + E[Z \AMsANs\] < y/EiM&y/ElNl]. (23.1)
2) Let M e M24. Then for any N e M2
EiM^N^} = E [ £ AMS ANS]. (23.2)
Moreoverf (Lt) = (MfNt — £ AM3ANS) is a uniformly integrable mar-
s<t
tingale.
3) MlnwcM24.

§4. Quadratic Variation 185
Proof. 1) By Schwarz inequality we have
e[\m0No\) + e\z\amsan3\\
(23.3)
< {E[M2} + £E(AMS)2}1/2WV02] + f?E(AAr3)2}i/2.
s s
(23.1) follows from (23.3) and (22.1).
2) At first we assume N € M2'd. By Theorem 6.22.1),
£[M°oAU = ^[(Moo + iVTO)2] - E[Ml\ - E[Nl}}
= ±E[Z(&MS + ANS)2 - £(AM3)2 - E(AiV3)2]
= e[z*msans].
Hence (23.2) holds for N € .M2'd. Now assume N € M2. Let Nd be
the purely discontinuous martingale part of N. Then N — Nd € M2'c,
MAIN - Nd, and ElM^N^ - JV*)] = 0. Thus
EiM^N^} = EiMocNi] = E[ZAMSANS].
(23.2) holds.
Let T be a stopping time. Applying (23.2) to MT and NT yields
.E[Lt] = 0. By Theorem 4.40, (Lt) is a uniformly integrable martingale.
3) Let M € M"q fl W. By Theorem 6.4, for any bounded martingale
N we have
^[McoiVoo] = E [ £ AM3AAT3]. (23.4)
Since the set of all bounded martingales is dense in M2 w.r.t. the norm
||M|| = \AE[M£J, by (23.1) we know that (23.4) holds for all AT € M2.
In particular, taking N = M,we have
e[mI) = e[u±ms)2].
Noting that M0 = 0, by Theorem 6.22.1) we know M e M24. □
§4. Quadratic Variation
6.24 Definition. Let M € M2. By Doob's inequality we know M^ =
sup | M^ | e L2. Hence M2 is a submartingale of class (D). By Doob-
t
Meyer decomposition theorem, there exists a unique predictable integrable
increasing proess, denoted by (M,M) or simply (M), such that M2 —

186 Chapter VI Martingales with Integrable Variation and Others
(M) € Mo- (M) is called the predictable quadratic variation or the angle
bracket process of M.
Let M,NeM2. Put
(M,N) = ±[(M + N)-(M)-(N)].
It is called the predictable quadratic covariation of M and N.
Remark. Let M e M2. Then (M) = 0 if and only if M = 0. By
Theorem 5.50 we know that if M is quasi-left-continuous, then (M) is
continuous.
6.25 Example. Suppose M = (M*) is a process with independent
increments and JE7[M$] = a is a constant. Then M is a martingale.
Furthermore, suppose M is a square integrable martingale. Then by
Theorem 2.69, M2 - (M02 + E[M2) - E[M$\) e Mq. Hence (M)t =
M02 + E[M2} - E[M$). In particular, if M0 = a, then (M)t = #[M2] is
non-random.
Remark. If M G M2, then M has orthogonal increments: for t\ <
f?[(M,3-M,2)(M,2-Mtl)|Ji2]
= (M,2-Mfl)f;[M,3-M,2|Ji2] = 0 a.s.,
E[(Mt3-Mt)(Mt2-Mtl)}=0.
The following theorem gives a characterization for (M, iV).
6.26 Theorem. Le< M,N eM2. Then (Af, AT) is the unique
predictable process with integrable variation such that MN — (Af, N) G .Mo-
Proof. We have
MN - (Af, AT) = i [(M + AT)2 - (M + AT) - M2 + (Af) - AT2 + (AT)].
Hence MN — (Af, Af) £ .Mo- The uniqueness can be easily deduced from
Theorem 6.3.2). □
An immediate consequence of this theorem is that (Af, AT) is a
symmetric bilinear form of M and N.
6.27 Definition. Let Af, N e M2. Put
[Af, AT], = M0AT0 + (Mc, Nc)t + £ &MSANS, t > 0, (27.1)
s<t
where Mc and Nc axe the continuous martingale parts of M and N
respectively. Then [M, N] is an adapted process with integrable variation

§4. Quadratic Variation 187
(Theorem 6.23.1)). [M, M] is also denoted simply by [M], and it is an
adapted integrable increasing process. [M] is called the quadratic variation
or square bracket process of M. [M, TV] is called the quadratic covariation
of M and TV.
6.28 Theorem. Let M,N eM2.
1) [M, TV] is the unique adapted process with integrable variation such
that MN - [M, TV] e M0 and A[M, TV] = AM AN.
2) (M, TV) is the dual predictable projection o/[M, TV].
Proof. 1) Let M = M0 + Mc + Md. We have MdN - [Md, N] e M0
(Theorem 6.23.2)), McNd e MQ(McALNd), and
MCN - [Mc, N] = NQMC + MCNC + McNd - (Mc, Nc) e M0.
Hence
MN-[M,N\ = MQN+McN+MdN-(M0No + [Mc,N} + [Md,N}) e M0.
A[M,N] = AM AN is obvious. The uniqueness follows from Theorem
6.3.2).
2) follows from 1) and Theorem 6.26. □
It is also easy to see from Theorem 6.28.1) that [M, TV] is a symmetric
bilinear form of M and N.
6.29 Corollary. Let M, TV G M2. The following assertions are
equivalent
1) MAIN,
2)[M,TV]e.Mo,
3) (M,TV) = 0.
6.30 Corollary. Let M, TV 6 M2. The following assertions are
equivalent:
1)[M,N] = 0,
2) MJ1TV and AM AN = 0 {i.e., M and TV have no common jump).
Proof. 1) =► 2). Let [M,TV] = 0. By Corollary 6.29, MiLTV. At the
same time, AMATV = A[M, TV] = 0.
2) =► 1). Let MAIN and AM AN = 0. By Corollary 6.29, [M,TV] =
(Mc, TVC) is a predictable martingale with integrable variation, null at
zero. Hence [M,TV] = 0 (Theorem 6.3.2) ). □
6.31 Theorem. Let M, TV 6 M2 and T be a stopping time. Then
(M,TVT) = (M,TV)T, (31.1)
[M,NT] = [M,N]T. (31.2)

188 Chapter VI Martingales with Integrable Variation and Others
Proof. Since (MN)T - (M,N)T is a martingale (Theorem 4.39), in
order to justify (31.1) it suffices by Theorem 6.3.2) to show that (MN)T-
(M, NT) is a martingale. But MNT — (M, NT) is a martingale, what
remains is to show that (MN)T - MNT is a martingale:
E[(MT -M^NrlFt] = E[(MT - M^N^^Tt]
= E[(MT - MtwT)NT\Ft} = E[(MT - Mt)NTI[T<t)\Ft]
= (MT - Mt)NTI[T<t] = (MtAT - Mt)NtAT-
Since (Mc, (NT)C) = (Mc, (NC)T) = (Mc, Afc)T, (31.2) follows from (31.1)
and the definition of [M, N]. □
6.32 Lemma. Let M,N e M2. Then for almost all u, for all 0 <
s < t < oo we have
\(M,N)t(u,)-{M,N)a(u)\
< [(M)t(u) - (M)a(u))^2[(N)t(u) - (NUu)]1'2, (32.1)
\[M,N]t(u;)-[M,N}s(u;)\
< {[M]t(u) - [MU^UNUu) - [N]3(u;)}V2. (32.2)
Proof. We only show (32.1). The proof of (32.2) is quite the same.
For given 0 < s < t < oo, for all rationals A
(M + XN)t -{M + XN)S > 0, a.s..
Denote (M,N)t - (M,N)3 by (M, JV)<. Then
(M)ts + 2\(M,N)ts + \2(N)ts>0, a.s.,
|(M,JV)il<{Wi}l/a{W.}1/a, a.s..
Because (M), (N) and (M, N) axe all right-continuous, then for almost all
u, (32.1) holds for all 0 < 5 < t < oo. □
Combining Lemma 6.32 and Theorem 1.45, we obtain immediately the
following Kunita- Watanabe inequalities.
6.33 Theorem. Let M,N e M2, and H,K be two measurable
processes. Then
L
\HsKs\\d(M,N)s\
[0,oo[
/
^[0,oo[
<(/ H2d(M)3Y/2(f K2d(N)sY/2 a.s., (33.1)
vJ[0,oo[ ' V./[0,oo[ '
\HsKs\\d[M,N]s\

Problems and Complements 189
<(/ Hld[M]a)l'\f Kld[N]sf2 a.s.. (33.2)
6.34 Corollary. Let p,q be a pair of conjugate indices, i.e., 1 < p <
oc, 1 < g < oo, —(-- = 1. Under the assumptions of Theorem 6.33 we
p q
have
E\ J \HsKs\\d(M,N)s\]
< 11// H2d(M)s\\ IIf K23d{N)t
\\\ J[0,oo[ \\p\\ y J[0,oo[
E\ I \HsKs\\d[M,N}s\]
(34.1)
< II f HU[M]s\\ \\Jf K2d[N]t
II V "'[0,oo[ llpll y J[0,oo[
(34.2)
Proof It follows from Theorem 6.33 and Holder's inequality (£J[|£t?|] <
Problems and Complements
6.1 Let M be a continuous uniformly integrable martingale, S and
T be two stopping times, S < T. If for almost all u, M.(u>) has finite
variation on [5(cj), T(u>) A (t V S(uj))] for each t > 0, then for almost all u
M.(lj) is constant on [S(u>),T(u)].
6.2 Mq and L2(J:q) are mutually orthogonal stable subspaces of A42,
and
M2 = Ml®L2(F0),
where © stands for the direct sum.
6.3 Let M e M24. If e[z |AM3|] < oo, then M e W0.
6.4 Let M e W0. If £[£(AMS)2] < oo, then M e M24.
6.5 Let M 6 Wo or M G M24, and T > 0 be a stopping time. If
[AM ^ 0] C [T], then M = A - A, where A = AMTJ[Too[.
6.6 Let
{T is a stopping time, £ G L2(Ft), \
aiTO°1' ^[T=o]=0,^/[0<r<oc]l^r-]=0 J'
M2'° = £(*), M2-s = XL.

190 Chapter VI Martingales with Integrable Variation and Others
1) M2'a and M2'3 axe mutually orthogonal stable subspaces of M2,
M2 = M2>(T®M2>3.
2) M2ida C M2'a C M24, where M%da is the subspace of all purely
discontinuous square integrable martingales having only accessible jump.
3) Let M e M2. M € M2'3 if and only if for all stopping time S
Msefs-.
4) The following assertions are equivalent:
i) For all stopping times T and S, [T < S] e FS-.
ii) M2 = M2>3.
iii) {Ft) is completely continuous, i.e., for all stopping time S
6.7 let M € M2, S and T be two stopping times, S < T. Then
Ms = MT if and only if (M)s = (M)T.
6.8 Let M, N € M2. Then
((M + N) + [M + JV])1/2 < ((M) + [M\fl2 + ((N) + [JV])1'2.
6.9 Let M, N € .M2. Then
< J[M-N],
< y/(M - N).
6.10 Let G = (Gt) be a filtration satisfying the usual conditions, and
for each t > 0, Gt C Ff Then the following statements are equivalent;
i) Every square integrable Gr-martingale is an jF-martingale;
ii) For each t > 0, Tt and Goo is conditionally independent given Gt\
iii) For every Goo * $(.R-j-)-measurable process A with integrable
variation and Ao = 0, the dual predictable projections of process A w.r.t. G
and F are indistinguishable.
In this case, for every G-stopping time T, Gt = Ft H £oo-

Chapter VII
Local Martingales
§1. The Localization of Classes of Processes
7.1 Definition. Let V be a class of processes. The localized class of
D, denoted by V\oc, is defined as follows: a process X = (Xt) belongs to
Dloc if and only if Xo € fo and there exists a sequence (Tn) of stopping
times with Tn ] oo such that for each n the stopped process
V. The sequence (Tn) is called a localizing sequence for X (w.r.t. V).
V C V\oc if and only if for each X eT>, X0 e Tq.
7.2 Theorem. LetV1 andV2 be two stable classes of processes. Then
(2)1nx>2)loc = l?LnAoc-
Proof. Let X G T>loc fl I^c, (Tn) be a localizing sequence for X
w.r.t. V1, and (5n) be a localizing sequence for X w.r.t. V2. Then by
the stability of V1 and X>2, (Tn A Sn) is a localizing sequence for X w.r.t.
D1 fl V2. Hence Dj1^ n Vfoc C (2?1 n X>2)ioc- The converse implication is
trivial. □
7.3 Theorem. Let V be a stable vector space of processes. Then so
is V\oc, and
(Aoc)loc = Aoc (3.1)
(i.e., it is no use iterating the localization procedure).
Proof. Only (3.1) needs to be proved. Let X e (Aoc)loc- For
simplicity we may assume Xq = 0. Let (Tn) be a localizing sequence for X w.r.t.
Ploc, i.e., XTn € X>ioc for each n. For each n let (SUik)k>i be a localizing
sequence for XTn w.r.t. V. Rearrange (Sn^ A Tn)n^>\ into a sequence
(Sn). Then sup5n = +oo and XSn e V for each n. Because for every
n
pair (S, T) of stopping times,
XSVT = XS + XT-(XS)T,

192
Chapter VII Local Martingales
we have XSlS/ "S/Sn e V by induction. Hence (Si V • • • V Sn) is a localizing
sequence for X w.r.t. Z>, i.e., X G V\oc. □
Remark. For validity of (3.1) it suffices to assume that V is stable
under stopping, i.e., for every X G V and stopping time T, X £ Z>. But
this needs a somewhat longer proof. Theorem 7.3 is enough for us to use.
The following lemma provides us a useful method to get localizing
sequences. Its proof is not difficult and left to readers.
7.4 Lemma. Let X = (Xt) be an adapted cadlag process. Put
Tn = inf{*>0: \Xt\ > n}, n > 1. (4.1)
Then (Tn) is a sequence of stopping times with Tn | oo.
7.5 Definition. Let V be the class of all bounded processes. Then
each process of Pioc is said to be locally bounded.
7.6 Theorem. Let X be a process with Xo G fo- Then X is locally
bounded if and only if there exists a sequence (Tn) of stopping times with
Tn | oo such that for each n, -X"J]o,Tn] is bounded.
Proof. Put Sn = 0[|x0|>n]- Then (Sn) is a sequence of stopping times
such that Sn | oo and for each n, XoI]ofsn] — ^o^[|x0|<n]^]o,oo[ *s bounded.
If X is locally bounded and (Rn) is a localizing sequence for X, then
XI]oyRn/\Sn] is bounded for each n. Conversely, if (Tn) is a sequence of
stopping times such that Tn | oo and XI^QTn^ is bounded for each n, then
(Tn A Sn) is a localizing sequence for X. D
Remark. Theorem 7.6 provides a more reasonable definition for
locally bounded processes. In fact, it is not natural to require Xq G Tq in
the definition of locally bounded process. Since all localization procedures
are applied to classes of adapted processes but not to the class of bounded
processes, we reserve the requirement Xq G T§ for the sake of unification.
The next theorem gives some frequently encountered examples of
locally bounded processes.
7.7 Theorem. Let X be an adapted cadlag process.
1) X- = (Xt-) is a locally bounded predictable process.
2) X is locally bounded if and only if AX is locally bounded too.
3) If X is predictable, then X is locally bounded .
Proof. 1) Let Tn be defined as in (4.1). It is obvious that |X_J]0)Tn]l ^
n. Thus X- is locally bounded by Theorem 7.6.
2) is an immediate consequence of 1) and Theorem 7.6.

§1. The Localization of Classes of Processes 193
The proof of 3) is quite the same as that of the second assertion in
Theorem 5.19. □
7.8 Definition. Recall that A (resp. A+) is the collection of all
adapted processes with integrable variation (resp. integrable increasing
processes). Then A\oc (resp. A^) is the collection all adapted processes
with locally integrable variation (resp. adapted locally integrable increasing
processes). This coincides with Definition 5.18, since we are concerned
with adapted processes.
For any adapted cadlag process X = (Xt) we denote by X* = (X£)
the supremum process of X:
XI = sup|Xa|, t>0.
s<t
Obviously, X* is an adapted increasing process, and A(X*) < |AX|.
7.9 Theorem. Let A = (At) be an adapted process with finite
variation. If A is locally bounded (or equivalently, A A is locally bounded ),
then A e A\oc.
Proof. Let (Sn) be a sequence of stopping times such that Sn | oo
and Al^Sn] ls bounded for each n. Put
Tn = inf{* > 0 : / \dAs\ > n} A 5n, n > 1.
Jo
Then Tn | oo, and for each n
f " \dAs\ = / \dAs\ + |AATn|/[Tn>0, < n + |AATn|/[Tn>0,
«/0 «/]0,T„[
is bounded. Therefore A € A\oc. □
7.10 Theorem. Let A = (At) be an adapted process with finite
variation. Then the following assertions are equivalent:
1) A e A\oc,
2) B = Z AA3 6 A>o
3) C= /EAA2 e A+ci
4) A* e A+c.
Proof. 1)=>2)=^3) is trivial.
3)=>4). Because |AA\ < C, A* < A*_ + \AA\ < A*_ + C. A*_ is locally
bounded (Theorem 7.7.1)), and C 6 A?oc. Then A* G Afoc.

194
Chapter VII Local Martingales
4)=>1). Let (5n) be a sequence of stopping times such that Sn | oo
and A^n — Aq is integrable for each n. Put
Tn = mf{t > 0 : / |<L43| > n} A Sn, n > 1.
Then Tn | oo and for each n
/Tn |<L4a| = / \dAs\ + |AATn| < n + 2(A*Sn - A*Aq)
Jo y]0,Tn[
is integrable. Therefore A € >tioc- n
7.11 Definition. Recall that M is the collection of all uniformly
integrable martingales. A process M € M\oc is called a local
martingale. Obviously, a local martingale is an adapted cadlag process. In the
same manner, Wioc is the collection of all local martingales with locally
integrable variation, Mfoc (resp. M^cc, resp. M^c) is the collection of
all (resp. continuous, resp. purely discontinous) locally square integrable
martingales. Obviously, M\oc, Wioo-MJLc all axe stable vector spaces.
Remarks. It is easy to see that
1) a (cadlag) martingale is a local martingale (by taking (n)n>x as a
localizing sequence);
2) a predictable local martingale is continuous (see Theorem 6.3.1) );
3) if M is a predictable local martingale with locally integrable
variation, then Mt = Mo a.s. for all £ > 0 (see Theorem 6.3.2) );
4) if A e A\oc, then its dual predictable projection A is the unique
predictable process with finite variation such that A — A G Wioc,o (see
Corollary 5.31);
5) if Mt = Mq £ To, M = (Mt) is a local martingale (by taking
(0[|A/0|>n]) as a localizing sequence).
7.12 Theorem. Let M be a local martingale. Then M € M if and
only if M is of class (D).
Proof. Only the sufficiency needs to be justified. Let M be of class
(D). Then Mo is integrable. If (Tn) is a localizing sequence for M, then
MTn e M for each n. For 0 < 5 < t < oo we have
E[MtATn \Fs) = M3ATn a.s.. (12.1)
Since (MtATn)n>i is uniformly integrable and lim MtATn = Mt,
E[MtATn \?s] -^ E[Mt\Ts] as n - oo.
Letting n -► oo in (12.1) yields
E[Mt\Ts] = Ms a.s..

§1. The Localization of Classes of Processes 195
This means M is a martingale. Then M € M. □
7.13 Theorem. Let M be a local martingale. Then AM has
predictable projection, and ^AM) = 0.
Proof It follows immediately from Theorem 4.41 and Theorem 5.8.2).
D
7.14 Theorem. Let A e A\oc and
A = AC + Ad" + Adi,
where Ac is the continuous part of A, A^ is the accessible jump part of
A, and Adt is the totally inaccessible jump part of A (see Theorem 4.25).
Then __
1) A*" is purely discontinuous, Adi is continuous,
2) A is continuous if and only if A*" is a local martingale,
3) A is purely discontinuous if and only ifAo = 0 and AC + Adl is a
local martingale.
Proof. 2) and 3) can be easily deduced from 1). Only 1) needs to
be justified. Let (Tn) be a sequence of predictable times exhausting the
jumps of A*1. Denote H = \JlTn]. By Theorem 5.22 we have
n
E\ I IHc(.,s)\dAf(-)\]<E\ ( IHc(;s)\dAf (.)|1 =0,
L./[0,oo[ J L^[0,oo[ J
i.e., for almost all cj measure dA^iu) has no charge outside (JPn(<*>)].
n
Hence A*" is purely discontinuous.
Because Adl is quasi-left-continuous, Adi is continuous by Corollary
5.28.3). D
7.15 Theorem. Let M e Wioc. Put
A = £ AM,.
Then A G A\oc, A is continuous, and
M = M0 + A-A.
(Hence M is also called a compensated sum of jumps.)
Moreover, if M has only accessible jumps, then
M = M0 + £ AM3.
Proof. The first part follows from Theorem 6.2. If M has only
accessible jumps, so does A. Since A is continuous, A = A*0, is a local martingale
by Theorem 7.14.2). Therefore A = 0, and M = M0 + A. □

196 Chapter VII Local Martingales
§2. The Decomposition of Local Martingales
7.16 Lemma. Let M be a local martingale, and e > 0. Put
A= £ AMsI[lAMal>e].
s<.
Then A e A\oc.
Proof. It is well known that A is an adapted process with finite
variation. Let (Sn) be a localizing sequence for M. Put
Tn = inf{t > 0 : \Mt - M0\ > n or £ \AAS\ > n} A Sn.
s<t
Then Tn | oo and
\&ATn\ < \AMTn\ <n + \MTn - M0|,
£ |AA3|< £ \AAs\ + \AATn\<n+\AATn\<2n + \MTn-M0\.
s<Tn 9<Tn
Since Tn < Sn, we have E[\Mrn - M0|] < oo. Hence A £ ^4i0c- D
The following theorem is the fundamental theorem for local
martingales. It plays an important role in the theory of local martingales.
7.17 Theorem. Let M be a local martingale. For any given e > 0,
M can be decomposed as follows:
M = M0 + U + V, (17.1)
where U is a locally bounded martingale1^ with Uq = 0 and \AU\ < e, V
is a local martingale with locally integrable variation and Vq = 0.
Furthermore, if M is quasi-left-continuous, U and V can be chosen quasi-left-
continuous, having no common jump.
Proof. Without loss of generality we may assume Mq = 0. Put
^=EAMs/[|AMa|>|].
Then A € ^loc by Lemma 7.16, and V = A - A € Wi0c,o, U = M-V e
A4ioc,o- F°r any predictable time T we have
£[AMr/[T<oo,|:FT_]=0,
AATI[T<oo] = E[AATI[T<oo]\TT-\ = E[(AAT - AMT)/[T<00,|fr].
But
\(AAT - AMT)I[T<oo]\ = |AMT/[|AMT|<§r<oo]| < £,
*> The class of locally bounded martingales is the localized class of the class of
bounded martingales. A locally bounded martingale is a local martingale which is
locally bounded at the same time.

§2. The Decomposition of Local Martingales
197
thus on [T < oo] we have
\AAT\ < ^, \AUT\ < |AMT - AAT\ + \&AT\ < e a.s..
For any totally inaccessible time T on [T < oo] we have A At = 0 a.s.,
and \AUt\ = |AMt - AAt\ < -• In a word, for any stopping time T,
\AUTI[T<oo]\ <£ a-S»,
i.e., |AC/1 < e. Hence U is a locally bounded martingale.
If M is quasi-left-continuous, so is A. Then A is continuous, U and
V are quasi-left-continuous. In addition, AV = AM/n^^^ei, AC/ =
AM/n^^^ei, C/ and V have no common jump. □
7.18 Corollary. Let M be a local martingale. Then its supremum
process M* € A^c.
7.19 Theorem. If M is both a local martingale and a process with
finite variation, then M G W\oc, i-e.,
•MiocnV = >Vioc.
Proof. Let M G M\oc fl V, and be decomposed as in (17.1):
M = M0 + U + V,
where V G Wioc,o and U is a locally bounded martingale with C/o = 0.
Since U G V is locally bounded , by Theorem 7.9 we have U G WiOc,0- At
last, we have M € Wioc- a
7.20 Corollary. Let A G V. A G -4ioc if and °nty if there exists a
predictable process BG V such that A — Bisa local martingale.
7.21 Definition. Let M be a local martingale. M is said to be
purely discontinuous if Mo = 0 and M can be decomposed as follows:
M = U + V,
9 rl J
where U e M^c and V e Wioc- We denote by Mfoc the collection of
all purely discontinuous local martingales. And the collection of all
continuous local martingales is denoted by A4foc. If we denote by Mc the
collection of all continuous uniformly integrable martingales, then M^ is
exactly the localized class of M°. Obviously, we have A1foc = Mx£c. It is
natural to define Md = M H Mfoc.
7.22 Lemma. Let M be a local martingale. If M is both continuous
and purely discontinuous, then M = 0.

198
Chapter VII Local Martingales
Proof. Since M € Mfoc, M = U + V, where U € M%£, V e Mkc,o-
On the other hand, M e Mfoc = M%£. Thus V e M?oc, and by Theorem
6.23.3) V e M%£. Then there is a localizing sequence (Tn) for M such
that for each n, MTn is both continuous and purely discontinuous square
integrable martingale. Hence MTn = 0 for each n, and M = 0. □
7.23 Corollary. Let M and N be two purely discontinuous local
martingales, and AM = AN. Then M = N.
7.24 Lemma. Let V e Wioc,o arid V" = Vc + V^ + Vdi {see Theorem
4.25). Then V^ is a local martingale, and Vc = -Vdi.
Proof. We have
0 = V = VC + Vda + Vdi.
But by Theorem 7.14, V^ is purely discontinuous and Vdi is continuous.
It must be that V^ = 0, i.e., V^ € Wioc,o. Then Vc = -V*. D
7.25 Theorem. Let M be a local martingale. M can be uniquely
decomposed as follows:
M = M0 + MC + M*" + Md\
where Mc G A4foc0, M^0 G Mdoc has only accessible jumps, and Mdi €
M^. has only totally inaccessible jumps.
Mc andMd = Mda+Mdi are called the continuous martingale pari and
purely discontinuous martingale part of M respectively. For any stopping
time T we have
{MT)C = (MC)T and {MT)d = (Md)T.
Proof. Existence. Put
M = Mo + U + V,
where U e MfocQ and V e Wioc,o- By Theorem 6.22.3), U has the
following decomposition;
U = Uc + U^ + Ud\
where Uc e M*£0, U^ e M*£ has only accessible jumps, and Ud% €
Mx^ has only totally inaccessible jumps. By Lemma 7.24
V = Vc + Vda + Vd\
where V*" e Wioc,o has only accessible jumps, Vc + Vdl e Wioc,o has only
totally inaccessible jumps. Put
Mc = C/c, M^ = U*1 + V*1, Mdi = Udi + VC + Vdi.

co The Decomposition of Local Martingales 199
Then Af = Afo + Mc + M^ + Mdl is the required decomposition.
Uniqueness. Let M = M$ + M +M +M be another decomposition
satisfying the requirements. Then by Lemma 7.22 Mc = M°. Thus M6*1 —
~fifda = M % — Mdl has neither accessible nor totally inaccessible jump,
i e. M**" ~~ M ^ continuous. Again by Lemma 7.22, M*" = M and
The last assertion follows from the uniqueness easily. □
The next Theorem is a supplement to Theorem 7.19.
7.26 Theorem. Let M be a purely discontinuous local martingale,
or all t > 0,
Proof. Put
If for all t > 0, E |AM3| < oo a.s., then M e W\oc
s<t
A=Z AM3.
By Theorem 7.17, we know A e A\oc. Put N = A- A. Then N e Wloc,o-
By Theorems 5.27.2) and 7.14, we have
AA = P(AA) = P(AM) = 0.
Hence AM = AN. By Corollary 7.23, M = N and M e W\OCi0. □
Now we start to define quadratic variation for local martingales.
7.27 Lemma. Let M be a local martingal. Then for all t > 0,
E(AM3)2<oo a.s..
s<t
Proof. Let M = Mo + U + V, where U e M?ocQ and V e Wloc,o- Let
(Tn) be a localizing sequence for M such that for each n, UTn € M2 and
VTn € W. Then
E (AM3)2<2{ E (AC/a)2+ E (AV^)2}
0<3<Tn K S<Tn 3<Tn J
<2{ E (AC/a)2+f E |AV3|l2}<oo a.s.. D
^s<Tn ls<Tn J >
7.28 Lemma. 1) For any M G M2oc there exists a unique predictable
locally integrable increasing process, denoted by (M,M) or simply (M),
such that M2- (M) e MfocQ.
2) For any M, N G M2oc there exists a unique predictable process with
locally integrable variation, denoted by (M, N), such that MN — (M, N) G
Proof. 1) is a special case of 2). We axe to show 2). Let M,N € M2oc,
and (Tn) be a common localizing sequence for M and N. For each n

200
Chapter VII Local Martingales
((M - M0)Tn,(N - N0)Tn) is well defined. Now piece them together into
a process:
(M,N) = M0N0 + g <(M - M0)Tn, (N - N0f»)I]Tn_liTn],
n=l
where To = 0. Then it is not difficult to verify that (M, TV) satisfies the
requirements. □
Remark. Similarly to the remark after Definition 6.24, if M € M^
is quasi-left-continuous, then (M) is continuous.
7.29 Definition. Let M and TV be two local martingales, Mc and
TVC be their continuous martingale parts respectively. Define
[M, TV] = M0TV0 + (Mc, TVC) + Z (AM3 ATV3). (29.1)
[M, TV] is an adapted process with finite variation. In particular, [M, M]
is an adapted increasing process. It is also denoted simply by [M].
[M] is called the quadratic variation of M. It is easy to see that
M = 0 if and only if [M] = 0, M G M\oc if and only if [M] is continuous,
M € M foc if and only if [M] is purely discontinuous.
[M, N] is called the quadratic covariation of M and N. If M, TV €
Xj2oc, by localization we know that [M, TV] G ^4ioc and (M, A^) is the dual
predictable projection of [M, TV], in particular, [M] G .Aj£c and (M) is
the dual predictable projection of [M], In general, for M, TV G A^ioc if
[M, TV] € -4ioc> we define (M, TV) as the dual predictable projection of
[M, TV]. (M, TV) is also called the predictable quadratic covariation of local
martingales M and TV, and (M) (if M G M^oc) the predictable quadratic
variation of M.
It is easy to see that [M, TV] (resp. (M, TV), if exists) is symmetric and
bilinear in M and TV, and for any stopping time T,
[MT,TV] = [M,TV]T (resp. (MT,TV) = (M,TV)T).
7.30 Theorem. Le£ M be local martingale. Then \/[M] is a locally
integrable increasing process.
Proof. Since
y/[M} - \M0\ < yl[M]-M% < sJ[M - MQ],
we may assume Mq = 0. Let
M = U + V,

§2. The Decomposition of Local Martingales
201
where U G M2ocQ and V G Wioc,o- Because Kunita-Watanabe inequality
still holds for quadratic covariation of local martingales, we have
sm<^/p]T^]+2^wi<ym+vw]
= \ + \[U] + jE(AFs)2<i + i[c;] + E|A^|.
Since [U] € A+c and £ \AVS\ € Ate #1 € Ate- °
7.31 Theorem. Let M and TV be two local martingales. Then
[M, TV] is the unique adapted process with finite variation such that MN —
[M,N] e Moc,o and A[M,TV] = AM AN.
Proof. At first, we show that MN — [M, TV] is a local martingale. To
this end, it suflSces to deal with the case of M = TV. We may also assume
Mo = 0. Let M = U + V, where U is a locally bounded maxtingaie with
U0 = 0 and V e Wioc,o- We have
M2 - [M] = U2 - [U] + V2- [V] + 2(UV - [[/, V\). (31.1)
We know U2 - [U] e M\ocQ by Lemma 7.28.1), and UV - [U, V] e Moc,o
by the localized Theorem 6.4. By the formula of integration by parts, we
have
vt2 - [v]t = v2-z (Avg2 = 2 / vs.dvs.
Since VL is locally bounded, V2 — [V] G A^iocO by the localized Theorem
6.5. Hence by (31.1) we obtain M2 - [M] e M\OCjq.
A[M, TV] = AMATV is trivial by definition.
Now let A be another adapted process with finite variation such that
MN - A e M\OCjo and AA = AM AN. Then A - [M, TV] is continuous
and A - [M, TV] e WioC,o. By Lemma 7.22, A = [M, TV]. □
7.32 Theorem. Let M be a local martingale. Then M G M2 if and
only if E[M]oo < oo.
Proof. Only the sufficiency needs to be justified. Let (Tn) be a
localizing sequence for M2 — [M]. Then for any stopping time T,
E[Mhrn} = E[M]TATn < EiM}^.
By Fatou's lemma we have
E[M$I[T<oo]] < ElMU < oo. (32.1)
This implies that M is of class (£>), and M e M by Theorem 7.12. Again
by (32.1) we have sup£?[Mt2] < oo, i.e. M e M2. □
t

202
Chapter VII Local Martingales
7.33 Definition. Let M and N be two local martingales. If MN €
M\OCiq, i.e., [M, N] £ A^iocO? we saY that M and N are mutually
orthogonal, and denote it by MALN.
If M e Mfoc, N e A4foc, then [M,N] = 0 by definition, i.e., every
purely discontinuous local martingale is orthogonal to every continuous
local martingale. Below we will show that this property can be used to
characterize purely discontinuous or continuous local martingales.
7.34 Theorem. Let M be a local martingale with Mq = 0. If M is
orthogonal to every continuous bounded martingale, M is purely
discontinuous.
Proof. Let M = Mc + Md, where Mc e -Mfoc0, Md e Mfoc. Put
Tn = inf{t > 0 : \MC\ > n}.
Then Tn | oo, and for each n, (Mc)Tn is a continuous bounded martingale.
By the assumption we have ((Mc)Tn) = (Mc)Tn = [M, (Mc)Tn] e .MiocO-
Since ((Mc)Tn) is non-negative, it must be ((Mc)Tn) = 0 (see Problem
7.6), i.e., (Mc)Tn = 0. Hence Mc = 0, and M = Md is purely
discontinuous. □
7.35 Theorem. Let M be a local martingale. If M is orthogonal to
every purely discontinuous local martingale, M is continuous.
Proof. Let M = M0 + Mc + Md. Then MMd = (M0 + Mc)Md+(Md)2.
By the assumption, MMd G A^iocO- But (Mo + Mc)Md G M\oc$. Hence
(Md)2 e M^o. It must be Md = 0. M = M0 + Mc is continuous. □
7.36 Theorem. Let M be a local martingale. If M is orthogonal to
every bounded martingale, then M = 0.
Proof. Apparently, Mq = 0. Let (Tn) be a localizing sequence for M
and TV be a bounded martingale. Then by the assumption we have
[MT\N] = [M,NT»]eMloCio,
i.e., MTnN e Moc,o- But MTnN is of class (/?), so MTnN 6 M0 and
E[MTnN00} = E[MoN0} = 0.
Since Mj-n £ F^ and N^ may be any bounded ^"oo-measurable r.v., we
have Mxn = 0 a.s.. Hence MTn = 0 for each n. Letting n —► oo yields
M = 0. D
7.37 Examples. 1) Let W = (Wt) be a standard Wiener process.
Then W is a continuous local martingale. By Theorem 2.69 we know that
(W2 — t) is a (local) martingale, therefore
[W]t = (W)t = t, t>0.

§2. The Decomposition of Local Martingales 203
2) Let P = (Pt) be a (homogeneous) Poisson process with parameter
1. Since (Pt — t) is a (local) martingale, the dual predictable projection of
Pis
Pt = t, t > 0.
Hence put
Nt = Pt-t, t > 0,
N = (JVi) is a martingale with locally integrable variation: N € Wioc,o (N
is also called a compensated Poisson process), and
[N]t = £ (AJVS)2 = £ (APS)2 = £ APS = Pt, t > 0,
s<t s<t s<t
(N)t = [N]t = Pt = t, t>0.
3) Put L = W + N, M = W -N. Then L,M € -Mioc.o, ^d
[L,Af]t = [W]t-[JV]t = *-Pt, t>0,
i.e., [L,M] 6 A^ioc.0, LALM. But
Lc = Mc = W, Ld = N = -Md.
This illustrates that even though L and M are mutually orthogonal,
neither Lc and Mc nor Ld and Md need be mutually orthogonal.
7.38 Theorem. Let M be a local martingale, T be a stopping time
and £ be a real TT-measurable r.v.. Then
N = f (M - MT)
is a local martingale, and for any local martingale L,
[N,L]=tt[M,L]-[M,L}T).
Proof, At first, we assume that M is a uniformly integrable martingale
and £ is bounded. Then for t > 0.
EiNaolft] = tfK(Moo - MT)\Tt] = tf[tfMoo - MT)\^T\Tt]
= E[Z(MtvT - MT)\Tt] = iI[T<t]{Mt - MtAT) = Nu
i.e., AT is a uniformly integrable martingale.
In the general case, without loss of generality we may assume Mo = 0.
Let (Tn) be a localizing sequence for M. Put
Sn = Tn A T[|j|>n], n > 1.
Then Sn | oo and (Sn) is a localizing sequence for N: for each n,
JVS» = [£(M - Mr)]5« = £%,<n,(M - MT)T«
= (Im<n}(MT»-MT^T)eM0.

204
Chapter VII Local Martingales
Finally, for any local martingale L, put
A = £([M,L]-[M,L]T).
We have AA = f A([M, L] - [M, L]T) = f A(M - MT)AL = ANAL and
NL - A = £(M - MT)L - f ([M, L] - [M, L]T)
= Z{(ML - [M, L\) - {ML - [M, L])T} - £MT/[r<oo](£ - LT)
is a local martingale by Theorem 7.31. Again by Theorem 7.31 we conclude
A = [N,L]. □
§3. The Characterization of Jumps of Local Martingales
7.39 Definition. An optional process X = (Xt) is said to be thin if
[X ^ 0] is a thin set. A typical example of thin process is the jump AX
of an adapted cadlag process X.
For any thin process X = (Xt), if for all t > 0,
£ \XS| < oo a.s.,
s<t
we define the summation process EX of X as follows:
EX = £ X5 or (EX), = £ X„ * > 0.
s<- s<t
EX is an adapted process with finite variation. In fact, if [X ^ 0] = UPnL
n
where (Tn) is a sequence of stopping times with disjoint graphs, then
EX = E ^Tn^[Tn,oo[-
n
In reality, we have already encountered the summation processes of thin
processes for many times before.
7.40 Theorem. Let M e Mfoc.
1) Me M^dc if and only if E(AM)2 <E A&..
2) M e Wioc,o if and only if E\AM\ e A^.
Proof. 1) follows from the localized Theorem 6.22 and Theorem 7.32.
2) follows from Theorems 7.15 and 7.26. □
7.41 Lemma. Let H be a thin process with Hq = 0. The following
assertions are equivalent:
1)A = VZIP<=A+C,
2)B = £(ff2/[|/f|<fc] + \H\Im>b]) e A& for any b > 0,

§3. The Characterization of Jumps of Local Martingales 205
4) D = E(l - y/TTH)2 € A&. ifH> -1.
Proof. Since
C < B < (1 + bWj^pfiH |<6] + ^E lfj^Wi * (1 + b + ^,
we have 2)^=»3) immediately. Noting that if y > -1,
{1, 2/->oo,
1/4, y-*0,
2, y-»-l,
we have two constants K\ > 0, if2 > 0 such that
Therefore, if H > -1, 3) ^=^4).
2) =>• 1). Let (5„) be a sequence of stopping times with Sn ] oo such
that E[BSn) < oo. Then A € V+. In fact, for all t> 0, £ |fl-|a/[|tf,|>j]
is only a sum of a finite number of terms. Put Tn = inf{t > 0 : At > n).
We have Tn | oo and
4r„ASn <n + &ATnASn <n + \HTnASn\<n + b + BSn.
Hence E[ATnASn] < oo, and A e Afoc.
1) => 2). Let (Sn) be a sequence of stopping times with Sn | oo such
that E[ASn] < oo. Then BgV+. In fact, for all* > 0, £ |#a|/|jj.|>&]
is only a sum of a finite number of terms. Put Tn = inf{t >0:Bt>n}.
We have Tn | oo and
BTnASn <n + A£TnASn < n + b2 + ASn.
Hence E[BTnASn] < oo, and £ e ^c. □
The following theorem describes the characterization for jumps of local
martingales. It is also a fundamental result of local martingales, and will
play an important role in the theory of stochastic integrals. Lemma 7.41
provides a useful technique when this characterization is concerned.
7.42 Theorem. In order that a thin process H be the jump AM of a
local martingale M it is necessary and sufficient that

206 Chapter VII Local Martingales
i) m = o,
Proof. The necessity follows from Theorems 7.13 and 7.30.
Sufficiency. At first, we assume EH2 G A*. Let (Tn) be a sequence of
strictly positive stopping times with disjoint graphs such that [H ^ 0] C
\J[Tn] and each Tn is either predictable or totally inaccessible. Put
n
An = HTnI[TnM, Mn = An-A".
Then as in the proof of Theorem 6.22, we know that the orthogonal series
EMn in M^. converges to an element M e M2,d and H = AM.
Now we assume T,H2 G A^c. Let (Tn) be a sequence of stopping times
with Tn | oo such that (EiJ2)Tn e A+. Then for each n there exists
Mn e M24 such that AMn = HI^Tny By Corollary 7.23 we have
(Mn+1)Tn = Mn, n > 1.
Then (Mn)n>i can be pieced together into a process
oo
M = EM\,,Tn| (T0 = 0).
71=1
Obviously, M e (AOioc = Mfoc andAM = H.
If E|#| € Aj^., then M = Etf - (EH) € Wloc,o and
AM = H -m = H.
Now we are in a position to deal with the general case. Put
A = EH2, K = HI[m>1], H" = K -PK, H' = H - H", B = E\K\.
It is not difficult to see that H" and H' are thin processes and P(H") =
^H') = 0. Since VEK2 < A, by Theorem 7.10 we have B € A^. On the
other hand,
E\PK\ < V[\K\) <Be A+c.
Therefore E\H"\ € A^.. There exists M" € .Mfc such that AM" = H".
Because |#'|2 < 2(|#|2 + \H"\2), E(H')2 € V+. Since PH = o',
H' = H-K + K = HI[]mi] -p(HIm<1}), \H'\ < 2.
Hence E(#')2 6 A^. and there exists M' € .M2^ such that AM' = H'.
Then M = M' + M" is the required local martingale. □
7.43 Corollary. Let H be a thin process such that PH = 0.
1) There exists M 6 M2>d (resp. M2^.) such that H = AM if and
only if EH2 € A+ (resp. A^.).

Problems and Complements
207
2) There exists M € Wo (resp. WW o) such that H = AM if and only
ifX\H\£A+(resp. A+c).
Problems and Complements
7.1 Let V be a stable vector space of adapted processes such that if £ is
a bounded .Fo-measurable r.v. and Xt = £, then X = (Xt) € V. X G X>ioc
if and only if there exists an increasing sequence (Tn) of stopping times
with Tn | oo such that for each n, XTnI[Tn>Q] G V.
7.2 Let A be a predictable process with finite variation, and a ^ 0.
If \AA = —a] is an evanescent set, then —— is a locally bounded
1 J a + AA J
process.
7.3 Let M be a local martingale. If M > 0 and jB[Mo] < oo, then M
is a supermartingale.
7.4 Let M be a local martingale. If AM > 0, M is quasi-left-
continuous.
7.5 Let M be a continuous local martingale, and T be a stopping time.
Then for almost all u 6 [T < oo] either there exists e > 0 such that M.(a;)
is constant on [T(u;), T(u;) + e], or there exist two sequences (tn) and (sn)
with £n | T(uj) and sn j T(u;) such that for all n, sn+i < tn+\ < sn < tn,
Mtn(u) > MT(uj)(u) > MSn(u>).
7.6 If M e M\OC}0 and M > 0, then M = 0.
7.7 Esich cadlag supermartingale X can be uniquely decomposed as:
X = M - A,
where M is a local martingale, and A is a predictable increasing process
with Aq = 0. Moreover, if X > 0, then A is integrable.
7.8 Let M e A4foc, M > 0 and lim*_>oo Mt = 0. Then for any a > 0,
P[sup|Mt|>a|Jo] = lA— •
7.9 Let W be a standard Wiener process and P be a Poisson process
with parameter 1. Put
Mt = Wt + Pt - t, t>0,
L = MS, S = inf{t > 0 : \Mt\ > 2}.
Then L is a bounded martingale, but Lc is not a bounded martingale.
7.10 Let M be a locally bounded martingale. Then for any local
martingale TV (M,N) exists.

208 Chapter VII Local Martingales
7.11 Let M be a local martingale and T be a stopping time. Set
Qt = Tt+u ~Mt = MT+*J[T<00], t > 0.
Then
1) M = (Mt) is a (^)-local martingale,
2) [M]t = ([M]T+t - [M]T + M?)/[T<oo],
3) M? =m+t -M$)I[T<oo],Mdt = (M*+t - M*)I[T<oo],
where [M], M° and M are all defined w.r.t. (Qt).
7.12 Let M, N € Moo S = ini{t > 0 : [M]t > 0} and T = inf{t >
0 : [N]t > 0}. Then [M,N]2 = [M][N] if and only if there exist two r.v.
£ € Ts and -q € Fr such that [5 V T < oo] = [S = T < oo] and
£^0, r/^0, £M-riN = 0 on [SvT<co].
7.13 Let M,iV € A^foc, S = inf{i > 0 : (M)t > 0} and T = inf{t >
0 : {N)t > 0}.
1) If (M, N)2 = (M)(N), then there exists a r.v. £ € ^s (resp. 7/ 6 -Fr)
such that [f ^ 0](resp. [r? ^ 0]) = [S V T < oo] = [S = T < oo] and
£M-N = 0 ( resp. M - r]N = 0) on [5vT<oo],
2) If there exists a r.v. £ 6 Fs- (resp. r/ € Ft-) such that [£ ^ 0]
(resp. [r/ ^ 0]) = [S V T < oo] = [S = T < oo] and
£M-N = 0 ( resp. M - 7}N = 0) on[SvT<oo],
then(M,JV)2 = (M)(JV).
7.14 Let M be a local martingale, and F be an optional set. Then
the following two assertions are equivalent:
1) M = Mx+M2, where M\M2 € Moc, AM1^ = 0 and AM2IF =
0,
2) p(AM7f) = 0.

Chapter VIII
Semimartingales and Quasimartingales
§1. Semimartingales and Special Semimartingales
8.1 Definition. A process X = (Xt) is called a semimartingale, if X
can be decomposed as follows:
X = M + A, (1.1)
where M is a local martingale, and A is an adapted process with finite
variation. Apparently, a semimaxtingale is an adapted cadlag process.
The class of all semimartingales is denoted by S. Obviously, S is a stable
vector space. In addition, if X is a semimaxtingale and T is a stopping
time, then
X ~ = A7[0 T[ + XT-I[t}oo[ = x - AXr/[r,oc[
is also a semimartingale.
By the fundamental theorem for local martingales (Theorem 7.17) we
know that in the decomposition (1.1) of a semimartingale X we may
assume that M is a locally bounded martingale (even the jump AM of M is
bounded). But the continuous maxtingale part Mc of M does not depend
on the decomposition (1.1), and is uniquely determined by X (Lemma
7.22). We denote it by Xc. Xc is also called the continuous martingale
part of semimaxtingale X. It is easy to see for any stopping time T
(XT)C = (XC)T, (XT-)C = (XC)T-
8.2 Definition. Let X and Y be two semimartingales. Define
[X, Y)t = Xo^o + (Xc, Yc)t + £ (AXSAF3), t > 0.
s<t
[X, Y] is an adapted process with finite variation, and is called the
quadratic covariation of X and Y. In fact, by Lemma 7.27 and (1.1) we know

210
Chapter VIII Semimartingales and Quasimartingales
that for any semimartingale X and t > 0
£(AX3)2<oo a.s..
s<t
[X, X], also denoted simply by [X], is an adapted increasing process, and is
called the quadratic variation of X. It is easy to see that for any stopping
timeT
[x,yt\ = [x,y\t, [x,yt-] = [x,y]t-.
If [X,Y] e A\oc, its dual predictable projection is denoted by (X, Y),
and is called the predictable quadratic covariation of X and Y. In this case
we say that (X,Y) exists. In particular, if [X] e -4j£c, its dual predictable
projection is denoted by (X), and is called predictable quadratic variation
oiX.
It is easy to see that Kunita-Watanabe inequality holds for
semimartingales.
8.3 Theorem. Let X and Y be two semimartingales, H and K be two
measureable processes, p and q be a pair of conjugate indices. Then
J \HsKs\\d[X,y\s\<{l H2sd[X\sf K2sd[Y)syl* a.s., (3.1)
y[0,oo[ •/[0,oo[ -/[0,oo[
E[f \HsKs\\d[X,Y}s\)<\\J[ #M*]J \\Jf K^d[Y]3\\ .
J[0,oo[ \\\ J[0,oo[ IIlpII y J[0,oo[ Wlv
(3.2)
Remark. If (X), (Y) and (X, Y) all exist, we have the corresponding
Kunita-Watanabe inequatity.
8.4 Definition. A semimartingale X is called a special
semimartingale, if it can be decomposed as follows:
X = M + A,
where M is a local martingale, and A is an adapted process with locally
integrable variation. If the special semimartingale X has another
decomposition: X = N + B, where N is local martingale and B is an adapted
process with finite variation, then B must be an adapted process with
locally integrable variation, too. In fact, B — A = M — TV is a local
martingale with finite variation. By Theorem 7.19, B — A e Wioc and
hence B G ^ioc- The class of all special semimartingales is denoted by Sp.
Obviously, Sp is a stable vector space.
8.5 Theorem. Every special semimartingale X can be uniquely
decomposed as follows:

§1. Semimartingales and Special Semimartingales
211
X = M + A, (5.1)
where M is a local martingale, A is a predictable process with finite
variation and Aq = 0. Hereinafter, we call this decomposition the canonical
decomposition of X.
Proof Let X = N + JB, where N e M\oc and B e A\oc,o- Putting
A = B and M = N + B — JB, we obtain the required decomposition (5.1).
The uniqueness follows from Remark 3) after Definition 7.11. □
The following theorem gives several useful characterizations for special
semimartingales.
8.6 Theorem. Let X be a semimartingale. Then the following
statements are equivalent:
1) X is a special semimartingale,
2) y/[X] is a locally integrable increasing process,
3) X* = (X£) is a locally integrable increasing process.
Proof 1)=>2). Let X be a special semimartingale, and X = M + A
be its canonical decomposition. By Kunita-Watanabe inequality
yJ\X\ < yj[M) + [A] + 2yJ[M)[A] = y/[M] + yl\A\.
Because y/\M] e A?oc (Theorem 7.30) and vPJ = ^X(AA)2 < E\AA\ e
2)=>3). Let y/\X] e Aoc Since (AXY ^ ^(AX)2 < y/\Z], X* <
{AX)* + (X-)*, and X- is locally bounded (Theorem 7.7.1)), we have
3)=>1). Let X* e Afoc, and X = M + A, where M e Moo A e V0.
Since M* e A^. (Corollary 7.18), we have A* e A^. Furthermore, by
Theorem 7.10 we know A e A\oc. Hence X e Sp. D
8.7 Corollary. A locally bounded semimartingale is a special
semimartingale. In particular, a semimartingale with bounded jump or a
predictable semimartingale is a special semimartingale.
8.8 Theorem. Let X be a special semimartingale, and X = M + A
be its canonical decomposition. If there exists a constant C > 0 such that
for any predictable time T > 0, \AXt\ < C a.s., then \AA\ < C.
Proof. Let T > 0 be a predictable time. By Theorem 7.13 we have
AAT = E[AAt\Pt-] = E[AXT - AMT\TT-} = E[AXt\Tt-] a.s..
Thus \AAT\ < C a.s.. But AA is predictable. Therefore \AA\ <C. □

212 Chapter VIII Semimartingales and Quasimartingales
8.9 Corollary. Let X be a quasi-left-continuous (resp. continuous)
special semimartingale, and X = M + A be its canonical decomposition.
Then A is continuous and M is quasi-left-continuous (resp. continuous).
8.10 Theorem. S and Sp are stable under localization, i.e., S\oc = S
and (<Sp)ioc = <Sp-
Proof. Let X G (Sp)\oc, and (Tn) be a localizing sequence for X.
Without loss of generality we may assume Xq = 0. Then each XTn is a
special semimaxtingale. Let
XTn =Mn + An
be its canonical decomposition. By the uniqueness of canonical
decomposition we have
(Mn+1)Tn = Mn, (An^)Tn = A71.
Put
oo oo
M = £ MnI]Tn_uTn], A=Z AnI]Tn_uTn] (To = 0).
71=1 71=1
Then M is a local martingale, A is a predictable process with finite
variation, and X = M + A, i.e., X is a special semimaxtingale. Hence (<Sp)ioc =
Sp.
Now let X £ 5ioc, and (Tn) be a localizing sequence for X. Since each
XTn £ 5, X is an adapted cadlag process. Put
vt=ZAXsi{\AXa>lh t>o.
s<t
V = (Vt) is an adapted process with finite variation. Then each (X —
V)Tn = XTn — VTn is a semimartingale, and its jump is bounded by 1.
Hence each (X — V)Tn is a special semimartingale (Corollary 8.7) and
X — V itself is a special semimartingale. Finally, X = (X — V) + V is &
semimartingale. Hence <Sioc = 5. □
Finally, we show that the semimartingale property is stable under
change of time.
8.11 Theorem. Assume that X is a semimartingale. Let r = (rt)
be a change of time, and for each t > 0, rt < oo. Put
Yt = xTt, gt = fTt, t>o.
Then Y = (Yt) is a semimartingale w.r.t. G = (Gt)-
Proof. Let X = M + A, where M is an (Ji)-local martingale with
Mo = 0, and A is an (^i)-adapted process with finite variation. Then
Y = N + B, Nt = MTt, Bt = ATt, t>0.

§2. Quasimartingales and Their Rao Decompositions 213
Obviously, B = (Bt) is a ((/^-adapted process with finite variation. It
remains to show that TV = (Nt) is a (^)-semimartingale. Let (Tn) be a
localizing sequence for M, i.e., for each n, MTn is a uniformly integrable
(.^-martingale. Put
Tn = inf {* >0:rt> T„}, n > 1.
Since for each t > 0, [Tn < t] = [rt > Tn] e TTt = Gt, each Tn is a
(£*)-stopping time, and Tn | oo. Put
A^ = M^=MTtATn,*>0.
By Doob's stopping theorem we know that each Nn is a uniformly
integrable (^)-martingale. Furthermore, we have [Tn > t] = [rt < Tn],
and
= N» + (NTn - N±n)I[TnM.
This means NTn is a (^)-semimartingale. By Theorem 8.10 TV is a (Gt)-
semimartingale. □
§2. Quasimartingales and Their Rao Decompositions
8.12 Definition. Let X be an adapted cadlag process. X is called a
quasimartingale, if for all t > 0, Xt is integrable and
Vu(X) = Bup{J:E[\Xu-E[Xu+l\Fti]\] + E[\Xtn\]}<+<x>t (12.1)
T ^ i=0 }
where r : 0 = to < ^1 < * • • < tn < +00 is a finite partition of [0, oo[, and
the supremum is taken over the set of all finite partitions of [0,00[.
If an adapted cadlag process X is not a quasimartingale, we denote
Var(X) = +00.

214 Chapter VIII Semimartingales and Quasimartingales
Let X be a uniformly integrable martingale. Then
Var(X) = sup JB[|Xt|] = £[|*oo|] < oo,
t
and X is a quasimartingale.
Let X be a non-negative cadlag supermartingale. Then
Vai(X) = E[X0] < oo,
and hence X is a quasimartingale, too.
Obviously, if X and Y are two quasimartingales, so is X + Y. In
addition, we have
Var(X + Y) < Vai(X) + Var(r). (12.2)
The following theorem is called Rao's decomposition theorem for
quasimartingales.
8.13 Theorem. Let X be an adapted cadlag process. Then X is
a quasimartingale if and only if X is the difference of two non-negative
cadlag supermartingales. In this case, X can be uniquely decomposed as
follows:
X = X' - X", (13.1)
where X' and X" are two non-negative cadlag supermartingales such that
Vai(X) = E[X'Q + Xf(}}. (13.2)
(13.1) is called the Rao decomposition of quasimartingale X.
Proof. The sufficiency is trivial. We are to show the necessity. Let
X be a quasimartingale, and r : t = to < t\ < • • • < tn < oo be a finite
partition of [£, oo[. Put
Ui(r) = E[J:{Xtt-E[Xtt+1\Ftl])++X+\rt],
1 1=0 J
u?(t) = E[n±\xti - E[xti+1\rti})- + xrM.
L 1=0 J
For t<s<u<vwe have
(Xs - E [XV\FS})+ = (X. - E[XU\FS\ + E[XU\FS\ - E[XV\?S])+
< (X. - E[XU\FS\)+ + (E[XU - E[Xv\Fu]\ra])+
< (Xa - E[Xu\Fa})+ + (E[XU - E[Xv\Fu])+\ra].
Hence
E[(Xa - E[Xv\Fs])+\rt] <
E[(Xa - E[Xu\ra])+\Ft] + E[(XU - E[Xv\ru])+\Ft].

§2. Quasimartingales and Their Rao Decompositions 215
In the same way we have
E[X+\ft] < E[(XU - E[Xv\Fu])+\Ft] + E[X+\rt\.
Therefore, if partition r' is a refinement of partition r,
U't{r) < Ui(r'), U?(t) < U'^r').
But for any finite partition r of [t, oo[,
E[Ui(r)] < Var(X), E[U?(r)] < Var(X).
Then in the direction of partly ordered set of all finite partitions of [t, oo[
U[(t) and U"(t) converge in Ll. Their Umits are denoted by U[ and U"
respectively (In fact, U't = ess supC/^r) and U" = ess supC//;(r).). For
r r
S<t,
E[U[{r) \FS] = E[nEQ(Xti - E[Xti+1\Tti\r + X+\Fa]
< E[(XS - E[Xt\F3})+ + n±\xti - EHX^fc])* +X£\F,\
= U'3{t)<U's, a.s.,
where r is a finite partition of [5,00[. Thus we have
E[tft\Fa]<U'„ a.s.,
i.e., (U't) is a non-negative supermartingale. By the same reason, (£/") is
a non-negative supermartingale, too. By Follmer's lemma there exist two
non-negative cadlag supermartingales (Xft) and (X") such that for almost
all u) we have
X't(u)= Km £/», X't'(u)= lim 0»
seQ+,siit seQ+Jsiit
for all t > 0. Since for any finite partition r of [£, oo[.
Xt = Ui(r)-Uif(r) a.s.,
we have Xt = U[ - U't' a.s.. By the right-continuity of X, Xt = X[ - X't'
a.s.. Again by the right-continuity of X, X' and X" we conclude that X
is indistinguishable from X' - X". Moreover,
Var(X) = supEp^r) + Ug{r)] = E[U'Q + U^}
T
> lim E[U'9 + U'J] > E[X'Q + X'J] (by Fatou's lemma)
seQ+,siio
= Var(X') + Var(X").
By virtue of (12.2), (13.2) holds.

216 Chapter VIII Semimartingales and Quasimartingales
What remains is to show the uniqueness. Let X = X1 — X2, where
X1 and X2 are two non-negative cadlag supermartingales such that
Vai(X) = E[XZ+X$}.
For s < t,
(E[X9-Xt\fs})+ = {E[Xl-Xl\Fa]-E[Xl-X2t\?a))+ < E[X]-Xl\?,\.
Whence it is easy to deduce
U't < Xl E[U'S - tft\Fa\ < E[Xl - Xl\Fa\.
Hence X1 — U' is a non-negative supermaxtingale, and X1 — X' is a non-
negative cadlag supermaxtingale. By the same reason, X2 — X" is a non-
negative cadlag supermaxtingale, too. By the assumption, E[XfQ + Xq] =
Var(X) = E[X& + Xg], thus Xl - X'0 = Xl - X'£ = 0 a.s.. Furthermore,
for all* > 0, X} - X[ = X2 - Xf( = 0 a.s.. Then X1 = X1 and X2 = X".
D
8.14 Theorem. Assume that X is a quasimartingale. Let r = (r*) be
a change of time, and for each t > 0, t* < oo. Put
Yt = XTt, Gt=fTt, t>0.
Then Y = (Yt) is a quasimartingale w.r.t. (Gt)-
Proof. By Theorem 8.13 we may assume that X is a non-negative
cadlag supermaxtingale (w.r.t. (Ft))- In this case, by Doob's stopping
theorem we know immediately that Y is a non-negative cadlag supermar-
tingale w.r.t. (Gt)- Hence Y is a quasimartingale w.r.t. (Gt). □
The quasimartingale property is stable not only under change of time
but also under reduction of filtration.
8.15 Theorem. Let (Gt) be a filtration satisfying the usual conditions
such that for each t > 0, Gt C Tt. Suppose X is an (Tt)-quasimartingale
and (Gt)-adapted. Then X is a (Gt)-quasimartingale.
Proof. For 0 < s < t < oo,
E[\XS -E[Xt\Gs]\] = E[\E[X3 - Xt\Fs\Gs]\]
< E[\E[XS - Xt\fs}\] = E[\XS - E[Xt\fs}\}.
By (12.1) we have
Var(X)(ft) < Var(X)(7i) < oo,
i.e., X is a (^)-quasimaxtingale. □

§3. Semimartingales on Stochastic Sets of Interval Type 217
§3. Semimartingales on Stochastic Sets of Interval Type
8.16 Definition. B C ft x il+ is called a set of interval type if there
is a non-negative r.v. T such that for each uj the section B^ is [0,T(u;)[
or[0,T(u;)] and £„ ^0.
8.17 Theorem. B is an optional set of interval type if and only if
Ib = IfI\q,t[ + ^Fc J[o,r]» (17-1)
where T is a stopping time, F G Tt and Tp > 0.
Proof. The sufficiency is trivial. We want to show the necessity. Put
T(o;) = inf{* : (u>,t) e Bc}, (17.2)
F = {u : T(cj) < oo, (w,r(cj)) € Bc). (17.3)
Then T is a stopping time. Since Ip = 1 ^^ ^bc(T')^[t<oo] = 1, we have
F G ^r- Now it is easy to verify (17.1) □
8.18 Theorem. The following statements are equivalent:
1) B is a predictable set of interval type.
2) Ib = IfI[qj\ + IfcI[q,T\, where T is a stopping time, F € Tt- and
Tp > 0 is a predictable time.
3) B = U[0>2n]> where (Tn) is an increasing sequence of stopping
71
times (it is called a fundamental sequence for B).
Proof. 1)=>2). Let T and F be defined by (17.2) and (17.3). Since B
is predictable, we have F G Tt- • Since [Tp] = [0, T] n Bc is predictable,
Tp is a predictable time.
2)=>3). We may suppose F C [T < oo]. Otherwise, we may replace F
by F[T < oo], and (17.1) remains true. Let (5n) be a sequence of stopping
times foretelling Tp. Put Tn = SnAT. Then it is straightforward to check
B = u[o,r„].
71
3)=>1) is obvious. □
8.19 Definition. Let B be an optional set of interval type, and X be
a process defined on B (i.e., XIb is an ordinary process). If there exists
an increasing sequence (Tn) of stopping times with Tn | T (T is the debut
of Bc) and a sequence of semimartingales (Xn) such that
U[0,Tn]DB and (XIB)T» = (XnIBf\ (19.1)
71
X is called a semimartingale on JB, and (Tn,Xn) is called a fundamental
coupled sequence for X. The collection of all semimartingales on B is de-

218 Chapter VIII Semimartingales and Quasimartingales
noted by SB. In the same manner we can define (SP)B, (M\oc)B, (A4foc)5,
(^foc)B,Mloc)B,VB,---.
8.20 Theorem. Let B be an optional set of interval type and X £ SB-
Let S be a stopping time such that [0, S] C B. Then Xs G S.
Proof Let (Tn, Xn) be a fundamental coupled sequence for X. Put
Sn = (Tn)[Tn<S]'
Since \J[Tn > S] = ft, Sn ] oo. It is easy to see from (19.1) that
71
(xs)Sn = xSATn = (xn)SATn e s.
Hence Xs e S\oc = S. □
Remark. The theorem holds for any class VB (e.g. {M\oc)B, (SP)B,
Mloc)B> ), whenever V is stable under localization: V\oc = V.
8.21 Theorem. Let B be a predictale set of interval type and X be a
process defined on B. Then the following statements are equivalent:
1) Xe SB,
2) For each stopping time S satisfying [0,5] C B, Xs G S,
3) There is a fundamental sequence (Tn) for B such that for each
n,XTn eS.
Proof 1)=>2) follows from Theorem 8.20. 2)=>3) it trivial. 3)=>1) is
easy, since in this case (Tn,XTn) is a fundamental coupled sequence for
X. □
8.22 Theorem. Let B be an optional set of interval type, and X be
a process defined on B. If X € SB, then there exists a predictable set of
interval type B D B and X G SB such that XIb = XIb, *•£•> X is the
restriction of X on B.
Proof Let (Tn, Xn) be a fundamental coupled sequence for X, and T
be the debut of Bc. Put
Ai = [T1=T< oo], Ak = [Tk = T < oo,Tfc_! < T], k > 2.
Then (Ak)k>i is a sequence of disjoint sets. Define
X = XI[Q Tr + J2 ^t^aJit.ooI
k=l
It is not difficult to see that X coincides with X on JB, and by induction,
xT> = (^Vnf1 +x1Ti[Tl=T<oo]ilT<ool = (x1)^,

§3. Semimartingales on Stochastic Sets of Interval Type 219
71+1
fc=l
= XT» + (X^1 J|0fT|)T-+1 - (^n+1/[0,T[)^
+ X^1(I[Tn+l=T<OQ] - lTn=T<oo])I[T,oo[
= XT» + (Xn+1)T»+1-(Xfl+1)T».
Thus for each n, XTn G <S, and X G <S5, where B = |J[0, Tn] D B. □
n
Remark. The theorem holds for any class VB whenever V is stable
under stopping.
Theorems 8.22 and 8.21 open a way to investigate semimartingales,
local martingales, • • • on an optional set of interval type. As an example,
we discuss the decomposition of local martingales.
8.23 Theorem. Let B be an optional set of interval type and M £
(M\oc)B. Then M can be uniquely decomposed as
M = M0 + MC + Md,
where Mc <E (A4foc0)B and Md <E (Mfoc)B.
Proof. In order to show the existence, by Theorem 8.22 we may
consider B as a predictable set of interval type. Let (Tn) be a fundamental
sequence for B. For each n,
MTn =M0 + Ln + Nn,
where Ln e M^ 0 and Nn e M(oc. By the uniqueness of decomposition
(Ln+l)Tn = Ln? (JVn+1)T- = Nn.
Then it is easy to see
oo oo
Mc=ZLnI]Tn,1>Tn] and Md = £ ^/]Tn_liTnJ (T0 = 0)
71=1 71=1
satisfy all the requirements.
We want to show the uniqueness. If M has another decomposition of
the same type:
M = M0 + Mc + Md.
By Theorem 8.22 we may consider Mc and Mc (resp. Md and Md) as
continuous (resp. purely discontinuous) local martingales on ^predictable
set of interval type B D B. Then N = (Mc + Md) - (Mc + Md) e

220 Chapter VIII Semimartingales and Quasimartingales
(M\OCio)B, and NIb = 0. Let (Tn) be a fundamental sequence for B, and
T be the debut of Bc. For each n,
NTn = NTl[Tn=T«x>]IlTM € Alfoc-
Hence 0 = (NTn)c = (MC)T» - (MC)T*. Therefore, MCIB = MCIB, and
the uniqueness is established. □
8.24 Remark. It is odd enough that a local martingale on an optional
set of interval type B with continouous trajectories on B need not be a
continuous local martingale on B. For example, let T > 0 be a totally
inaccessible time with P(T < oo) > 0, and B = [0,T[. Put A = I[t,oc[
and M = AT[o,t[- Then M € (M\oc)B, and all the trajectories of M are
continuous on B. But M is not a continuous local martingale on B.
By the same argument used in the proof of Theorem 8.23 one may
show the following theorems. The details are left to readers.
8.25 Theorem. Let B be an optional set of interval type, and M,N G
(M\oc)B. Then there exists a unique process [M, N] G VB such that MN-
[M,N] € (A<ioc,o)B and A[M, N] = AM AN on B.
8.26 Theorem. Let B be an optional set of interval type, and X G
(SP)B. Then X has the following unique canonical decomposition
X = M + A,
where M € (A^ioc)^ and A € (-4ioc,o)5 is predictable (i.e., A is the re-
striction of a predictable process on B).
In particular, for any A € (.4ioc)5 there exists a unique predictable
A € (A\oc)B such that A — A e (M\OCio)B'. A is also called the dual
predictable projection or compensator of A.
§4. Convergence Theorems for Semimartingales
8.27 Definition. Let X = (Xt)t>o be an adapted process. Denote
[X —>] = {lj : lim Xt(uj) exists and is finite.}.
£—►00
It is natural to define Xoo = lim Xt on [X —►].
t—>oo
8.28 Theorem. LetX = M + B, where M e Moc.o and B 6 A^

§4. Convergence Theorems for Semimartingales 221
If for every stopping time T, E[X£ A (AXt)* I[t«x>]] < °°> then
[M -+][B ->] = [X -+} = [sup |Xt| < oo] = [snpXt < oo] a.s..
t t
Proof. Clearly, we have
[M -+][B ->] C [X ->] C [sup \Xt\ < oo] C [snpXt < oo].
t t
Put Sn = inf{t >0:Xt> n}. Then Sn > 0 and
X5" < n + [X+ A (AX5n)+]/[sn<oc] = ^,
where Un > 0 and jB[C/n] < oo. Let F^71^ be a uniformly integrable
martingale such that Yt{n) = JB[I7n| Ji] a.s.. Then
Z(n) = yin) _ XSn > 0
By the assumption there exists a localizing sequence (Tfc) for M and B, i.e.,
for each A;, MTk e M and BTk e A+. Thus (Z(n))Tfc is a supermartingale.
But E[Z^n)] < oo, by means of Fatou's lemma we know that Z^ is a
non-negative supermartingale, and
P([Z<n> _>]) = l.
Since P([YW ->]) = 1, we obtain P([X5" ->]) = 1.
Obviously, we have
[supX* < oo] C \J[Sn = oo] C [X -►] a.s.. (28.2)
t n
Since JB > 0,
M5n = XSn - BSn < XSn < Un.
Applying the above argument to W^ = Y^ — MSn > 0 yields
[supXt < oo] C \J[Sn = oo] C [M ->] a.s.. (28.3)
t n
Combining (28.3) with (28.2) gives
[supXt < oo] C [X ->][M ->] = [M -+][B ->] a.s..
t
Hence (28.1) holds. □
8.29 Theorem. Let X = M + B, where M is a local martingale,
and B is a predictable increasing process with Bq = 0. If X > 0 and
E[X0] < oo, then
[B ->] = [X ->][M ->] a.s.. (29.1)
Proof Put Tn = inf {t >0: Bt>n}. Then Tn > 0 is predictable, and
y(») = -MT"- + X0 = BTn" - XT-~ + X0 < n + X0.

222 Chapter VIII Semimartingales and Quasimartingales
Since Y^ is a local martingale and for every stopping time T, E[(Ypn')+
I[T<oo]] < °°, applying Theorem 8.28 to Y^ yields
P([MTn~ ->]) = p(sup(-MfTn-) < oo) = 1.
Clearly, we have
[£HcU[T„ = oo]c[M->], a.s.,
n
[fl-]c[Af->][*">] a.s..
The reverse imphcation is trivial. Hence (29.1) holds. □
8.30 Corollary. Let B be an adapted locally integrable increasing
process. Then
1) [Boo <oc] C [Boo < oo].
2) If for every stopping time T, E[ABtI[t<oo]] < °°> then
[Boo < OO] = [Boo < OO].
Proof. Without loss of generality, we may assume Bo = Bo = 0. Since
B = M + B, M = B-B e Moc,o, applying Theorem 8.29 to B gives 1)
and applying Theorem 8.28 leads to 2). □
8.31 Corollary. Let M be a local martingale. If for every stopping
time T, E[(\MT\ A |AMt|)/[t<oo]] < oo, then
[M —>] = [supM* < oo] = [inf Mt > — oo] a.s.,
t t
i.e., for almost all lj either lim M*(oo) exists and is finite, or we have
t—►oo
both lim sup M*(u;) = +oo and lim inf Mt(uj) = —oo.
*—oo *->°o
Proof. We may assume Mq = 0. Then it suffices to apply Theorem
8.28 to M and -M (B = 0) respectively. □
8.32 Theorem. Let M be a locally square integrable martingale. Then
[(M) ->] C [M ->]. a.s..
Moreover, if for every stopping time T, E[(AMt)2I[t<oo]] < °°> then
[<M> -] = [[M] ->] = [M -]. a.*..
Proof. We may assume Mo = 0. Put
Tn = mi{t > 0 : (M)t > n}.
Then Tn is predictable, and MTn~ is a local martingale. Because
(MTn-) = (M)T"- < n,

Problems and Complements 223
MTn~ is a square integrable martingale, and P([MTn~ —>]) = 1. Hence
[(M) ->] C \J[Tn = oo] C [M ->] a.s..
n
If for every stopping time T, jE[(AMx)2 Jpr<00]] < oo, by Corollary
8.30, [(AT) ->] = [[M] ->] a.s.. Put
Sn = inf{*>0: \Mt\ >n}.
Then
Ms»<n+\AMSn\I[Sn<oo}.
MSn is a square integrable martingale, and (M)sn = (M5n)oo < oo a.s..
Hence
[M^]c\J[Sn = oo]c[(M)-+] a.s.. D
n
8.33 Theorem. Let X = M + B, where M is a local martingale and
B is a predictable increasing process. If AX is bounded, then
[X ->] = [M -+][B ->] = [(M) + BH = [(X) + B-+] a.s..
Proof. We may assume Mo = Bo = 0. By Theorem 8.8 we know that
AB is bounded, and so is AM. Hence M is a locally square integrable
martingale. By Theorems 8.28 and 8.32 we have
[X ->] = [M -+][B ->] = [(M> + B-+] a.s..
On the other hand, [X] = [M] + 2[M,B] + [B], [B] unpredictable, [M,B] =
(AM).B is a local martingale ([M, B] e A\oc and [M, B] = P(AM).B = 0),
we have
(X) = (M) + [B).
Evidently, on [B ->], £ AB3 < B^ < oo, and [£]«, = £ (ABS)2 < oo.
3>0 3>0
Hence
[(M> + B ->] = [(X) + B ->] a.s.. D
The above convergence results are also available for local martingales
and semimartingales defined on stochastic sets of interval type discussed
in the previous paragraph.
Problems and Complements
8.1 Let X be an adapted cadlag process. If there exists a sequence (Tn)
of stopping times with Tn | oo and a sequence (X^) of semimartingales
such that for each n XTn- = (X^)Tn-, then X is a semimartingale.

224
Chapter VIII Semimartingales and Quasimartingales
8.2 Let X be a special semimartingale and T be a stopping time. Then
AXtI[t<oo] ls cr-integrable w.r.t. Tt--
8.3 Denote by V the collection of all optional processes of class (D).
Then5p = 5nDioc.
8.4 X is a predictable semimartingale if and only if X = M+A, where
M is a continuous local martingale and A is a predictable process with
finite variation.
8.5 Let X e S. If JB[[X]oo] < oo, then X e Sp. In addition, if
X = M + A is the canonical decomposition of X,then M € M^.
8.6 Put <S* = {X e S : X = M + A,M e M\oc,A e V0}. 1) Let
X e S. Then X e S* if and only if for each t > 0, £ \AXS\ < oo, a.s.. 2)
3<t
X £ S* if there exists a sequence (Tn) of stopping times with Tn | oo and a
sequence (X^) of elements in S* such that for each n, XTn~ = (X^)Tn-.
In particular, (<S*)ioc = S*.
8.7 Let X be an adapted integrable increasing process. Then X is a
quasimartingale. Find its Rao decomposition.
8.8 Let X be a cadlag supermartingale. Then X is a quasimartingale
if and only if supE[Xf] < oo, and Vai(X) = E[X0] + 2supE[Xf].
t t
8.9 Let X be an adapted cadlag process.
1) If S > T are two stopping times, then Vax(Xs) > Vai(XT).
2) If (Tn) is a sequence of stopping times such that Tn | oo, then
Var(X) = supVar(XT").
n
8.10 Let 7J, be the collection of all bounded stopping times. Let M be
an optional process. Put ||M||i = sup{jB[|MT|] : T e Tb}. If ||M||i < oo,
we say that M is bounded in L1.
Let M be a local martingale. Then M is a quasimartingale if and only
if M is bounded in L1. In this case, Var(M) = ||M||i.
8.11 Let X be an adapted cadlag process. Then X is a quasimartingale
if and only if X = M + A, where M is a local martingale, bounded in L1,
and A is a predictable process with integrable variation and Aq = 0. In
addition, such decomposition of a quasimartingale is unique.
8.12 Denote by Q the class of all quasimartingales. Then Sp = Q\oc-
8.13 Let M be a local martingale, bounded in L1. Then M can
be uniquely decomposed as follows:M = M' — M", where M' and M"
are non-negative local martingales and ||M||i = ||M'||i + ||M"||i. (This
decomposition is called Krickeberg-Kazamaki decomposition.)
8.14 Let M be a local martingale, bounded in Ll. Let (Gt) be a
filtration satisfying the usuall conditions such that for each t>0,ftC ^i-
If M is (^)-adapted, then M is also a (^)-local martingale, bounded in

Problems and Complements
225
L1.
8.15 Let X be a local martingale. Let (t$) be a continuous change of
time such that for each t > 0,t* < oo. Put Yt = XTt, Gt = TTt, t > 0.
Then Y = (Yt) is a (</*)-local martingale.
8.16 Let B be a predictable set of interval type, and M be a local
martingale on B. Then there exists a fundamental sequence (Tn) for B
such that for each n, MTn is a uniformly integrable martingale.
8.17 Let (Tn) be a sequence of stopping times, supTn = T, and M
71
be a process defined on [0,T[. If for each n,M is a local martingale on
[0,Tn[, then M is a local martingale on [0,T[.
8.18 Let X = M + JB, where M is a local martingale and B is a
predictable process with finite variation and JBo = 0. If X > 0 and
E[X0] < oo, then [B+ ->] = [X -+][M -+][B~ ->].
8.19 Let X = M + B, where M is a local martingale and B is a
predictable process with finite variation and J3o = 0. If AX is bounded,
then [M ->, B++B- ->] = [inf Xt > -oo][J8+ ->] = [supXt < oo][J3" ->].
8.20 Let X G«S and X = M + A be a decomposition of X, where
M e Moc and AeV. Set
in(MfA) = «[lA(^[A^ + yP \dA\)]+SUPE[lA\AMT*n\],
where T runs through the set of all stopping times,
Il*ll5,n= fof >(Af,.4),
where infimiiTn is taken over all decompositions of X, and
oo
11*115 = E 2-n||X||5,n.
n=l
Then 1) S with d(X, F) = ||X — y||,s is a complete metric space (the
topology induced by this metric is called Emery topology),
2) Let Xn,X G S. If there exists a sequence (T^) of stopping times
with Tk T oo such that for each jfc, (Xn - X)*Tk_ A 0 then \\Xn - X\\s -> 0,
3) Let Xn, X € 5. If ||Xn - X\\s -> 0, then there exists a subsequence
(X71') and a sequence (Tk) of stopping times with Tk T oo such that for
each it, (X71' -X)*Tk_-^0.

Chapter IX
Stochastic Integrals
The stochastic integrals we will define axe of the form / HsdXs or
Mt]
/ HsdXs, where both the integrand (Ht) and the integrator (Xt) are
Jo
stochastic processes. In 1944, K. Ito first defined the stochastic integrals
of adapted measurable processes w.r.t. a Brownian motion. The key
character of this kind of stochastic integrals is that the processes produced
by integration axe martingales (or, more generally, local martingales). In
1967, H. Kunita and S. Watanabe defined the stochastic integrals of a class
of adapted measurable processes w.r.t. general squaxe integrable
martingales, and took a crucial step in developing modern theory of stochastic
integrals. In 1970, C. Doleans-Dade and P. A. Meyer investigated the
stochastic integrals of locally bounded predictable processes w.r.t.
local martingales or semimaxtingales. In 1976, P. A. Meyer discussed the
stochastic integrals of optional processes w.r.t. local martingales. In 1979,
J. Jacod found the reasonable way to define the stochastic integrals of
non-bounded predictable processes w.r.t. semimaxtingales. In this
chapter we introduce the definition and fundamental properties of stochastic
integrals (§1—§3). In §4 we present the very useful Lenglart's inequality,
and by means of it study the continuity of stochastic integrals w.r.t.
integrand processes. In §5 we deal with the change of vaxiables formula (Ito
formula) and Doleans-Dade exponential formula for semimartingales. In
§6 we introduce local times of semimaxtingales and a generalization of Ito
formula. In §7 a short discussion on stochastic differential equations, by
using Metivier-Pellaumail's approach, is given.

§1. Stochastic Integrals of Predictable Processes w.r.t. Local Martingales 227
§1. Stochastic Integrals of Predictable Processes with
Respect to Local Martingales
In this paragraph, we will define (indefinite) stochastic integrals of
predictable processes w.r.t. local martingales such that the resulted integrals
are still local martingales. At first, for elementary predictable processes
we can define stochastic integrals in a natural manner. And it is easy to
find the characterization for this kind of stochastic integrals. Then, based
on this characterization, the definition of stochastic intergals for general
predictable processes is given.
Let S and T be a pair of stopping times, and S <T. Let £ be a real
.Fs-measurable r.v.. Put H = £I]syT\' Then H is a predictable process.
Let M be a local martingale. The stochastic integral of H w.r.t. M,
denoted also by H.M, should be defined reasonably as
(H.M)t = f (MtAT - MtAS), t > 0.
By Theorem 7.38 we know that H.M is a local martingale, and for every
local martingale N we have
[H.M, N] = £([M, N]T - [M, N]s) = H.[M,N],
where H.[M, N] is an indefinite Stieltjes integral. In addition, by Theorem
7.31 we know that H.M, defined above, is the unique local martingale L
such that for every local martingale N,
[L,N] = H.[M,N\. (1.1)
Enlightened by this example, we introduce the following definition of
stochastic integrals.
9.1 Definition. Let M be a local martingale, and H be a predictable
process. If there exists a local martingale L such that (1.1) holds for
every local martingale TV (this tacitly implies that H is integrable w.r.t.
[Af, N]), then we say that H is integrable w.r.t. Min the domain of local
martingales (or simply, integrable), and L (it is uniquely determined by
Theorem 7.31) is called the stochastic integral of H w.r.t. M, and denoted
by H.M. The collection of all predictable processes which are integrable
w.r.t. M is denoted by Lm(M).

228 Chapter IX Stochastic Integrals
The following theorem gives the characterization for elements in the
Lm(M).
9.2 Theorem. Let M be a local martingale, and H be a predictable
process. Then H e Lm(M) if and only if y/H2.[M] G A^.
Proof Necessity. Putting N = H.M in (1.1) yields
[H.M] = H.[M,H.M] = H2.[M\.
Since H.M e Mioc, by Theorem 7.30 y/H2\M) = y/\HM] e A^.
Sufficiency. It suffices to show that there exist V G M^q and L" €
Mf^ such that for every TV £ Mioc,
[L\N] = H.[MC,N], (2.1)
[L",N] = H.[Md,N\. (2.2)
Then L = HqMq + V + V is the required local martingale.
By Theorem 7.42 and Corollary 7.23, there exists a unique L" G Mf^
such that AL" = HAM. Hence (2.2) holds.
We want to show the existence of V. At first, we assume
E[{ti1.[Me\)00\ < oo.
By Kunita-Watanabe inequality (Theorem 6.33), for every N £ MQ'C
E[J~\H,\\d[Me,N]a\] < (E[f~ H*d[M*\a)*(E[N]00)1'2.
Thus <p(N) = E[f°° Had[McfN]a] is a bounded linear functional on
the Hilbert space M$ . By Riesz representation theorem, there exists a
unique V e M%° such that for all N e Ml'c,
E[L', Ml,* = EiL'^Nn] = E[j°° Hsd[Mc,N]s\ (2.3)
Let T be a stopping time. Replacing TV by NT in (2.3) yields
rp
E[L',N]T = E[J Had[Mc,N]a].
By Theorem 4.40 we know A = [L',N] - H.[MC,N] € M. But A is a
continuous adapted process with finite variation and Aq = 0. By Theorem
6.3.2) A = 0, i.e.,
[L',N] = H.[MC,N].
Now it is easy to see that (2.1) holds for every N € Mioc-

§1. Stochastic Integrals of Predictable Processes w.r.t. Local Martingales 229
Generally, we have a sequence (Tn) of stopping times such that Tn | oo
and for each n, E[(H2.[Mc])tJ < oo. Applying the result obtained above
to each local martingale (Mc)Tn, we have L^ e M%c such that for all
N € Mioc,
[L(n),N] = H.[Mc,N]Tn.
In view of the uniqueness, for each n, (z/71*1))7* = Z,(n). By "piecing
together", we have V e M2^ and (2.1) holds for all N G Mioc. □
The following theorem summarizes the fundamental properties of
stochastic integrals.
9.3 Theorem. Let M be a local martingale, H,K e L^M).
1) Lm(M) = Lm(Mc) n Lm(Md), (H.M)o = HQMQ, (H.M)C = H.MC,
(H.M)d = H.Md.
2) A(H.M) = HAM.
3) H + K e Lm(M), and (# + K).M = H.M + K.M.
4) Le£ H' be a predictable process. Then H1 G Lm(H.M) if and only
if(HH') £ Lm(M). In this case, we have
H'.(H.M) = (H'H).M.
5) Let T be a stopping time. Then
(H.Mf = H.MT = (HI[QjT]).M.
Proof. 1) and 2) have been included in the proof of Theorem 9.2. 3)-5)
are apparent. □
As usual, we also use the following notations for stochastic integrals:
for t > 0,
/ HsdMs = {H.M)U
( H3dM3 = f HsdMs = {(HI]QM).M)t.
Jo J]o,t]
The concept of stochastic integral will be generalized below, but we always
use the same notations for stochastic integrals.
9.4 Examples. 1) Let M be a local martingale, and A be a predictable
process with finite variation. Then A A € Lm(M) and
(AA).M = [M, A] - MqAq. (4.1)

230
Chapter IX Stochastic Integrals
In fact, AA is locally bounded (Theorem 7.7), AA is integrable w.r.t.
[M, N] for all N e Mioc- On the other hand, we have already known that
[M, A] — MoAo is a local martingale (see the proof of Theorem 8.33). It
is easy to see for all TV £ Mioc,
[[M,A] - M0Ad, TV] = Zi&MAAAN) = (AA).[M,N\.
By Definition 9.1, (4.1) holds.
This result is called Yceurp 's lemma.
2) Let M be a local martingale, and T > 0 be a predictable time.
Then
I[T\M = AMT/[T,oo[
In fact, it is well known that AMx/pr,oo[ *s a local martingale and for
all N € Moo [AMTI[TM, N] = AMTA/Vr/p\ool = Im\M, N].
9.5 Theorem. Let M be a local martingale with locally integrable
variation, and H be a predictable process.
1) //£ \HAM\ e A^, then H <E Lm(M) and
(H.M)t(w)= [ Hs(uj)dMs(u), t>0, (5.1)
where the right hand side of (5.1) is a Stieltjes integral. For the sake of
clarity, we denote it by H-SM sometimes.
2) IfZ\HAM\ e V+ and H e Lm(M), then (5.1) holds as well.
Proof. 1) Since £ \HAM\ e Af^, bY Theorem 6.2 the Stieltjes integral
H-SM exists, and H*SM £ Mioc by Theorem 6.5. On the other hand,
slH\[M\<T.\H^M\€Atoc,
thus H.M exists, and A(H.M) = HAM = A(H$M). Because H.M -
(H.M)o and H'SM — (H$M)o are all purely discontinuous local
martingales, and (H.M)o = HqMq = {H-sM)q, we have H.M = H-SM.
2) Let (T„) be a sequence of stopping times such that Tn | oo and for
each n, e[ / £ ^2AMS2] < oo. Put
5n = inf{* > 0 : J2 \HaAM3\ > n} A T„.
s<t
Then S„ | oo and for each n, £?[ £ |#SAMS|] < oo. Hence £ |iJAM| G
.4^, and 2) follows from 1). □

§2. Compensated Stochastic Integrals of Progressive Processes 231
Theorem 9.5 illustrates that our predictable stochastic integrals (i.e.,
the integrals of predictable processes) axe just Stieltjes integrals when
integrators axe local martingales with finite vaxiation and the corresponding
Stieltjes integrals exist. This is another reason which justifies our
definition of stochastic integrals.
§2. Compensated Stochastic Integrals of Progressive
Processes with Respect to Local Martingales
We will extend the class of integrands to progressive processes.
9.6 Lemma. Let M be a continuous local martingale, and H be a
progressive process. Then there exists L G Mioc such that (1.1) holds for
all N £ Mioc if and only if H2.[M] G V4". In this case, there exists a
predictable process K £ Lm(M) such that K.M = L. We say that H is
integrable w.r.t. M, and L is called the stochastic integral of H w.r.t. M,
denoted by H. M.
Proof. The necessity is easy (putting TV = L in (1.1)). Now
assume H2.[M] G V+. Let °H be the optional projection of H. We have
(°H)2.[M] = H2.[M] (refer to Problem 5.9). By Theorem 3.20 there exists
a predictable process K such that [K ^ °H] is a thin set. Thus we have
K2.[M] = (°H)2.[M] = H2.[M\. Put L = K.M. Then for all N e Mloc
we have
[L, N] = K. [M, N] = °H. [M, N] = H. [M, N]. □
9.7 Definition. Let M be a purely discontinuous local martingale,
and H be a progressive process. If HAM has predictable projection, and
there exists a purely discontinuous local martingale L such that AL =
HAM — ^HAM), we call L the compensated stochastic integral of H
w.r.t. M, and denote L = H^M.
It is easy to see that if H is predictable, Definition 9.7 coincides with
Definition 9.1. In general, the predictable projection of HAM (if exists) is
not zero. We have no longer A(H^M) = HAM, but A{H^M) = HAM-
^HAM). The following is a typical example of compensated stochastic
integrals.
9.8 Lemma. Let M be a purely discontinuous local martingale. Put

232
Chapter IX Stochastic Integrals
H = /[am^o]- Then the compensated stochastic integral of H w.r.t. M
exists, and HCM = M.
Proof. We have if AM = AM, and *>[H AM] = 0. Then by definition
M = HtM. □
9.9 Definition. Let M be a local martingale, and H be a progressive
process. If H2.[MC] € V+, p{HAM) exists and
JUHAM-p(HAM))* € A^,
define
HtM = HqMq + H.MC + HtMd.
H-CM is called the compensated stochastic integrl of H w.r.t M.
Evidently, the compensated stochastic integral is a generalization of
the predictable stochastic integral in Definition 9.1. The conditions, given
above, for the existence of compensated stochastic integrals are the most
general ones, but they axe hard to be verified. Besides, we have no
characterization for compensated stochastic integrals. The following theorem
remedies these two defects to some extent. In fact, it is the definition of
compensated stochastic integrals given first by P. A. Meyer.
9.10 Theorem. Let M be a local martingale, and H be a
progressive process. If yJH2\M] € A^, then H-CM exists, and it is the unique
local martingale L such that for every bounded martingale N, [L, N] —
H.[M,N]eMloc,0.
Proof. Without loss of generality, we may assume Mq = 0. Put
W = HAMI[lHAM>lh U = HAMI^tM^.
Then A = £ W e Aioc. We have *\W) =P(AA) = A(AP). Since HAM =
W + U, P(HAM) exists. At the same time, B = £(C/2) € A^ and
A(BP)=P(U2),
E(pU)2 < EP(U2) <EA(Bp) < Bp.
Put Z = HAM - *(HAM). Then
UZ2) < 2{H2.[M] + UP(HAM))2}
< 2{H2.[M] + 2Z(pW)2 + 2Z(pU)2},
\/Tffi] € A^. On the other hand, H2.[MC] < H2.[M] € V+. Hence
H-CM exists.

§2. Compensated Stochastic Integrals of Progressive Processes 233
Now let iVbea bounded martingale. By Kunita-Watanabe inequality
we know
V = [HbM, N] - H\M, N] e Aioc.
Evidently, AV = -*{HAM)AN. Thus A(V*) = p(AV) = 0, i.e, W
is a continuous process with finite variation.Since Vc = [H.MC,NC] —
H.[MC, Nc] = 0, V = X)(AV) is a purely discontinuous process with finite
variation having only accessible jumps (note that P(HAM) is a predictable
thin process). By Theorem 7.14.1) Vp should be a purely discontinuous
process with finite variation. Hence, it must be Vp = 0, i.e., V G Mioc$.
Finally, by Theorem 7.36 the local martingale satisfying this requirement
is unique. □
Remarks. 1) In the theorem, if H2 is integrable w.r.t. [M], then the
condition y/H2. [M] G A^c is also necessary for the existence of H^M.
The details are left to the reader.
2) Hereinafter compensated stochastic integrals are also called simply
stochastic integrals, and the notation H-CM is replaced by H.M.
To conclude this paragraph, we expound how to define the stochastic
integrals of adapted measurable process w.r.t. a class of continuous local
martingales. They generalize Ito's stochastic integrals w.r.t. a Brownian
motion.
9.11 Theorem. Let M be a continuous local martingale with Mq = 0,
and a = (at) be a continuous increasing (non-random) function such that
for almost allu, d[M](u) <C da. Let H be an adapted measurable process.
Then there exists L € Mioc such that (1.1) holds for all N G Mioc if and
only if H2.[M] € V+. In this casef there exists a predictable process K
such that K G Lm(M) and K.M = L. L is called the stochastic integral
of H w.r.t. Mf and denoted by H. M.
Proof. Only the sufficiency is required to be proved. Let H be an
optional modification of H (i.e., H is optional and W € R+ Hf = Ht a.s.,
refer to Problem 5.10). Since d[M] is absolutely continuous w.r.t. da, by
Fubinfs theorem we know
H2.[M] = H2.[M].
In fact, let (Tn) be a sequence of stopping times such that Tn f oo and
" H*d[M]s] < oo, Vn > 1.

234 Chapter IX Stochastic Integrals
Then for every stopping time T,
E[fTnHMM}3} = f E[H?I]0,TATn]d^±}das
= E[j*ATnH2ad[M)s\.
By Lemma 9.6 H.M exists, and for all N G Aiioc
[H.M, N] = H.[M, N] = H.[M, N],
where the second equality follows from Kunita-Watanabe inequality and
(if - H)2.[M] = 0 (again by Fubini's theorem). The other assertions
follow directly from Lemma 9.6. □
§3. Stochastic Integrals of Predictable Processes with
Respect to Semimartingales
Since every semimartingale can be decomposed as the sum of a
local martingale and an adapted process of finite variation, naturally the
stochastic integral w.r.t. a semimartingale can be considered as the sum
of integrals w.r.t. the two parts. The crucial point lies in that the sum of
integrals should be independent of the decompositions of this
semimartingale. The following lemma guarantees it.
9.12 Lemma. Let X be a semimartingale, and H be a predictable
process. Let X = M + A and X = N + B be two decompositions of X,
where M,N e Mloc and A, B e V0. IfHe Lm(M) n Lm(N), H-SA and
H'SB exist, then
H.M + HhA = H.N + HhB. (12.1)
Proof. Since M-N = B-A e W^o, by Theorem 9.5.2) H.(M-N) =
Hs(B - A), i.e., (12.1) holds. □
Based on Lemma 9.12, we may give the following definition.
9.13 Definition. Let X be a semimartingale, and H be a predictable
process. If there exists a decomposition X = M+A, where M G Mioc and
A £ Vo, such that H G Lm(M) and H-SA exists, we say that H is integrable

§3. Stochastic Integrals of Predictable Processes w.r.t. Semimartingales 235
w.r.t. X in the domain of semimartingales (or simply H is X-integrable),
and X = M + A is an H-decomposition of X. At this time, put
H.X = H.M + HkA. (13.1)
H.X is independent of ^-decompositions of X, and is called the stochastic
integral of H w.r.t. X.
9.14 Remarks. 1) Let X be a semimartingale and X = M + A be a
decomposition of X, where M e Mioc and A e Vo- Then for any locally
bounded predictable process H, H is X-integrable, and X = M + A is an
^-decomposition of X.
2) Definition 9.13 of stochastic integrals of predictable processes w.r.t.
semimartingale is a natural generalization of Definition 9.1 of stochastic
integrals of predictable processes w.r.t. local martingales. In fact, if M is
a local martingale, H is a predictable process, and H is integrable w.r.t.
M in the domain of local martingales, then H is also integrable w.r.t.
M in the domain of semimartingales, and the stochastic integrals in the
two senses coincide. But if H is integrable w.r.t. M in the domain of
semimartingales, in general we cannot assert that H.M is still a local
martingale, i.e., H need not be integrable w.r.t. M in the domain of local
martingales. For example, let M G Wioc,o and if be a predictable process.
If the Stieltjes integral H-SM exists, but H-SM 0 Aioc, then H & Lm(M).
The next theorem summarizes the fundamental properties of stochastic
integrals of predictable processes w.r.t. semimartingales. Its proof is easy.
Hereforth we denote by L(X) the collection of all predictable processes
which are integrable w.r.t. a semimartingale X.
9.15 Theorem. Let X be a semimartingale, and H e L(X).
1) (H.X)C = H.XC, A(H.X) = HAX, (H.X)0 = HqXq.
2) For any stopping time T,
(H.Xf = H.XT = (HIl0,n).X, (H.Xf- = H.XT~.
3) For any semimartingale Y,
[H.X,Y] = H.[X,Y].
4) If Y is a semimartingale and H G L(Y), then H G L(X + Y) and
H.(X + Y) = H.X + H.Y.
5) If K is a predictable process and \K\ < \H\, then K G L(X).

236
Chapter IX Stochastic Integrals
9.16 Theorem. Let X be a special semimartingale, and X = M + A
be its canonical decomposition. Assume H G L(X). Then H.X is also a
special semimartingale if and only if X = M + A is an H-decomposition
ofX.
Proof. The sufficiency is trivial. We are to show the necessity. Let
X = N + B be an //-decomposition of X, where N G Mioc and B £ Vo.
Then H.X = H.N + H.B e Sp and H.B e Aioc. Because A = B, by
Theorem 5.23.2) H is integrable w.r.t. A and H.A = H.B. We have
y]H*\M\ < y/H*.[X] + JH*.[A] < y/ilLX] + E \HAA\.
Since yflinZ] € A^ (Theorem 8.6), y/H2.[M] e A^, i.e., H e Lm(M).
In a word, X = M + A is an //-decomposition of X. □
The next theorem is an important consequence of the above theorem.
9.17 Theorem. Let X be a semimartingale and H G L(X). Let U be
an optional set such that U D [|//AX| > 1 or \AX\ > 1] and for almost
all (j, {s : (u;, 5) £ U} fl [0, t] contains at most a finite number of points
for each t > 0. Put
At=Z AXsI{(.}3)eU}, Zt = Xt- At, t > 0.
8<t
Then H € L(Z), and the canonical decomposition Z = N + B of the
special semimartingale Z is an H-decomposition of Z.
Proof. Under the assumption of the theorem, the definition of (At) is
meaningful, and A is a process with finite variation whose trajectories are
all step functions. Hence H is integrable w.r.t. A, and thus integrable
w.r.t. Z. In addition, we have \AZ\ < 1, \A(H.Z)\ = \HAZ\ < 1. Thus
Z and H.Z axe special semimartingales. By Theorem 9.16 the canonical
decomposition Z = N + B is an //-decomposition of Z. □
Remark. In the theorem, if U = [\HAX\ > 1 or |AX| > 1], then
X = N + (B + A) is an //-decomposition of X, where TV € Mioc. But
\AN\ < 2 (since |AB| < 1), so TV is a locally bounded martingale.
Below we continue to investigate the properties of stochastic integrals
by making use of Theorem 9.17.
9.18 Theorem. Let X be a semimartingale.
1) H,K e L(X) => H + K e L(X).

§4. Lenglart's Inequality and Convergence Theorems for Integrals 237
2) Let H £ L(X) and K be a predictable process. Then K G L(H.X)
if and only if KH e L(X). In this case, we have K.(H.X) = (KH).X.
Proof. 1) In Theorem 9.17 put U = [\HAX\ > 1 or \KAX\ > 1 or
|AX| > 1]. Then X = N + (A + B) is both an //-decomposition and a
/^-decomposition. So it is an (H + K^decomposition, i.e., H + K e L(X).
2) The necessity is easy. We want to show the sufficiency. Let KH G
L{X). In Theorem 9.17 put U = [\HAX\ > 1 or \KHAX\ > 1 or
|AX | > 1]. Then X = N + (A + B) is both an //-decomposition and
an Z//f-decomposition. This implies that H.X = H.N + H.(A + B) is a
AT-decomposition of H.X. Hence K G L(H.X) and
K.(H.X) = K.(H.N) + K.(H.(A + B))
= (KH).N + (KH).(A + B) = (KH).X. □
9.19 Theorem. Let X be a semimartingale and H be a predictable
process. If there exists a sequence (Tn) of stopping times such thatTn | oo
and H e L(XTn) for each n, then H e L(X).
Proof. Since H2.[X]Tn = [H.XTn], H2.[X] is an increasing process.
Put
A = £(A-*7[|tf ax|>i or |AX|>i))> Z = X - A.
Then A is a process with finite variation whose trajectories are all step
functions. Hence H.A exists. It remains to prove H € L(Z). Let Z = N +
B be the canonical decomposition of Z G Sp. For each n, ZTn = NTn +BTn
is the canonical decomposition olZTn. Since |A(//.ZTn)| = \HAZTn\ < 1,
H.ZTn e Sp. By Theorem 9.16, H.NTn and H.BTn exist. Therefore //.TV
and H.B exist, i.e., H e L(Z). □
§4. Lenglart's Inequality and Convergence Theorems for
Stochastic Integrals
In this paragraph we first introduce Lenglart's inequality, then by
means of it we study the continuity of stochastic integrals w.r.t.
integrands.
9.20 Definition. Let X be an optional process, and A be an adapted
increasing process. It is said that X is dominated by A if for any bounded

238 Chapter IX Stochastic Integrals
stopping time T,
E[\XT\] < E[AT\. (20.1)
In this case, (20.1) holds also for any finite stopping time T.
9.21 Examples. 1) Let M be a locally square integrable martingale.
Then M2 is dominated by [M] or (M). In fact, if (Tn) is a localizing
sequence for M, for any bounded stopping time T,
E[M$ATn] = E[[M]TATn] = E[{M)TATn\. (21.1)
Letting n —► oo in (21.1) yields by Fatou's lemma,
E[M$\ < E[[M]T] = E[(M)T]-
2) Let A be an adapted locally integrable increasing process. Then A
and its dual predictable projection A are dominated by each other.
9.22 Lemma. Let X be an adapted cadlag process, dominated by an
adapted increasing process A. Then for any constant C > 0 and stopping
time S we have
P(X*S >C) = ±E[AS]. (22.1)
Moreover, if S is predictable, we have
P(X*S_ >C)< ^EAS-. (22.2)
Proof. Put T = inf{< > 0 : \Xt\ > C) A S A n. Then
E[AS\ > E[AT] > E[\XT\] > f \XT\dP > CP(X*SAn > C).
•/[^5A„>C]
Letting n —► oo yields
P(X*s >Q< ^E[AS]. (22.3)
Replacing C by C - e in (22.3) and letting e j 0, we find (22.1).
If S is predictable, take a sequence (5n) of stopping times foretelling
5. Since
P(X'Sn >C)< ±E[ASn}.
Letting n —> oo, we get
P(X*S_ >C)< ±E[As-}.
In the same manner we can obtain (22.2). □

§4. Lenglart's Inequality and Convergence Theorems for Integrals 239
9.23 Theorem. Let X be an adapted cadlag process, dominated by an
adapted increasing process A. Then for arbitrary constants C > 0, d > 0,
stopping time T and measurable set H we have
P{H n [XJ > C]) < ^E[AT A (d + (AA)*T\ + P(H n [AT > d\). (23.1)
Moreover, if A is predictable, we have
P(H H [X£ > C]) < ^E[AT Ad\ + P(H n [AT > d\). (23.2)
(23.1) or (23.2) is called Lenglart's inequality.
Proof. Put S = inf{t > 0 : At > d}. Then As < d + AAS, As <
A00A(d+(AA)*00),and
H n [X^ > C] c [X*s > C] u (H n [S < <»])
c [^>C]u(^n[Aoo>d]).
By Lemma 9.22 we have
p(h n [X^ > c\) < P(x*s >C) + p(h n [4TO > d])
< ±E[As] + P(Hn[Aoo>d\)
< ijE?[i4oo A (d + (AA)^)] + P(H n [Aoo > d]).
(23.3)
Replacing X and A by XT and /lr respectively yields (23.1).
If A is predictable, so is 5. Similarly, we have
Hn[x^>c\c [X*s_ > c\ u (^n [Aoo > d])
P(H n [Xi > c]) < P(X|_ > c) + p(ff n [a^ > d])
< ip[As_] + P(ffn[i4oo>d])
< ^£7[>loo A d] + P(# D [Aoo > d]).
Replacing X and A by XT and AT respectively yields (23.2). □
9.24 Corollary. Let X be an adapted cadlag process, dominated by an
adapted increasing process A. If \AA\ < a (constant) (or A is predictable),
then for arbitrary constants C > 0, d > 0, stopping time T and measurable
set H,
P(H n [XT > C\) < ^-^ + P(H n [AT > d])

240 Chapter IX Stochastic Integrals
(or
P(H n [Jtt. >c])<- + P(H n [AT > d\)).
c
9.25 Corollary. For each n let X^ be an adapted cadlag process,
dominated by a predictable increasing process A^nK Let T be a stopping
time and H be a measurable set. If IhAt ~~* ^> then J/j sup \X\n \ —> 0.
t<T
Proof. By Corollary 9.24 for e > 0 and 6 > 0,
P(H fl [(XW)*T > e]) < - + P(H fl [AP > 6}). (25.1)
Letting n —> oo and 5 | 0 consecutively in (25.1), we obtain immediately
the required assertion. □
Now we turn to study the continuity of stochastic integrals. Let M
be a locally square integrable maxtingaie, and H be a predictable
process such that H.M is a, locally square integrable maxtingaie. Since
[H.M] = H2.[M], we have (H.M) = H2.(M). Thus (H.M)2 is
dominated by H2.(M). By Corollary 9.25 we obtain the following theorem
immediately.
9.26 Theorem. Let M be a locally square integrable martingale, T be
a stopping time, and B be a measurable set. Assume H, H^ 6 Lm(M),
n > 1, and (H - H^).M e M2loc, n>\. If
then
Ib f (Hs-HW)2d(M)3Zo,
IBsup \(H.M)3 - (#<»>.M),| 4 0.
3<T
The next theorem is a convergence theorem for stochastic integrals
w.r.t. semimaxtingales.
9.27 Theorem. Let X be a semimartingale, T be a finite stopping
time, B be a measurable set, and H,H(n\n > 1, be all locally bounded
predictable processes. If for almost all u G B, (H(n\u))n>\ is uniformly
bounded and convergent to H.(lj) on [0, T(uj)], then
IBSup\(H^.X)t - (H.X)t\£o. (27.1)
t<T

§4. Lenglart's Inequality and Convergence Theorems for Integrals 241
Proof. Let X = M + A, where M is a locally bounded martingale and
A is an adapted process with finite variation. Under the assumption of
the theorem, by Lebesgue dominated convergence theorem we have
Is I (H[n) - Ht)2d(M)t -> 0 a.s..
J[OyT\
Then by Theorem 9.26
IB sup \(H<n>.M)t - (H.M)t\ £ 0.
t<T
Again by Lebesgue dominated convergence theorem we have
IBsxip\(H^lA)t-(H.A)t\<IB I \H(tn) - Ht\\dAt\-+0 a.s..
t<T JO
Thus (27.1) follows. □
As an application of Theorem 9.27 we obtain Riemann-Stieltjes
approximations for a class of stochastic integrals.
9.28 Definition. Let T be a finite stopping time, and (Tn)n>o be
an increasing sequence of stopping times with To = 0 and supTn = T.
n
We say that r : 0 = To < T\ < • • • is a stochastic partition of interval
[0,T], if for almost all cj, the sequence (Tn(o;)) is stationary (i.e., there
exists a natural number n{u) such that Tn(u) = T(u) when n > n(cj)), in
other words, for almost all cj, (Tn(u>)) forms a finite partition of interval
[0,r(u;)]. Put
«(r) = sup|ri+i-ri|.
3
6(t) is a finite r.v., and is called the mesh of partition r.
9.29 Theorem. Let X be a semimartingale, H be an adapted cadlag
or left-continuous process, and T be a finite stopping time. Let
T(n) . 0 = T(n) < ri(n) < • • •, n > 1,
be a sequence of stochastic partitions of [0,T] such that lim6(T^) = 0
O.5.. Then
snp\ZHT(n)(XT{n)At-XTin)J-ftHs.dX3\^0, n -> oo. (29.1)
t<T ' t i« i»+iAf J» At JO '
Proof Put H^ = H0IlQ] + E /fT(n)/]T(n) T(n)r Then H^ is a locally
bounded predictable process and
(H^.X)t = E#r(n)(XT.(„)At - *T<n>At) + H0X0.
*i+V

242
Chapter IX Stochastic Integrals
Since H.(u) is bounded on the finite interval [0,T(u;)], (if(n)(cj))n>i is
uniformly bounded on [0,T(u;)]. In addition, for almost all a; we have
Urn H[n\u) = Ht-(u>)
71—>00
for all t e [0,T(cj)]. (29.1) follows from Theorem 9.27. D
The following theorem is the dominated convergence theorem for
stochastic integrals.
9.30 Theorem. Let X be a semimartingale, H G L(X), K^ and
K be predictable processes such that \K^\ < \H\, \K\ < \H\. Let B e
T and T be a finite stopping time. If for almost all u € B we have
lim K[n\u) = Kt(u>) for all t e [0,T(u/)], then
n—>oo
IB sup \(K^.X)t - (K.X)t\ $ 0, n -> oo. (30.1)
t<T '
Proof Without loss of generality we may assume Xq = 0. Put
A = £(A-*7[|//ax|>i or |ax|>i])> Z = X - A.
Let Z = N + B be the canonical decomposition of Z £ c?p. By Theorem
9.17 X = TV + (B + A) is an //-decomposition of X. Besides, we have
\AZ\ < 1, \A(H.Z)\ = \HAZ\ < 1. By Theorem 8.8 we know |A7V| < 2,
\A(H.N)\ < 2. In particular, N,H.N G Mf^o- BY tlie assumption
\K^\ < |//|, \K\ < |//|, thus K<<n\N, K.N e A^L,o- The remainders of
the proof are similar to that of Theorem 9.27. □
Remark. Let X be a semimartingale, H e L(X). Put H^ =
HI[\H\<n]- Then by the theorem we know for all t > 0 ,
sup 0, n —► oo.
Finally, we should point out that the stochastic integrals of predictable
processes w.r.t an adapted process with finite variation in the domain of
semimartingales need not be Stieltjes integrals. But we have the following
result.
9.31 Theorem. Let A be a predictable process with finite variation,
and H be a predictable process. If H is integrable w.r.t. A in the domain of
semimartingales, then in the sense of Stieltjes integrals H is also integrable
w.r.t. A, and the two kinds of integrals coincide.

§5. Ito Formula and Doleans-Dade Exponential Formula
243
Proof. Put #(n) = fl7[|jy|<n]. Evidently, the conclusion of the theorem
is valid for each H^n\ In particular, each H^n\A is a predictable semi-
martingale. By Theorem 9.30 H. A is also a predictable semimartingale,
thus it is a special semimartingale (Corollary 8.7). Then the conclusion
of the theorem follows from Theorem 9.16. □
§5. Ito Formula and Doleans-Dade Exponential Formula
In this paragraph we will establish the change of variables formula for
semimartingales, that is, the famous Ito formula. It is the most powerful
tool in stochastic calculus. As its applications, we show the strong law of
large number for semimartingales and Doleans-Dade exponential formula.
9.32 Lemma. Let M be a locally bounded martingale, and A be a
predictable process with finite variation. Then MA — (M-).A is a local
martingale.
Proof. By localization we may suppose that M is a bounded martingale
and A is a predictable process with integrable variation. Put L — MA —
(M-).A. By Theorems 5.32 and 5.33 we know that for any stopping time
T, E[LT] = 0. Then by Theorem 4.40, L e M. □
9.33 Theorem. Let X and Y be a pair of semimartingales. Then
XtYt = f Xs-dYs + / Ys.dXs + [X, Y]u t > 0. (33.1)
Jo Jo
(33.1) is called the formula of integration by parts.
Proof. By polarization it suffices to prove (33.1) for the case of X = Y,
i.e.,
X2 = 2(X-).X - 2Xl + [*]• (33-2)
To this end, put A = X2 — 2(X-).X + 2X$. First, we want to show that
A is an increasing process and A A = (AX)2. Let t > 0 and
rn:0 = tf <*?<-"<C(n)=*
be a sequence of finite partitions of [0, t] with 6(rn) tending to zero. Then
Xt - XQ = Y,(Xtn^ — Xtn)
= 2£Xt»(Xt«+1 - Xt?) + Z(Xtn+i - Xt?f.

244
Chapter IX Stochastic Integrals
By Theorem 9.29 we know
At = X% + P- lim £(.Yt« - Xtn)2, a.s.. (33.3)
In particular, by (33.3) A is an increasing process. By the definition of A,
we have
AA = A(X2) - 2X_AX = (X. + AX)2 -X2_- 2X.AX = (AX)2.
In order to show (33.2) we first assume that X is bounded. In this case
X is a special semimartingale. Let X = M + A be its canonical
decomposition, where M is a locally bounded martingale and A is a predictable
process with finite variation and Aq = 0. Put
B = X2- 2{XJ).X + 2Xl - [X] = A - [X].
It is shown above that B is a continous process with finite variation and
Bq = 0. On the other hand, noting Xq = Mo, we have
B = (M + A)2 - 2(M_ + A-).(M + A) + 2M02 - [M + A]
= (M2 - [M]) - 2(M_.M - M02) + 2(MA - M-.A)
-2A_.M-2[M,A]. (33.4)
Here we have made use of the formula of integration by parts for Stielt-
jes integrals (Lemma 1.39). By Lemmas 9.4 and 9.32 we know that each
term in (33.4) is a local martingale. Hence B is a local martingale.
Consequently, it must be B = 0, i.e., (33.2) holds.
In general cases, put Tn = inf{£ > 0 : \Xt\ > n}. Then XTn"/[Tn>0] is
a bounded semimartingale, and
(X2)T»- = (XT»-)2 = 2X7}*-.XT»~ - 2X1 + [XTn~]
= (2X-X - 2X1 + [X])Tn~-
Since Tn | oo, (33.2) remains valid. □
Remark. In the above proof we have shown for t > 0,
x* + z(xthi-xtn)2£>.[x]t
i
This is why we call [X] the quadratic variation of X.
9.34 Corollary. Let X be a semimartingale and A be a predictable
process with finite variation. Then
XA = A.X + {XJ).A - X0A0. (34.1)

§5. ltd Formula and Doleans-Dade Exponential Formula
245
Proof. Let X = X0 + M + B, M € .Mioc,o, B € V0. By Yoeurp's
lemma (Example 9.4.1)) we have
[X, A] = X0Ao + [M, A] + [B, A]
= X0Ao + {AA).M + {AA).B
= X0A0 + (AA).X. (34.2)
Then (34.1) follows from (33.1) and (34.2). □
9.35 Theorem. Let X1, • • •, Xd be semimartingales, and F be aC2-
function on R? (i.e. F has continuous partial derivatives of the first and
the second orders). Put Xt = (Xl,---,Xf) {(Xt) is also called an d-
dimensional semimartingale). Then
F(Xt)-F{X0)=E f* DjF{X.-)dX> + E ri.(F) + Ut(F), (35.1)
j=l JO 0<s<t *
where
V3(F) = F(XS) - F(XS-) - E DjF(Xs-)AXi, (35.2)
3=1
MF)= E /'AjF(Xa_)d((X')c,(X')% (35.3)
dF d2F
DjF = ——, DijF = ——-—, and the series ^2 r]s(F) is absolutely
OXj OXiOXj 0<s<t
convergent
(35.1) is the famous ltd formula1^.
Proof. We adopt the line of C. Dellacherie and P. A. Meyer to show
Ito formula, starting from the formula of integration by parts.
We may suppose all X1, • • •, Xd axe bounded: \X^\ < C, j = 1, • • •, d,
where C > 0 is a constant. Otherwise, put Tn = inf{t > 0 : \Xl\ > n
or \Xf\ > n, • • •, or \Xf\ > n}, and deal with XTn~/[7'n>o] as in Theorem
9.33. If (35.1) holds for each XTn-I[rn>o], then (35.1) holds for X, since
Tn | oo. Under the boundedness assumption we can choose a squence (Fn)
of polynomials on Rd such that Fn, DjFn and DijFn uniformly converge
to F,DjF and D^F, i,j = 1, •• • ,d, on [-C,C]d respectively. If (35.1)
holds for each Fn, then by Theorem 9.30, (35.1) holds for F as well. Hence
we may suppose F is a polynomial on Rd.
It is easy to see that F may be a complex-valued function in Ito formula.

246 Chapter IX Stochastic Integrals
If F(xl,---,xd) = xixi, (35.1) reduces to (33.1). By induction it
suffices to show the following statement: if (35.1) holds for a polynomial
F on K*, then (35.1) holds also for G(xl, ■■-,xd) = xiF(x1, • • •, xd). Since
T)S(G) = G(X.) - G(Xa.) - £ DjG{Xa-)*Xi
j=i
= Xis_ns(F) + A[Xi,F(X)]s
and (Xl3_) is locally bounded, the series £ Vs(G) is absolutely conver-
0<3<t
gent. In addition, we have
l-MG)=\ fxi_dAs(F)+t f DjF(Xa.)d((X%(XJ)e)a
l l Jo j=i Jo
= \Io X°-dMF) + <(Xi)C' (F™C>< - *o*TO, (35.4)
£ f DjG(Xs-)dXi
j=lJ0
= /' F(Xa.)dXi + £ f xi_DjF{Xa-)dXi. (35.5)
Jo j=i Jo
Then from (33.1), (35.4) and (35.5) we obtain
G(Xt) - G(X0) = X\{Xt) - XtF(X0)
= f X\_dF{Xa) + f F(Xa-)dXi + [X\ F(X)]t - Xi)F(Xo)
Jo Jo
= t f DjG{Xs.)dXi+ £ Vs(G) + \At(G),
j=lJO 0<s<t *
i.e., (35.1) holds for G. □
The next theorem is a refinement of Theorem 9.35, its proof is
completely analogous, and is omitted.
9.36 Theorem. Let X1, • • •, Xn be semimartingales, and Xn+1, • • •,
Xn+m be adapted processes with finite variation. Let F be a continuous
junction on i£n+m, of class C2 w.r.t the first n variables and of class
C1 w.r.t. the last m variables (it may be n = 0 or m = 0). Put Xt =
{Xl---,X^m). Then
n+m ft
F(Xt)-F(X0)= £ / DiF(X,-)dX>+ E Vs(F)
j=l JO 0<s<t
+1 E [*DijF(Xa..)d((Xi)c,(Xir)t
* t j=l Jo

§5. ltd Formula and Doleans-Dade Exponential Formula 247
9.37 Theorem. Let X be a semimartingale and A be a predictable
increasing process with A^ = oo a.5.. // lim I- r-^0 exists and is
finite a.s., then
lim — = 0 a.s..
t-KX> At
Proof. Put Y = j^—iX. Then (1 + A).Y = X (Theorem 9.18.2))
1 + A
and
(1 + A)Y = (1 + A).Y + (Y.).A - A0Yo = X + (Y.).A - A0Y0
by Corollary 9.34. Thus
v^a V I (Yt-Ya.)dA.
v Yt + AoYo , Jjo^]
+
1 + At 1 + At 1 + At
the second term on the right hand side is a Stieltjes integral.
For almost all u> and any e > 0 there exists te(w) such that
\Yt(u)-Y00\<e, t>te(u).
Then for t > te we have
l-i-/ (Yt-Y.-)dA.\
11 +At J[o,t] I
^ 7T^-{ / ly* " Y-\dA, + f\\Yt - Fool + |n_ - Y^dA,}
1 + At <• J[o}tc] Jte '
< 2Y^Ate + 2eAt
1 + At 1 + At
Letting t —► oo and e | 0 consecutively, we find
—l— I (Yt-Ys.)dAs-+0, t->oo.
1 + At J[o,t]
Evidently,
Yt + AoYo\ YZJl + Ap)
< — ► 0, t —> oo.
1 + At
1 + At
Hence
lim — = lim — = 0 a.s.. □
*—oo At t->oo 1 + At
9.38 Corollary. Let M be a locally square integrable martingale with
(M)oo = oo a.s.. Then
lim .-... = 0 a.s..
*-oo (M)t

248 Chapter IX Stochastic Integrals
Proof. Since
(tTW))2 '<M> - (l + (M»(1 + (M)-)-<M> £ '■
by Theorem 8.32 lim ( TTrr.Af ] exists and is finite a.s.. Now apply-
ing Theorem 9.37 to M and (M) leads to the required assertion. □
Theorem 9.37 is the strong law of large number for semimartingales in
the general form. It is a primary application of the formula of integration
by parts — a special case of Ito formula. Now we give another important
application of Ito formula — Doleans-Dade exponential formula.
9.39 Theorem. Let X be a semimartingale. Put
Vt= n (1 + AXs)e-AX- (Vb = 1). (39.1)
0<s<t
Then for almost allu the infinite product on the right hand side o/(39.1)
is absolutely convergent for all t > 0, and V = (Vt) is an adapted purely
discontinuous process with finite variation. Put
Zt = exp [Xt - X0 - \(Xc)t) J] (1 + AXs)e-AX>. (39.2)
0<3<t
Then Z = (Zt) is the unique semimartingale satisfying the following
stochastic integral equation
Zt = l+ f Zs-dXs. (39.3)
Jo
Proof Put
Obviously, the product defining V( is of a finite number of terms. Since
e~x2 < (1 + x)e~x < e~\x\ N<^
the infinite product defining V" is absolutely convergent. Hence the
infinite product in (39.1) is absolutely convergent. Apparently, (logV^) and
(logVy7) are adapted purely discontinuous processes with finite variation,
so are (V() and (V") by Ito formula. Because Vt = Vt'V", again by Ito
formula V is an adapted purely discontinuous process with finite variation.

§5. ltd Formula and Doleans-Dade Exponential Formula 249
Denote F(x, y) = exy and K = X - X0 - \{XC). Then Z = F(K, V)
and Kc = Xc, AK = AX. Noting Za = Zs_(l + AX3), by Ito formula
we have
Zt = 1 + / Za.dKa + I eK-dVa + I f Zs-d(Kc)a
Jo Jo * Jo
+ £ (AZs-Zs-AK3-e-K>-AVs)
0<s<t
= 1+/ Z3-dX3,
Jo
i.e., Z satisfies (39.3).
Now assume that Y = (Y$) is another semimartingle satisfying (39.3).
Then AY = Y-AX, (XC,YC) = (Y-).(XC). Put
W = e~KY. (39.4)
By Ito formula we have Wq = 1 and
Wt = 1 - / W3.dK3 + / e-K°-dY3 + J / W3_d(Xc)3
Jo Jo * Jo
- [te-K>-d(Xc,Yc)s+ £ (A^ + ^-AX5-e-^-Ays)
JO 0<s<t
= 1+ £ aw;.
0<s<*
On the other hand, by (39.4) Wt = Wt-.e-AXt(l + AXt),
AWt = Wt-[e-AXt(l + AXt) - 1].
Put At = Y, [e~AXa(l + AX3) - 1]. A = (At) is an adapted purely
0<3<t
discontinuous process with finite variation. W satisfies
Wt = l+ [ WM-dA*.
Jo
This means W is an adapted purely discontinuous process with finite
variation. Of course, V = e~KX also satisfies the same equation. Put
U = V — W. Then U satisfies the homogeneous equation
Ut = ( U3-dA3. (39.5)
Jo
Put Bt = / \dA8\. Making use of the formula of integration by parts and
Jo
by induction, it is not difficult to show
f{Ba-)ndBa < -±-(Bt)n+1. (39.6)
Jo n +1

250
Chapter IX Stochastic Integrals
By iterating (39.5) and induction, from (39.6) we obtain
\Us\<±Ut*(Bt)n, \Ua.\<-Ut*(Bt)n, s€[0,t}.
n\ n
Hence Ut = 0 a.s. for each t > 0. Consequently, U = 0, W = V, and
Y = eKW = eKV = Z. □
(39.2) is called Doleans-Dade exponential formula, semimartingale Z
is called the exponential of semimartingale X and denoted by £(X). By
(39.3) we know that if X is a local martingale (resp. a special
semimartingale, resp. an adapted process with finite variation), so is £(X). Besides,
£(X) = £(X - X0) and for any stopping time T, £(X)T = £(XT).
It is easy to check that if X is a standard Brownian motion, then
£(X)t = exp | Xt — -t >, and if X is a Poisson process, then £(X)t = 2Xt.
A process Z is called an exponential semimartingale if there exists
a semimartingale X such that Z = Zq£(X), i.e., Z is the unique
semimartingale satisfying
Zt = Zq + / Zs-dXs.
Jo
We want to characterize the class of exponential semimartingales.
9.40 Lemma. Let X be a semimartingale. Put
T = inf{*>0: AXt = -1},
S = inf{t > 0 : £(X)t = 0 or £{X)t- = 0}.
ThenT = S a.s..
Proof. If T < oo, then AXt = — 1. By the exponential formula
€ (X)I{TM = 0.
Hence S <T a.s.. On the other hand, if t < T, from the proof of Theorem
9.39 one can see V( ^ 0 and V[_ ^ 0. But V" and V"_ axe always positive.
Thus £(X)t = VlVl'eK* ^ 0, €(X)t- = V(JV{LeK^ ? 0, and T < S a.s..
Therefore, S = T a.s.. □
9.41 Theorem. Let Z be a semimartingale, and T = inf {t > 0 : Zt =
0 or Zt- = 0}. Then Z is an exponential semimartingale if and only if
the following conditions are satisfied:
1) Z = ZI^t\,
2) H = -^—I[z_^o] is integrable w.r.t. Z.

§6. Local Times of Semimartingales
251
Proof. Necessity. Let Z = Z0€(X), X € S. On [Z0 = 0] we have
Z = 0 = Z7[o(T[. On [Z0 > 0] we have T = inf{* > 0 : AXt = -1} and
£(X)Ip>,oo[ = 0- By Lemma 9.40, again we get Z = ZI^t\-
Obviously, \HZ-\ < 1, HZ. € L(X) and Z. € L(X). By Theorem
9.18.2) H € L((Z-).X). But Z = Z0 + (Z_).X - Z0X0, so H € L(Z).
Sufficiency. Put X = H.Z. Then ZoXo = ZoI[Zo^o] = ^o, and
(Z.).X = (HZ.).Z = I[Z_^0].Z.
Put i? = T[^r_=o)o<T<cx>]- R is a predictable time, because it is foretold
by Rn = inf {t > 0 : t < T, \Zt\ <-} An. Now
I[Z-=o]-z = (7[oIr=0]] + h*] + hTM)z
= ZqI[T=0] + I\R\Z
= AZRI[RM = 0 (due to 9.4.2)).
Hence Z = (Z.).X = Z0 + (Z-).X - ZQX0 = ZQ£(X). D
§6. Local Times of Semimartingales
9.42 Lemma. Let X be a semimartingalef f be a continuous convex
function on R, and f be its left derivative. Then f(X) is a
semimartingale and
f(Xt) = f(X0)+ f f'{Xs.)dXa
Jo
+ E [f(Xs)-f(Xs-)-f'(Xs-)AXs} + Ct, (42.1)
0<s<t
where C = (Ct) is a continuous adapted increasing process with Cq = 0.
Proof. Take a non-negative function <p G C°°(R) such that [—a,0]
/oo
<p(s)ds = 1. Put
-oo
fn(t) = n I f(t + s)ip(ns)ds = j f(t+ -)<p{s)ds.
Then fn(t) is convex, fn(t) € C°°(R) and as n —> oo we have
fn(t) - f(t),
m = J°af(t+^)<p(s)dsu'(t)
(f(t) is left-continuous and monotone increasing).

252 Chapter IX Stochastic Integrals
Apply Ito formula to fn and X,
fn(Xt) = fn(Xo) + f f'n{Xs.)dXs + B[n\ (42.2)
Jo
where
B\n) = E [fn(X3) - fn(Xs-) - ti(Xs.)AXs] + \f f':(Xa.)d(Xc)t
is an adapted increasing process in view of the convexity of fn.
We may suppose X-hQoo^ is bounded. Otherwise, we may deal with
XTn, where Tn = inf{£ > 0 : \Xt\ > n}. Since /' is bounded on each finite
interval, letting n —> oo in (42.2) yields
f(Xt) = f(X0) + f nXs.)dXs + Bt
Jo
(for all t > 0, Bt = P-lim£jn)), where B = (Bt) is an adapted increasing
n
process with Bo = 0 and
ABt = Af(Xt) - f\Xt.)AXt = f(Xt) - f(Xt-) - f'(Xt-)AXt > 0.
Let C be the continuous part of B. Then (42.1) follows. □
9.43 Theorem. Let X be a semimartingale and a G R. Then
{Xt-a)+ = (X0-a)+ + / I[x9_>a]dX8+ £ [I[Xa_>a](Xs - a)~
Jo o<s<t
+ I[Xa-<a)(Xs-a)+]+l-L«(X), (43.1)
(Xt-a)~ = (Xo-a)~ - I[x <a]dXs+ £ [I[Xa->a](Xs - a)"
Jo "- J 0<3<t
+ I[x,-<a](Xs - o)+] + \l?(X), (43.2)
where L^(X) is a continuous adapted increasing process with Lq(X) = 0.
It is called the local time of X at a. (43.1) or (43.2) is called Tanaka-Meyer
formula.
Proof. Let f(x) = (x - a)+. Then / is convex, f'(x) = /]a)00[(x) and
[ (y — a)~, if x > a,
f(y)-f(x)-f'(x)(y-x) = {
{ (y - a) > if x < a.
Applying (42.1) to (x - a)+, we have
(Xt - a)+ = (X0 - a)+ + jT I[Xa_>a)dXs

§6. Local Times of Semimartingales
253
+ £ [I[Xa->a](Xs-a)- + I[Xa_<a](Xs-a)+] + Ct, (43.3)
0<3<t ~
where (Ct) is a continuous adapted increasing process with Co = 0.
Similarly, applying (42.1) to (x — a)"", we have
(Xt -a) = (X0 -a) - / I[Xa_<a]dXs
jo
+ £ [I[Xs->a](Xs-a)-+I[Xs_<a](Xs-a)+} + Dt, (43.4)
0<s<t ~
where (Dt) is also a continuous adapted increasing process with Do = 0.
Subtracting (43.4) from (43.3) yields
(Xt - a)+ - (Xt - a)' = (X0 - a)+ - (XQ - a)~ + / </X3 + Ct - A-
Jo
Hence C = D. Denote them by -La(X), and therefore we have (43.1)
and (43.2). □
9.44 Theorem. Let X be a semimartingale and a € R. Then for
almost allu the measure dL?(X)(u>) does not charge the set {t : Xt~(u) ^
o} and the interior of {t : Xt-(u)) = a}.
Proof Let S and T be two stopping times, and 0 < 5 < T. If
]5,T[C [X- < a], then [5,T[c [X < a] and }S,T} C [X_ < a]. Prom
(43.1) we know on [T < oo]
(XT - a)+ - (X5 - a)+ = (XT - a)+ + \l«t(X) - \l%(X) a.s., (44.1)
L$(X) = Las(X) a.s.. (44.2)
Similarly, if ]5,T[c [X- = a], then [5,T[c [X = a] and ]S,Tj C
[A"_ = a]. (44.1) and (44.2) remain true.
Let r > 0 be a rational. Put
Xr_ < a,
_ J r, if
I oo, if
S{r)=- - x >
Ar_ > a,
T(r) = inf{t > S(r) : Xt_ > a},
tf = Uj5(r),T(r)[.
r>0
For each u the section i/^ is the interior of {t : Xt-(ui) < a}. Prom
(44.2) we know that for almost all cj, dLa(X)(u) does not charge H^.
Analogously, it can be shown that for almost all u dLa(X)(u) does not
charge the interior of {t : Xt-(w) > a}. Apparently, {t : Xt-(uj) < a}

254
Chapter IX Stochastic Integrals
and {t : Xt~(u>) > a} differ from their interiors by at most countable sets.
Thus for almost all cj, dLa(X)(u;) does not charge {t: Xt-(uj) ^ a}. Now
put
U(r) =
{r, if Xr- = a,
oo, if Xr- ^ a,
V(r) = mf{t > U(r) : Xt- ^ a},
W= U|^(r),V(r)[.
r>0
For each u the section W^ is the interior of {t : Xt-(uj) = a}. By the
same argument, for almost all a;, dL?(X)(u) does not charge W^. D
Integrating 7[x_=a] and ?[X-<a] w.r.t. the two sides of (43.1), we obtain
the following two formulae for local times.
9.45 Corollary. Let X be a semimartingale and a G R. Then
L?(X) = 2[[tI[Xs_=a]d(Xs-a)+- £ /[Xa_=a](*5-aH,
lJ0 0<s<t J
L?(X) = 2f fl[Xa_<a]d(Xs-a)+- £ /[xa_<a](*5-aH.
Below we will give another expression for (C*) in (42.1) by means of
local times, and obtain a generalization of ltd formula.
9.46 Theorem. Let X be a semimartingale, fbe a continuous convex
function on R, and f be its left derivative. Then
f(Xt)=f(X0)+ f f'(Xs.)dXs
JO
+ E \f(Xs)-f(Xa.)-f'(Xs.)AXs]+l f°° LUX)p(da),
0<s<t L J ^ J-oo
(46.1)
where p is the second order derivative of f in the sense of generalized
functions (p is a Radon measure).
Proof. First assume that the measure p is finite and has compact
support. Put g(x) = (x — a)^p(da). Then / and g have the same sec-
J—oo
ond order derivative, and hence f(x) = a + bx + g(x), a,b e R. Evidently,
(46.1) holds for f(x) = a + bx. Hence we may assume f(x) = g(x). Thus
/oo
I[x>a)p(da). Since I[x,_>a\(Xa - a)~ + I[x,_<a](X3 - a)+ =
-OO

§6. Local Times of Semimartingales 255
(Xs - a)+ - (Xs- - a)+ - I[Xa_>a]AXs, (46.1) follows by integrating the
two sides of (43.1) w.r.t. p(da) on (—00,00).
For an arbitrary convex function /, put
( f(n) + f(n)(x — n), x > n,
fn(x) = I f(x), -n<x<n,
{ f(-n) +f(-n)(x + n), x < -n,
Tn = inf{* > 0 : |Xt| > n or |Xt_| > n}.
Then pn(da) = 7j_nin[(a)p(da), and /n(X) = f(X) on [0,Tn[. In addition,
when \a\ > n, we have L^n(X) = 0. Thus
/oo /*oo
L?(X)pn(da) = / L?(X)p(da), t < Tn.
-OO ' J — OO
Applying the result obtained above to /n, we know that (46.1) holds on
[0,Tn[. Then (46.1) follows by letting n -► 00. □
Remark. In the proof of Theorem 9.46 we need to deal with the
stochastic integrals with parameters and the results of Fubini theorem
type. The treatments are ordinary, and the details are omitted (cf.
Strieker and YorW).
9.47 Corollary. Let X be a semimartingale, and g be a non-negative
or bounded Borel function. Then
f g(Xs)d(Xc)s = f°° L?(X)g(a)da. (47.1)
JO J-oo
Proof. Let / G C2(R). Comparing (46.1) with ltd formula, we obtain
f f"(Xs)d(Xc)a = f f"(Xa-)d(Xc)3 = f°° L?(X)f"(a)da. (47.2)
JO Jo J—oo
Then (47.1) follows from (47.2) by the monotone class argument. □
9.48 Remark. Let A be a Borel set in R. From (47.1) we get
flIA(Xs)d(Xc)s = / Lat(X)da. (48.1)
^0 J A
Thereby we have the following interpretation for local times. Taking d(Xc)
as the measure of the "intrinsic" time of X, (48.1) means Lf(X) is the
density of the "intrinsic" occupation time of X at a up to moment t.

256
Chapter IX Stochastic Integrals
§7. Stochastic Differential Equations:
Metivier-Pellaumail's Method
In this paragraph we will investigate stochastic differential equations
by using Metivier-Pellaumail's method. This method is based on a stopped
Doob's inequality (51.1), due to M. Metivier and J. Pellaumail.
9.49 Lemma. Let (fi,^7, P) be a probability space and Q be a sub-
a-field of J7. Assume that A G T and X is a Q V {A}-measurable square
integrable r.v. such that E[X\Q\ = 0. Then
E[IAcX2) = E[IAE[X2\g)). (49.1)
Here Q V {A} stands for the a-field generated by QU {A}.
Proof. Put
a = E[IA\g\, b = E[IAc\Q\.
Then a + b = 1. Since X can be expressed as £IA + wIac with f, i) 6 L2(Q),
we have of + brj = 0, and
E[IAE[X2\G}\ = E[IA(ea + V2b)} = E[£2a2 + V2ab],
E[IAcX2} = E[rj2b] = E[rj2b(a + b)] = E[r)2b2 + Tj2ab\.
Thus we get (49.1), because £2a2 - rj2b2 = (fa + T]b)(£a - t]b) = 0. □
9.50 Lemma. Let M be a square integrable martingale. For any
stopping time T we have
E[(E[AMt\Ft-})2} < E[(M)T-\. (50.1)
Proof. If M is quasi-left-continuous, then
E[(E[AMt\Tt-})2} < E[(AM%)} < E[M]T
= E[(M)T] = E[(M)T-].
If M is accessible, we take a sequence (5n) of predictable times with
disjoint graphs such that \J[Snl D [Taj where Ta is the accessible part of
T. We may further assume Sn < T on [Sn < oo], otherwise we can replace
Sn by (Sn)[sn<T\- Under this assumption we get
(E[AMT \Ft-])2 = (E[AMTI[T=Ta]\FT-])2
= E(E[AMT\TT.])2I[T=Sn] = UE[AMSn\TT.})2I[T=SnV
n n

§7. Stochastic Differential Equations: Metivier-Pellaumail's Method 257
Since Tr- n [T = Sn] = Tsn- n [T = Sn] = (*s»- V {[T = Sn]})n[T = Sn]
and JFS„_ V {[T = Sn]} = ?sn- V {[Sn < T]}, we have
(E[AMsn\rT-})2I[T=sn) = (E[AMSn\fsn- V {[Sn < T]}])2I[T=Sn].
Set
Xn = E[AMSn\FSn-V{[Sn<T}}).
Since AMSnI[T<sn] = 0, we have X2/[T=Sn] = X*/[r<s«]- Therefore, by
Lemma 9.49
E[X*I[T=Sn]] = E[XlI[T<Sn]\ = E[E{X^sn-]I[Sn<n]
< E[E[AMln\TSn-}I[Sn<T\} = E[A(M)SnI[Sn<T]],
whence
£p[AMT|fr-])2] < EE[A(M)SnI[Sn<T\} < E[(M)T-}.
n
Thus we have proved (50.1) for the quasi-left-continuous case and the
accessible case. For general cases, let Mi = Mo + Mc + Mdi and Ma =
M^ (see Theorem 6.22.3)), T* (resp. T°) be the totally inaccessible part
(resp. accessible part) of T. Then (50.1) follows from the facts that
(M) = (AP) + (Ma) and
{E[AMT\TT-\? = {E[AMT\7TJi)'zI[T=Ti] + (B[AMr|fr-])2/[T=ri
= (E[AM*.\TT.]))2 + (E[AMf\fT-])2. a
9.51 Theorem. Let M be a square integrable martingale. Then for
any stopping time T we have
£?[(Mf_)2] < 4E[{M)T- + [Af^lr-]. (51.1)
Proof. Put
M = M- (AM? - E[AM£\fT-])I[TM. (M° = M**.)
Then M is a square integrable martingale (see Problem 5.3) and M
coincides with M on [0, T[. By Doob's inequality we have
E[Mfi_] = E[Mtf_] < E[M}2] < 4E[M$\ = 4E[[M]T]
= 4E[[M]T - (AMf)2 + (E[AM£\FT-])2],
which together with (50.1) gives (51.1), because
E[[M]T - (AM?)2] = E[[Ma]T. + [AP]T]
(Mi = M0 + MC + Mdi), and
£[[¥%] = EiiM^r] = EKM^t-}. □

258
Chapter IX Stochastic Integrals
The following theorem, due to M. Metivier and J. Pellaumail again, is
essential for studying stochastic differential equations.
9.52 Theorem. Let X be a semimartingale. There exists an adapted
increasing process A which controls X in the following sense: for any
stopping time T and any bounded predictable process H,
E[(H.X)#_] < E[AT-(H2.A)T-}. (52.1)
Proof If X is an adapted process with finite variation, then X is
controled by its variation process A: At = \dXs\. In fact, by Schwarz
J[o,t]
inequality we have
(ff.X)r- < (l#M)r- < AT-{H2.A)T-.
If X is a locally square integrable martingale, then by Theorem 9.51 we
have
E[(H.X)#_] < E[(H\B)T_],
where B = 4([X] + (X)). Let A = \/2JB, then A controls X, because
dB < AdA.
E[(H.X)*T2_} < E[((H2A).A)T-] < E[AT-(H2.A)T-}.
Finally, for a semimartingale X, let X = M + V, M G M2loc and V G V0.
Put A = y/2 I \dVs\+4([M] + (M))2, then it is easy to see that A controls
Jo
X. In fact, if A' controls X', and A" controls X", then y/2(A' + A")
controls X' + X". U
Remark. Metivier-PellaumaiTs inequality (52.1) indeed gives a
characterization for semimartingales (see Problem 9.29).
Now we are in a position to study the following stochastic differential
equation:
X = H+Z(FiX).Z\ (53.1)
where Z = (Zl, • • •, Zn) is an n-dimensional semimartingales with Zo =
0, H = (H1,*",Hm) is an m-dimensional cadlag adapted process (i.e.,
each component Hl is a cadlag adapted process) and Fj, 1 < i < n, are
mappings from the set of all m-dimensional cadlag adapted processes to
the set of all n-dimensional locally bounded predictable processes such
that for each stopping time T, Fi(XT~) coincides with F{X on ]0, Tj. X =
(X1, • • •, Xm) is the unknown process. For instance, let fi(uj, s, xi, • • •, xm)

§7. Stochastic Differential Equations: Metivier-Pellaumail's Method 259
be an n-dimensional measurable function on ft x R+ x Rm such that 1) for
fixed xi, • • • xm and s, /»(•, s, x\, • • •, xm) is .^-measurable; 2) for almost
all uj and for fixed xi, • • •, xm, /i(u;, •, xi, • • •,xm) is left-continuous with
right limits; 3) for almost all u and all s, fi(u;,s,-) is continuous. Put
{FiX)t = fi(uj,t,Xl_, • • •,-XT™). Then Fj meets the above requirements.
Obviously, the exponential equation (39.3) is a special case of (53.1).
9.53 Theorem. If each F{ satisfies the following Lipschitz condition
_ r m . -i
E[(FiX - FiY)2] < CIS £ (X* - n2] , (53.2)
w/iene C is a constant, then equation (53.1) has a unique solution.
Proof. For the notational simplicity we only prove the theorem for the
case n = m = 1. Let A be an adapted increasing process with Aq = 0 such
that A controls the semimartingale Z. Put Tq = 0 and define a sequence
(Tn) of stopping times as follows:
Tn+1 = inf {t > Tn : A\ - A$n > ± }.
Then Tn | +00, and ^rn+1_ - A\n < — on [Tn < 00]. In order to
prove the theorem it suffices to show that if equation (53.1) has a unique
solution on [0, Tn], it also has a unique solution on [0, Tn+i]. To this end,
set *(X) = H+(FX).Z. Assume that X is a solution of (53.1) on [0, T„],
and X is unique up to Tn. Let Y and W be two cadlag adapted processes
such that YTn = WT" = XTn. We have
J5[((«(y) -*(w)yTn+i_)2} = e[((fy - FW).zyTn+xj>
<E\ATn+l- I (FY - FW)2sdAs\
= E[ATn+l- [ (FY - FW)23dAs]
1 J]Tn,Tn+i[
< E[ATn+l.(ATn+1. - ATn)(FY - FW)%+1_]
< \e\(y - wyTn+l_)\
Then by the fixed point theorem there exists a cadlag adapted process
Y such that YTn = XTn and $(F)Tn+1~ = yTn+1~. Moreover, such a
process Y is unique on [0,Tn+i[. Put
Y = YT^~ + (AHTn+1 + (FY)Tn+1AZTn+1)IlTn+uoo[.

260 Chapter IX Stochastic Integrals
Then Y is a solution of (53.1) on [0,Tn+i] and Y is unique on [0,Tn+iJ.
Thus we are done. □
9.54 Corollary. Let W = (W1, • • •, Wn) be an n-dimensional
standard Wiener process, i.e., W1,*** ,Wn are independent standard Wiener
processes, £ be an n-dimensional To-measurable r.v.. Letb?(t,x), <r/(£,x),
i = 1, • • •, n, j = 1, • • •, m, be measurable functions on i£+ x Rm such that
m n m
E |6>(*,s) -V(t,y)\+ E E K(«,x)-(7f(«,y)|
j = l 1=1 j = l
<c(g|**-|f»-|2)1/a, (54.1)
171 I . _ 71 171 I . / 171 . „\
E V(*,x)|2+ E E k('^)l2<c2(1+ S M )> (54-2)
j=i' t=ij=i' v j=i 7
where C > 0 is a constant Then the equation
Xt = i + f b(s, X3)ds + f (7(5, X3)dW5, t > 0, (54.3)
Jo Jo
has a unique continuous solution, where b(t,x) = (&*(£, x), • •• , 6m(£,x)),
*(«,*) = (of (t,x)).
Usually, equation (54.3) 25 cai/ed an /£o equation.
Proof. It suffices to deal with the equation
Xt = £+ [ b(s,Xs-)ds+ f a(s,Xs-)dWs,t>0. (54.4)
Jo Jo
The condition (54.2) guarantees that b(t,Xt~) and a(t,Xt-) are locally
bounded. Then by Theorem 9.53 (54.4) has a unique solution, which
is continuous. Hence (54.3) has the same unique continuous solution as
(54.4). □
9.55 Example. Let W be a Wiener process, £ be an To-measurable
r.v. and a > 0. Then the equation
Xt=Z-a f Xads + Wu t>0 (55.1)
Jo
has a unique solution
Xt = £e"°< + / e-^-^dWs. (55.2)
Jo
Indeed, (55.1) is the famous Langevin's equation, and its solution
(55.2) is called an Ornstein-Uhlenbeck process, which is used to model
the velocity process of a Brownian motion.

Problems and Complements
261
Problems and Complements
9.1 Let M be a local martingale, and H be a progressive process.
1) If H2\M] e A+, then HtM is the unique L e M2 such that for all
N € M2 E[{H\M, 7V])oc] = E[L„N„\.
2) If H2\M] € A^ then HtM € A^-
9.2 Let M be a quasi-left-continuous local martingale, and H be a
progressive process such that y/H2.[M] G A^- Then for all N G A^/oc we
have [#ftM] = #.[M.AT] and A(HtM) = ffAM.
9.3 Let X be a semimartingale, and M be a local martingale. The
compensated stochastic integral (AX)^M exists if and only if (X, M)
exists. In this case we have (AX)^M = [X, M] — (X, M).
9.4 Let M be a local martingale, and H be a progressive process. If
HtM exists and H2\M] e V+, then \/#2.[M] e ^
9.5 Let H be a closed subspace of M2. H is stable if and only if for
any M GH and predictable process i? with jB[(/f2.[M])00] < oo we have
H.M e W.
9.6 Let M, TV £ Mf^. Then there exists a predictable process if and
L e -Mj^ such that N = H.M + L, and L AL M.
9.7 Let X be a semimartingale, and H be a predictable process,
integrate w.r.t. X. Let X = M + A, where M 6 Mioc and A e Vq. This
decomposition is an //-decomposition of X if and only if H G Lm(M) and
for each £ > 0 £ |//3AA3| < oo a.s..
s<t
9.8 Let M be a local martingale and H e L(M). If H.M is a special
semimartingale, then H £ Lm(M).
9.9 Let X be a semimartingale and if € Z/(-X"). Then the process
H.X — HX is predictable.
9.10 Let X be a cadlag adapted process, dominated by a predictable
increasing process A. Then [A^ < oo] C [X^ < oo] a.s., and for every
stopping time T, [At = 0] C [Xf = 0] a.s..
9.11 Let X and Y be a pair of semimartingales. Let T be a finite
stopping time, and rn : 0 = Tft < T" < • • • be a sequence of stochastic
partitions of [0,T] with 6(rn) tending to zero. Then
sup ]£(Xr?» At — XT?At)(YTp At - Yr?At) + ^o*o - [^,*1t —>0.
KT ' t + '
9.12 Let M and TV be a pair of continuous local martingales with
Mq = No = 0. If M and N are independent, then (M, AT) = 0.

262
Chapter IX Stochastic Integrals
9.13 Let X and Y be a pair of semimartingales. Let t > 0 and
rn :0 = to <t" < - - < *JJj/n\ = t be a sequence of finite partitions of [0, t]
with £(rn) tending to zero. Show that as n —► oo
JO ^
Denote the limit by / Ys o dXs, and call it the Stratonovich integral of V
Jo
w.r.t. X. Find the chang of variables formula for Stratonovich integrals.
9.14 Let W = (Wt) be a standard Wiener process.
1) For each n > 1
JO 71+1 I Jo
2) If if = 5PT V^o] and M* = W? - M > 0, then H.M = W.
9.15 Let TV be a Poisson process and Tn be its n-th jump time. Let
1 fl
H be a, bounded predictable process. Then lim - / H3ds exists and is
*-KX> t JO
1 n
finite a.s. if and only if lim — T] Htl. exists and is finite a.s.. In this
n-*°° n k=i
case, the two limits are identical a.s.. This property is called "Poisson
arrivals see time average" in queueing theory.
9.16 Let X and Y be a pair of semimartingales. If [Y = 0 or VL = 0]
is evanescent, X/Y is a semimartingale.
9.17 Let X,Y e S. Then £(X)£(Y) = £(X + Y + [X,Y]).
9.18 Let X e So. If £(X) = 1, then X = 0.
9.19 Let X,Y eS. If £(X) = £(Y) and [£(X) = 0] is an evanescent
set, then X - X0 = Y - YQ.
9.20 Let X G «Sp,o and X = M + A be its canonical decomposition. If
[AA = —1] is an evanescent set, then there exists N G Mioc,o such that
£(X) = £(N)£(A).
9.21 Let X, y £ 5o. If [Ay = — 1] is an evanescent set, then there
exists Z e So such that £(X + Y) = £(Z)£(Y), and there exists a unique
Y' e <S0 such that £(Y)£(Yf) = 1.
9.22 Let H £ S and Z € <S be continuous. Then the unique solution
of the equation
Xt = Ht+ f Xs-dZ3, t>0,
Jo

Problems and Complements 263
is given by
Xt = £(Z)t{H0 + J*£(Z);ld(Hs - [H,Z]3)}.
9.23 Let X € Sq. Suppose AX is bounded, and there exists e > 0
such that for each A e]0, e[ £(XX) e Mice- Then X e Mloc-
9.24 Let X be a complex semimartingale, i.e., X = X' + iX", where
X' and X" axe real semimartingales. Then
Zt = exp [Xt -X0- \{Xc)t) 11(1 + AXs)e-AX°
s<t
is the unique complex semimartingale satisfying
Zt = \+ I Z3-dXs,
Jo
where (Xc) = ((X')c) - ((X")c) + 2i{(X'f, (X")c).
9.25 Let Z = X + iY be a continuous complex semimartingale such
that [X, Y] = 0, and / be an analytic function. Then
f(Zt) = f(Z0) + ftf'(Z3)dZs + I f f"(Z3)d(Z)3.
J0 4 Jo
In particular, if Z is a complex Brownian motion, i.e., X and Y axe
independent standard Brownian motions, then f(Z) is a local martingale.
9.26 Let X be a semimartingale. Then for alH > 0 / I[xa_=o]dX^ =
0 a.s..
9.27 Let X be a continuous local martingale with Xq = 0. Then
|A"| = M + L°(X), where M is a continuous local martingale with Mo = 0
and L°t(X) = supa<j(—Ms). Moreover, if X is a standaxd Wiener process,
so is M.
9.28 Let X be a semimartingale. Denote
Lt(X) = ±[Lt(X) + Lr{-X)], a€R.
1) For any continuous convex function f on R
f(Xt) = f(X0)+ f f'(Xs.)dXs
Jo
+ E [f(Xs) - f(Xs.) - f'(Xa.)AXa] + \[°° Lt(X)p(da),
0<3<t * J-OO
where p is the second order derivative of / in the sense of generalized
functions.
2) If L°(X) ^ 0, then for all /? e (0,1), \X\0 is not a semimartingale.

264 Chapter IX Stochastic Integrals
3) Assume that / is a non-negative continuous convex function on R
and f(x) = 0 <=> x = 0. Put f* = -(/+ + fL), where f'+ and f'_ are the
right and left derivatives of / respectively. Then
£?(/(*)) = f"(0) \ f I[x.-=o]dXa - £ I[x.-=o)X.] + lP({0})Lt(X).
lJ0 0<3<t * 1 I
9.29 Let X be an adapted cadlag process. If there exists an adapted
increasing process A such that (52.1) holds for any bounded elementary
n-l
predictable process H (i.e., J/ = ]£ CiJ]r1i,7;+l]> where Tq <T\ < • <Tn
i=0
are stopping times and & G bTT^i = 0, • • •,n — 1), then X is a semi-
martingale.
9.30 Let X, Y € S. We have the following formulae:
Lt(x v y) = Lt(*+ - y+) + jT /[xs<o]^(y),
L*(X V y) = jf /[y^ojdL^X) + jf* I[Xs<0]dLs(Y),
+ £l[x,_=Y.-=o]dLa(X+-Y+),
Lt(X V y) + Lt(X A Y) = Lt(X) + Lt(Y).
If L(X - Y) = 0, then
M^vy)= [tI[ys_<0]dLs(X)+ fI[x,_<0]dLa{Y).
«/0 «/0
9.31 Let I G5 and / be the difference of two continuous convex
functions on R. For any a € R we set
B(a) = {* : f{x) = a, |/;(s)| + |//(x)| > 0},
where f'T (resp. //) stands for the right (resp. left) derivative of /. Then
B(a) is at most countable and we have
Li(f(x))= E [f'r(x)+L?(X) + fl(x)-LT*(-X)}.
x£B(a)
9.32 Let W be a Wiener process and a > 0. Then Xt = e-atW2at,
t > 0, is an Ornstein-Uhlenbeck process.

Chapter X
Martingale Spaces Hl and BMO
The contents of this chapter belong to the fine and difficult parts of
modern martingale theory. The terms, spaces H1 and BMO, are borrowed
from modern analysis. H comes from Hardy and BMO is the abbreviation
of "bounded mean oscillation".
§1. T^-Martingales and BMO-Martingales
10.1 Definition. Let M be a local martingale. Put
||M||wi = E[y/[M]J. (1.1)
Denote
H1 = {M € Mloc : ||M||W. < oo}.
Each element of H1 is called an H1-martingale. Obviously, H1 is a linear
space.
10.2 Remark. 1) It is easy to check || • ||^i is a norm on H1. In
particular, if ||M||^i = 0, then M = 0.
2) Let M e M\oc and 2S[|M0|] < oo. Then by Theorem 7.30 we know
that there exists a sequence (Tn) of stopping times with Tn | oo such that
for each n, MTn e W1.
3) Let M e M2. Since E[yf[M]^\ < y[E\M]^ < oo, we have M eH1
and ||M||wi < ||M||^2.
4) Let M € W. Since 0M]^ = /M02 + £(AM3)2 < |M0|+£ |AM3|,
we have M 6 H1 and ||M||wi < ||M|U = E[ f \dMs\).
J[0too[
5) Let M eH1. Then for any stopping time T we have MT € Hl and
l|MT||wi<||M||wl.

266 Chapter X Martingale Spaces Hl and BMO
The following lemma is a refinement of the fundamental theorem for
local martingales (Theorem 7.17). By means of it, many problems
concerning local martingales can be reduced to ones for bounded martingales
and the compensations of single step processes with integrable variation,
then solving these problems is simplified greatly.
10.3 Lemma. Let M be a local martingale with Mq = 0. Put
A = U*MI[lAm>l]), V = A-A,U = M-V. (3.1)
Then there is a localizing sequence (Tn) for M such that for each n, UTn
is a bounded martingale and VTn is the sum of a finite number of
compensations of adapted single step processes with integrable variation. Here
we call the processes of form €Iit,oo[ single step processes.
Proof. Let Si = inf{f > 0 : \AAt\ > 1}, and define (Sk)k>2 by
induction as follows:
Sk+1 = inf{t>Sk:\AAt\>l}.
Then each Sk is a stopping time and Sk | +oo, [Sk] fl [Sj] = 0 when k ^ j.
In addition, we have
A= ZAMSkIlSkM.
By Theorem 7.17 there is a localizing sequence (Rn) for M such that for
each n, U1*™ is a bounded martingale and A1^ is a process with integrable
variation. Put Tn = Sn A R^. Then Tn | oo and
AT« = t AMskI[Sk<Tn]I[sk,ool VT" = AT» - IT».
But
E \±Aa\< £ \AMSk\I[Sk<Tn],
S<Tn fc=l ~
AMSkI[sk<Tn] is integrable and ^-measurable, i.e., AMSkI[sk<Tn]I[sklool
is an adapted single step process with integrable variation, 1 < k < n.
□
10.4 Lemma. Let MgW1 and (Tn) be a sequence of stopping times
such that Urn Tn = oo. Then Urn \\M - MTn\\Hi = 0.
71—+00 71—>00
Proof. Since £„ = J[M - M^ = ^[M]TO - [M]Tn -> 0 and £n <
JiM}^, we have E[£n] -» 0, i.e., lim \\M - MT» \\ni = 0. □
V 71—►OO
10.5 Theorem. The collection of all bounded martingale ( denoted
by M°°) is dense in H1.

§1. ^-Martingales and BA40-Martingales
267
Proof Since M°° is dense in M2 and the norm of M2 is stronger than
the norm of W1, it suffices to show that M2 is dense in H1.
Let M e H1. We want to show that there is (M^) C M2 such that
||M(n) - M||wi -► 0. Obviously, we may assume that Mo = 0 and M is
the compensation of a single step process by Lemmas 10.3 and 10.4.
Let T > 0 be a stopping time and f € Tt be an integrable r.v.. Put
A = £/[t,oo[, M = A-A.
Then M<n) € X2 and by Remark 10.2.4) we have
||M<"> - M\\w < ||M<"> - M|U < 2||A<»> - A\\A < 2E[^Im>n]] - 0,
where the second inequality follows from Theorem 5.22.2). □
10.6 Definition. Let M be a square integrable martingale. Put
/£7(M0O - Mr_/[T>0])2
\\M\\bmd = sup J \>i! , 6.1
where T is the collection of all stopping times and - = 0 by convention.
Put
BMO = {MeM2: \\M\\emo < <»}. (6-2)
Each element of BMO is called a SA4C?-martingale. It is easy to check that
BMO is a linear space, || • ||hmc> is a norm on BMO and ||M||^2 < ||M||ha4D
for M e M2.
10.7 Lemma. Let M € M2. Then for any stopping time T
EKMoo - MT-I[T>0])2\FT)
= E[[M]oo\Tt]-[M}t + M2I[t=0] + (AMt)2 a.s.. (7.1)
Proof. We have
E[(Moo - MT-I[t>o))2\^t} = £[(Moo - MT- + M0I[T=0])2\fT}
= E[Ml\fT] + (Mr_)2 + M02/[T=0] - 2MrMT-
= E[Ml - M$\TT] + M02/[r=01 + (AMT)2
= EilM}^ - [M)T\FT] + M2I[T=0] + (AMT)2, a.s.. D
10.8 Lemma. Let M € M2. Then M € BMO if and only if there
exists a constant c > 0 such that for any stopping time T
^(Moo-Mt-Z^o])2!^]^2 a.s.. (8.1)

268 Chapter X Martingale Spaces Hl and BMO
Proof. Necessity. Let M e BMO and c = ||M||ha40- Then for any
stopping time T
£;[(Moo_ - Mr_/[r>0])2] < c2P(T < oo). (8.2)
Lei A € TT- Replacing T by TA in (8.2) yields
E[(MX. - Mr_/[T>0])2] < c2P(A n [T < oo]) < c2P(A).
Hence (8.1) follows.
Sufficiency. Because (8.1) holds for any stopping time T, we have
E[(M^ - MT-/[T>o])2|^r] = E[(M^ - Mt-/[t>o])2|^t]/[t<oc]
<c2/[T<oc], a.s.. (8.3)
Taking expectations in (8.3), we obtain (8.2). Then M e BMO. □
10.9 Theorem. Let M be a local martingale. Then the following
statements are equivalent:
1) Me BMD,
2) There exist constants c\,C2 > 0 such that \Mq\ < c\ a.s., and for
any stopping time T, |AMx| < c\ a.s. and
E{[M\oo - [M]T) < 4P(T < oo), (9.1)
3) There exist constants c\,C2 > 0 such that |Mo| < c\ a.s., and for
any stopping time T, |AMx| < c\ a.s. and
E[[M]J^T] - [M]T < 4 a.s., (9.2)
4) There exists constants C\,C2 > 0 such that \Mq\ < c\ a.s., \AM\ <
c\ and for all t>0
E[Moo\?t]-[M]t<<$ «•«•• (9-3)
In particular, BMO-martingales are locally bounded martingales.
Proof 1)^=>3) can be seen from Lemmas 10.7 and 10.8. The proof of
2)«<=^3) is similar to that of Lemma 10.8. 3) => 4) is trivial. It remains to
show 4)=>3). Let (Xt) be the cadlag modification of (^[[M]^!^]). Then
by (9.1) and the right-continuity of X — [M], for almost all u we have
Xt(u) - [M]t(u>) < 4
for all t > 0. In particular, for all stopping time T
Xt — [M]T < c\ a.s.,
i.e. (9.2) holds. □
10.10 Remarks. 1) Let M e BMO. By Theorem 10.9.2) we have
\\Mc\\bmd < \\M\\bmd and WM'Wbmd < \\M\\bmd-

§1. ^-Martingales and &MO-Martingales 269
2) Let £ be an a.s. bounded r.v.. Denote by ||f ||l°° the a.s. supremum
of f. It can be seen from Lemmas 10.7 and 10.8 that for M € M2
\\M\\Imd = SUP \\E[[M]oo - [M]T_7[r>0]|^T]||L~
TeT
= sup \\E[(Moo- ~ Mt-/[t>o])2|^t]||l-.
In particular, by Lemma 10.7 for M e BMD, \Mq\ < \\M\\bmo as- and
|AM| < ||Af Hbmo- K M € M\oc and M 0 BMD, we define ||M||aMC? =
+oo.
Below we give some useful examples of BMO martingales.
10.11 Theorem. Let M be a bounded martingale. Then M € EMO
and
\\MWbmo < 2HMOOHIOO.
Proof Let c = ||Moo||l<». Then for any stopping time T
|MT|H^[Mx>|^r]|<c a.s.,
£?[(Afoo - MT-I[t>o])2\^t] < (2c)2 a.s..
Hence ||M||aMO < 2c. □
10.12 Theorem. Let A = (At) be an adapted integrable increasing
process, and M = (Mt) be the cadlag modification of (ElA^^t]). If there
is a constant c > 0 such that 0 < M — j4_ J]0)Oo[ < c, then M G BMO and
||AfHbmo < V3c.
Proof Let X = M - i4_/j0ooj. Then 0 < X < c and for any stopping
time T
E[(Moo - MT-I[t>o])2\^t] < EKAn - AT.I[T>0])2 + X$_I[T>Q]\TT)
<[(A^-AT.I[T>Q])2\TT] + c2 a.s.. (12.1)
For any increasing function (at) and t>0we have
aoo ~~ at- = / as-das + / asdas > 2 / as-das,
./[*,oo[ «/[t,oo[ -/[*,oo[
(aoo - at-)2 < 2 / (aoo - as-)das.

270 Chapter X Martingale Spaces H1 and BMO
Noting that (Xt) is the optional projection of (Aqo — At-I[t>o]), we get
EiiA^ - At-I[t>o])2\Ft] < 2E[J (Aoo - As-I[s>Q])dAs\rT]
= 2E\ I X3dAs\TT] < 2cE[A00 - AT-I[t>o]\^t)
= 2c[MT - AT-I[t>o]] < 2c2 a.s..
Hence, by (12.1) we find
E^Moo - MT-I[t>o))2\Ft) < 3c2 a.s,
and therefore by Remark 10.10.2) we have ||M||fl/\4c> < V%c. D
10.13 Theorem. Let A = (At) be a predictable integrable
increasing process with Aq = 0, and M = (Mt) be the cadlag modification of
(ElAoolTt]). If there is a constant c > 0 such that for allt > 0, Mt—At < c
a.s., then M e BMO and ||M||hmc? < 2\/3c.
Proof. Let Y = M—A. Then 0 < Y < c. Y is a special semimartingale
and Y = M + (-A) is its canonical decomposition. By Theorem 8.8 we
have 0 < AA < c. Hence, M - A_ Jj0)OO[ = M - A- = Y + AA < 2c.
By Theorem 10.12 we know M G BMO and ||M||bmc? < 2>/3c. D
10.14 Theorem. Let M £ M2, B be an adapted increasing process
and L = (JB_).M. If for all t > 0,|JB*Mt| < 1 a.5., then L e BMO and
\\L\\bmo < 2.
Proof. Let T be a stopping time. Since [L] = B2_ .[M], by the formula
of integration by parts we have
Woo - [L]t-I[t>o] = j[Too[Bs-diMl
= J[TJlMl°° - \M\s)dB2s + ([Mlc - [M]T_)B2_J[T>0]. (14.1)
Because [M] - M2 € M, ([M]^ - [M]t) and (M^ - Mt2) have the same
optional projection. By Theorem 5.16.1) we obtain
E[f ([M)00-[M]s)dB2a\TT\=E[j (Ml-M*)dBl\FT\
lJ[T,oo[ J lJ[T,oo[ J
<E[Ml(Bl-B*_I[T>0])\FT)
—I[T>0] as-
(14.2)

§1. ^-Martingales and &MO-Martingales 271
On the other hand,
E[([M]oo-[M]r_)fl2_J[T>0]|.Fr]
= JEKMoo - [M]t)\*t\Bt-Iit>o] + AM*B*_I[T>0]
= E[Ml - M\\TT\B\_LT>a + AM*B*
< E[M^T]B2_I[T>0] - M*B*_I[T>0]
+2(M|. + M$._)B%._I[T>0]
< E[Ml\FT]B%._I[T>0] + M^B\_ + 2Ml_B\_
< E[Ml\FT]Bl_I[T>0] + 3. (14.3)
By (14.1)-(14.3) we get
E[[L}oo-[L}T-I[T>o]\fT}<4 a.s..
Hence L € BMO and ||I-||hmo < 2. D
10.15 Corollary. Let M € M2 and H be a predictable process. If
there exists an adapted increasing process B such that \BM\ < 1 and
\H\ < B-, then H. M € BMO and \\H. M||hmo < 2.
Proof. Let L = B-.M. Then for any stopping time T
[H. M}^ - [H. M]T_I[T>0] = j^^H2sd[M)a
< f B*_d[M\a = [L]^ - [L]T_I[T>0].
Since L e BMO, we know H. M e BMO and \\H. M\\emd < ||£||flM0 < 2.
D
Finally, we conclude this paragraph with an interesting property of
BMCJ-martingales.
10.16 Theorem. Let M 6 BMO. Then for any stopping time T
\\Mt\\bmo < ||M||aMO, \\M - MT\\BMO < \\MWbmo. (16.1)
Moreover, if stopping times Tn | oo a.s., \\MTn\\s\4D T H-^llaMO-
Proof. Let 5 be a stopping time.
Moo - [MT)s_I[s>o] = ([M]T - [M]5_7[s>o])7[s<t]
<[M]M-[M]S./|S>0],
[M-Mr]00-[M-MT]s_/[s>o]
= [ML - [M)S_I[S>0] - [M]T + ([M)T)s-I[s>0]
< [M]TO - [M]S_J[S>0].
Thus (16.1) holds.

272 Chapter X Martingale Spaces H1 and BMO
If stopping times Tn | oo a.s., for any stopping time 5
[MTnloo - [MT%_I[s>o] T [M]^ - [M]S_I[S>0].
By Remark 10.10.2) it is easy to see ||MTn||aM0 T ll^llfiMO. n
Remark. We have \\M — MTti\\b\4D I- But it need not be \\M -
MTn \\emd I 0 in general. In fact, Dellacherie, Meyer and Yor[l] showed
that if M°° ^ BMD, then M°° is neither a closed set nor a dense set in
BMO.
§2. Fefferman's Inequality
Fefferman's inequality, given in the following theorem, is the most
important result concerning H1- and BA4C?-martingales. It is much deeper
than Kunita-Watanabe inequality.
10.17 Theorem. Let M and N be a pair of local martingales, and U
be a progressive process. Then
Elf \Us\\d[M,N}s\]<V2E[([ U^Miy^WNWBMo. (17.1)
lJ[0,oo[ J LV./[0,oo[ ' J
In particular, putting U = 1, we have
E[ I \d[M,N]s\] < VtWMWniWNWa*,. (17.2)
Proof. We may assume that the right-hand side of (17.1) is finite. Put
Ct= I U?d[M]s.
J[o,t]
We define two non-negative optional processes H and K as follows:
Thus
H?K2t > ±U*IlCi>o],
2
and by the formula of integration by parts
Hfd[M\t = I[Ct>Q)— * = I[Ct>o]dVct.
VCt + yJCt-I[t>Q]

§2. Fefferman's Inequality 273
Since (by Kunita-Watanabe inequality)
/ \U3\I[Ca=0]\d[M,N}s\<([ U2l[c,=0]d[M]3y/2([NU^
= ([ I[c.=0]dC3y/2([NU^ = 0, a.s.,
'[0,oo[
we have
<
where
^E[J[QM\UaMM,NU = >[/0^.>o,^IMA^U
Elf \HaKa\\d[M,N\a\<y/E^y/E;, (17.3)
Ly[0,oo[
Zz = e{[ KjdlN],} = E[ f ({N]«, ~ im-Wl (1T.4)
LJ[0,oo[ J L*/[0,oo[ J
Because the optional projection of {[N]^ - [N]t_) is (E^N^Ti) - [N]t_)
and the latter is bounded by HNH^^ on ]0,oo[, we have
E2 = E[f (E[[NUTs]^[N}sJdK2s<\\N\\l^ (17.5)
Then (17.1) follows from (17.3)-(17.5). D
The following theorem is a strengthened form of Fefferman's inequality.
10.18 Theorem. Let M and N be a pair of local martingales, U be
an optional process and T be a stopping time. Then
e[[ \Us\\d{M,N}s\}<V2E[(f fE#W.)1/a]||JV||HMD. (18-1)
e[[ |c/3||d[M,iV]j^r]<^£;[(/ uMMlY'VrlWNWi
lJ[T,oo[ J LV«/[T,oo[ ' J
\\BMD-
(18.2)
Proof Replacing U by t/ipr,oo[ in (17.1) yields (18.1). For any A e TT,
replacing T by TA in (18.1) gives (18.2). □
As an application of Fefferman's inequality we will show that H1-
martingales are uniformly integrable martingales.
10.19 Theorem. Let M e Hl. Then M is a uniformly integrable
martingale, and
||Moo||i<2V2||M||w». (19.1)

274
Chapter X Martingale Spaces H1 and BMO
Proof. First of all, assume that M is the sum of a bounded martingale
and a martingale with integrable variation. Then for any bounded
martingale N by Theorem 6.4, Theorem 6.28.1) and Fefferman's inequality
the inequality
\E\MeoNooH = \E[M,NU < V2\\M\\h4N\\sjw
< 2V5||Af||«i||JV00|Uoo, (19.2)
comes from Theorem 10.11. Putting A^ = sgnMoo in (19.2) yields
||Moo||i<2V2||M||wi,
where sgn(rr) = 1, 0 or -1 while x > 0, x = 0 or x < 0.
For an arbitrary M G H1 there exists a sequence (5n) of stopping
times such that Sn | oo and for each n, MSn is the sum of a bounded
martingale and a martingale with integrable variation. Then
\\MSn - MsJ\i <2>/2\\Ms» - Ms™\\Hi. (19.3)
Since ||M5n - M\\ni —► 0 as n —► oo, (19.3) means that (Msn) is a
fundamental sequence in L. Hence M$n —> £ € L1. Let (&) be the cadlag
modification of (JE[f I-FJ). For any given m
6A5m = E[Z\TtASrn] = L1- Vbn^ E[MSn \FtASm] = MtASm a.s..
Thus for all t > 0, Mt = & a.s.. In particular, M is a uniformly integrable
martingale and M^ = £ a.s.. In addition, we have
HMoollx = Urn ||M5n||i < 2v/2 Urn ||MS"||W1 = 2>/2||M||wi. □
71—»00 7i—>00
§3. The Dual Space of H1
First of all, we give a useful characterization for jKMO-martingales.
10.20 Theorem. Let N eM2. Then N e BMO if and only if there
is a constant c > 0 such that for all M G M2 ^
|£[M,Ar]J<c||M||w,. (20.1)
In this case, ||Af||flA40 < y/Ec.
Proof The necessity comes directly from Fefferman's inequality by
taking c = v^ATKaa/IP- We want to show the sufficiency.
J) It is easy to see that this is equivalent to that (20.1) holds for any bounded
martingale M.

§3. The Dual Space of H1 275
First of all, we are to show |AT0| < c a.s.. Let B = [|7V0|] > c]. Suppose
P(B) > 0. Put £ = S-^Ib- Then |(| = ^IB,EM = 1. Put
Mt = £, t > 0. Then M € A<2, ||M||wi = E[\t\] = 1, and
Wj#,«UI-*IW«-*[|^i^]
>c = c||M||wi.
This contradicts the assumption. Hence, it must be P(B) = 0.
Secondly, we want to show |A7V| < 2c. Let T be a predictable time or
totally inaccessible time, and T > 0. Suppose P([\ANt\ > 2c]) > 0. Put
sgn(A7VT) T
5 " P([|AATT|>2c])i«A^l>2c'*
Then £ e 6Jr, and £/[T=oo] = 0. Put M = £/[Too[ - (£/[T,oc[)p. Then
M £ A4 2 and only at T, M has a jump:
f £ - JE[f |.Ft-.], if T is predictable,
AMT = \
{ £, if T is totally inaccessible.
At the same time, we have ||Af ||wi < \\M\\A < 2||£J[T)00[||-4 < 2E[\£\] = 2.
Because E[ANt\Ft~] = 0 for predictable T, in any case by Theorem 6.28
we have
E[[M,N]oo\ = E[AMTANT] = E[£ANT]
-[Sf^]>-NIM|,
This contradicts the assumption. Hence, it must be P(\ANt\ > 2c) = 0,
i.e., \ANt\ < 2c a.s.. Therefore, for all stopping time T, lATVrl < 2c a.s..
Finally, Let T be a stopping time, M = N - NT,£ = [N]^ - [N]T.
Then M G M2 and [M]^ = [M, 7V]oo = £• By the assumption we have
E[£) = E^NU < c\\M\\ni = cE[^ = cE[y/Zl[T<oo]]
< c(E[^/2[P(T < oo)]1/2,
E[[N]oo - [N]T] = E[£] < c2P(T < ex.).
Now by Theorem 10.9 we know N € BMO and
\\N\\bm3 < y/c2 + (2c)2 = y/Ec. □
Now we are ready to show the dual space of H1 is &MD. More
precisely, we have
10.21 Theorem. Let (H1)* be the Banach space formed by all bounded
linear junctionals on Hl (i.e., (H1)* is the dual space ofH1). Let N €
BMO. Put
<pn(M) = e[[m,n]00], Men1. (21.1)

276 Chapter X Martingale Spaces W1 and BMO
Then N »-► ipx is a one to one linear mapping from BMO onto H1 and
^IIvjvII < IMIbmd < >/5\\<pnI (21.2)
where \\ip\\ denotes the norm of bounded linear function ip.
In particular, BMO with norm || • \\b\4D is a Banach space.
Proof Let N e BMO. By Fefferman's inequality
\<Pn(M)\ = \E[M,NU < V^WNWbmoWMW^, M e U\
Thus <pN e {HlY and \\ipN\\ < >/2\\N\\bmd.
If (pN = 0, taking account of the fact that BMO C M2 C W1, we
have ^[[A/'loo] = <pn(N) = 0 and N = 0. This means TV ■-► pN is a
one to one linear mapping from BMO into (H1)* (linearity is trivial).
It remains to show the image space of <p is the whole space (H1)* and
\\N\\bmo < V5\\<pnI
Let <p € (H1)*. Since M2 C H1 and || • ||„i < || • \\M2, for all M e M2
\<p(M)\ < \\<p\\\\M\\Hi < \\<P\\\\M\\M2.
This means that <£, restricted on Hilbert space M2, is a bounded linear
functional on M2. Hence, there exists a unique N G M2 such that for all
M eM2
\V{M)\ = WMaoNaoH = |^?[[AT,JV]cx>]| < IMH|M||wi.
By Theorem 10.20 we know N e BMO and ||JV||aM0 < >/5|M|.
Furthermore, <pn coincides with <p on M2. But M2 is dense in H1 (Theorem
10.5), so <ps and <p axe the same element of (H1)*. Hence
(W1)* = {<Pn:N€ BMO},
and BMO with norm \\(Pn\\ is isomorphic to (W1)*. By (21.2) \\ipn\\ and
||N||fiMC> axe equivalent norms. Thus BMO with norm || • ||ba40 is complete.
D
Remark. By the theorem we know that each bounded linear
functional on H1 has the form of (21.1). Replacing M2 by Mq in the proof of
Theorem 10.21, we obtain immediately the next theorem.
10.22 Theorem. Let N e BMO0. Put
<pN(M) = E([M,N}00), Me Hi
Then N »-► ips is a one to one linear mapping from BMOo onto (Hi)*
and
^llwll < ||JV||aMO < V5\\<Pn\\.

§4. Davis Inequalities 277
§4. Davis Inequalities
10.23 Lemma. Let M be a local martingale, and H be a progressive
process such that H. M G Hl and
Then for any N e BMD, [H. M, N] — H. [M, N] is a martingale with in-
tegrable variatoin. In particular, E[H. M,N]oo = E\ Hsd[M, N]s].
Proof. Since N G BMD, N is a locally bounded martingale. By
Theorem 9.10 [H. M, N]-H. [M, N] is a local martingale. By Fefferman's
inequality we have
E\ I \d[H. M,N]S\] < y/2\\H. MWwWNWbmo < oo.
E\ I \H3\\d[M,N}s\] < y/2E\( f H*d[M}s)l/2]\\N\\aMD < oo.
Hence [H. M, N] - H. [M, N] e W0. □
The following theorem is the first Davis inequality. In fact, its form
given here is a generalization of its ordinary one.
10.24 Theorem. Let M be a local martingale, H be a progressive
process such that y/H2.[M] is locally integrable. Then
E[(H. M)^] < 2V6E[(J^^d[M}sy/2}. (24.1)
In particular, we have
E[M^]< 2^1^11*,. (24.2)
Proof Evidently, we may assume E \( / H^d[M]s) < oo. In
this case (H. M)q is integrable. Thus we may assume further H. M GH1.
Otherwise, we may take a sequence (Tn) of stopping times with Tn | oo
such that for each n, H. MTn = (H. M)Tn eH1. At the same time, we
have^.M^^T^-M)^.
Let 5 be a non-negative finite r.v. and B = sgn(iJ. M)sI[s,oo[' Let A
be the dual optional projection of JB. Then for any stopping time T
E\ ( \dAa\\rT] <E\ f \dBs\M < 1 a.s..
lJ[T,oo[ ' J lJ[T,°o[ ' J
Hence
E[A^ - A+_I[T>0]\TT] < 1, E[A^ - A^_I[T>Q]\?T\ < 1 a.s.,

278 Chapter X Martingale Spaces H1 and BMD
where A+ and A" axe the positive and negative variation parts of A
respectively. Let N = (A^) be the cadlag modification of (jEfAool^]). By
Theorem 10.12 we know N e BMD and ||AT||aM0 < 2\/3.
Write L = H. M GW1. Let (Tn) be a sequence of stopping times such
that Tn | oo and for each n, NTn is a bounded martingale and LTn is the
sum of a bounded martingale and a martingale with integrable variation.
Thus we have
E[NTnLTn] = E[L, N]Tn = E[L, N7"}^ (24.3)
By Lemma 10.23 and Fefferman's inequality we have
E[LsATnsga(Ls)]
= E\ f L^dBs] =E\ f Lj"dAs] =E\ I LTndA3]
lJ[0,oo[ J L./[0,oo[ J lJ[0,oo[ J
= ElL^Aoo} = EILtM = E[LTnNTn} = E^N^U
= E\ j Hsd[M, NT«]S] <V2E\([ Hld[M]s)1/2} ||JVr»||aM0
LJ[0,oo[ J LW[0,oo[ J J
< 2V6E[(J^ H23d[M]s)1/2]. (24.4)
The last inequality holds, because WN^Weuw < \\N\\ba«d < 2\/3
(Theorem 10.16). On the other hand,
£sATnsgn(Ls) = \Ls\I[s<Tn] + LTnsgn(Ls)I[Tn<s]-
Since [Tn < S] j 0 and (£rn) ls uniformly integrable (since L e H1, we
have L e M by Theorem 10.19), E[LTnsgn(Ls)I[Tn<s]] —> 0 as n —► oo
and
Ji^ jE(LsATnsgn(Ls)] = lim^ E[\Ls\I[s<Tn]] = E[\LS\].
By (24.4) we have
E[\LS\] < 2V6E[(Iq^ H2sd[M]s)1'2]. (24.5)
Now for any given e > 0 put S(u>) = inf{£ > 0 : \Lt(v)\ > L^uj) -e}.
Since L e M and L^ < oo a.s., 5 > 0 takes on finite values, and \Ls\ >
L^ - e a.s.. Applying (24.5) to it yields
£[Z4] - e < E[\LS\] < 2V6E[(Jo^ H^d[M]s)1/2]. (24.6)
Letting e j 0 in (24.6) gives (24.1). □
The next theorem gives a strengthened form of the first Davis
inequality (24.2).

§4. Davis Inequalities 279
10.25 Theorem. Let M €Hl. Then for any stopping time T
E[M^ - Mi_I[T>0]\fT] < 4v/3£[v/[M]00 - [M]t-/[t>o]|^t]. (25.1)
Proof. Put Mt = (MT+t - Mt_/[T>0])/[t<oo] and ft = Ft+u t > 0.
Then M = (Mt) is a (ft)-local martingale (refer to Problem 7.14), and
[M\t = [M]T+t - [M]t-/[t>o]- Hence E[yJ[~M]00] < oo and M is an
^-martingale w.r.t. (ft). By Davis inequality (24.2)
iSpC] < 2V6E[yJ[MU - [M]T_/[T>0]].
Evidently, M^^W^^ M$_I[T>0]. Thus
E[M* - Mf_/[T>0]] < 2y/6E[y/[M]00 - [M]t-I[t>o]}. (25.2)
For all A e TT, replacing T by TA in (25.2) yields (25.1). □
The next theorem is an easy generalization of Theorem 10.24.
10.26 Theorem. Let M be a local martingale and H be a progressive
process such that y/H2.[M] is locally integrable. For any stopping time T
we have
E[(H. M)*T] < 2V6E[([ H23d[M]s)1'2]. (26.1)
Proof Since (H. M)*T = (H. MT)^ = (HI^.M)^ (26.1) follows
from (24.1). □
As an application of Theorem 10.26, we obtain the following
convergence theorem for stochastic integrals.
10.27 Theorem. Let M be a local martingale. Denote
L°(M) = {H : H is an optional process with Jh2\M\ € -4j£c}.
Let (ffW) C L°(M),H e L°(M) and T be a stopping time.
1) //«[(/o (Hin) - H3)2d[M}s)l/2] - 0, then
E[sup\(H(n\M)t - {H. M)t\] -+0.
2)IfEE\(f (#<»> - Hsfd[M]s)ll2] < oo, then
n=l LVJ[0,T] J
sup \(H^.M)t - (H. M)t\ -+ 0. a.s..
t<T ' '
The following theorem is the second Davis inequality.

280 Chapter X Martingale Spaces H1 and BMO
10.28 Theorem. Let M be a local martingale. We have
||M||wi<(7 + V2)£![AC]. (28-1)
Moreover, for any stopping time T
EyiMU - [M]T-I[t>0]\ft] < 2(7 + 4y/2)E[iC>\FT]. (28.2)
Proof We may suppose jB[M^] < oo. First we deal with the case of
Mo = 0. For given e > 0 put M = M + e. Then M^ > e, =5- < -.
Applying Corollary 10.15 to the bounded martingale Ht = 2£ =5-!^]
1V± 00
and adapted increasing process Bt = Mt, we know L = (M-).H 6 BMD
and ||L||hm0 < 2. By Fefferman's inequality
E\ I \d[M,L]s\\ < v^llMll^HLllaMO < 2>/2pf||wi. (28.3)
1 ./[0,oo[ J
Put K = M2 - [M] = 2(M_/]0,oo[).M. Then
[if, #] = 2(M_/|0iOo[).[M, tf ] = 2/,o,00[.[M, (M.).H\
= 2([M,L]-[M,L]0)
and by (28.3) we have
E[[ \d[K,H}3\] <W2||M||wi. (28.4)
L-/[0,oo[ J
Now we assume M, K € H1. Let (5„) be a sequence of stopping times
such that Sn | oo and {KH)Sn - [K, H]s" € M0. By (28.4)
\E[KSnHSn}\ = \E[K,H]Sn\ < 4n/2||M||w,.
But if is bounded and K e M, so KsnHsn —► Koo#oo- Thus
[g[Moo"[M]oo]| = ^[A-coHooll < W2||J7||wi.
Since —^ < M* we have
M2
Mi _
By Schwarz inequality
-]<f;[M^] + 4v/2||M||Ti,.
||M||W1 = E[yJ[MU\ < (E[lT„))^(E[^}y/2
AC
< (EiMl^/HEKo] +4V2||M||wl)1/2.

§5. Burkholder-Davis-Gundy Inequality 281
Solving this inequality, we get
\\M\\Hi < (2n/2 + 3)£;pC]- (28-5)
In order to relax the assumption that M, K € Ti1, we take a sequence
(T„) of stopping times such that Tn T oo and Af7", KTn € H1. Then by
(28.5)
||MT"||W1 < (2v^ + 3)£[(MT,X] < (2y/2 + Z)E\St00[.
Letting n —► oo, (28.5) remains valid. Noting that [M]oo = [Af]oo + e2,
from (28.5) we have
\\M\\Hi < ||M||wi < (2V2 + 3)(E[M^] + e).
Letting e 1 0 yields
||M||wi<(2^ + 3)£?[Mi]. (28.6)
Finally, we relax the assumption Mo = 0. Put M' = M — Mq. We
have (M')^ < M^ + |M0| < 2M^ and by (28.6)
||M||wi = E[y/[M]^} = E[y/[M%o + M^ < \\M'\\ni + E[\MQ\]
< (2x/2 + 3)E[(Mfyoo] + E[M^\ < (7 + iy/2)E[M^\.
The proof of (28.2) is similar to that of Theorem 10.25. □
As an important consequence of Davis inequalities, we have
10.29 Theorem. Let M be a local martingale. Then M GW1 if and
onlyifE[M^[ < oo. Furthermore, ||M||^i and ||Af£,||i are two equivalent
norms on H1. In particular, H1 with norm || • ||^i is a Banach space.
§5. Burkholder-Davis-Gundy Inequality
At first, we give two results from pure analysis. One is the classical
Young inequality, and another is concerning moderate convex functions.
10.30 Definition. Let $(t) be a non-negative monotone increasing
convex function on R+, and $(0) = 0. It is easy to see that there is
a non-negative right-continuous increasing function <p on R+ such that
$(£) = / <p(s)ds. We call <p the right derivative of $. Put
■/[o,t]
il>{t) = inf{s > 0 : <p(s) >t}, t> 0. (30.1)

282 Chapter X Martingale Spaces H1 and BMO
Then i/j is also a non-negative right-continuous increasing function on ii+
(the right-inverse function of </?, see Lemma 1.37). Put
tf(i) = / il>{s)ds. (30.2)
J[o,t]
Then #(t) is a non-negative monotone increasing convex function on R+)
, and is called the conjugate convex function of $(£).
10.31 Lemma. Le£ $(£) be a non-negative monotone increasing con-
vex function on i£+ with $(0) = 0, and #(t) be its conjugate convex
function. Then for all ix, v > 0 we have the following Young inequality
uv < *(u) + tf (v). (31.1)
Proof. We adopt the notations in Definition 10.30:
$(ix) + *(v) = / </?(s)ds + / i>(s)ds.
J[0,u] J[0,v]
If <^(ix) > v, then V>(v) < ^- By Lebesgue's lemma (Lemma 1.38) we
have (noting that sup{s : ip(s) < u} = <p(u))
$(ix) + #(v) = i/(/?(ix) - / sdip(s) + / ip(s)ds
J[0yu] J[0,v]
= mp(u) — / ip(s)ds
y{s:^(s)<tx}n]v,oo[
> wp(u) — u(ip(u) — v) = uv.
If </?(ix) < v, then t/>(v) > u. (31.1) can be shown in the same manner.
10.32 Definition. Let $(t) be a non-negative monotone increasing
convex function R+ with $(0) = 0. $(t) is called a moderate convex
function if there is a constant c > 0 such that for all £ > 0
*(2t) < c*(t).
Let $(£) be a moderate convex function and ip be its right derivative.
We introduce another constant p:
p = sup^. (32.1)
The next lemma summarizes the main properties of moderate convex
functions.
0) If <p(oc) = t0 < -foo, then rp(t) = -foo when £ > t0. Consequently, $(t) = +oo
when t > t0. But it is possible that ty(t0) < -foo. In this case, $(t) is only left-
continuous at to.

§5. Burkholder-Davis-Gundy Inequality 283
10.33 Lemma. Let $(t) be a moderate convex function on R+, #(t)
be its conjugate convex function, <p and x/j be the right derivatives o/$ and
^ respectively. Then
1) c>2, 1 <p< c-\,
2) for allt>\, u> 0, ${tu) < tp$(u),
3) for all u > 0, V(ip(u)) <{p- l)$(ix).
Proof. 1) We have
$(ix) = / ip(s)ds < uip(u) < l <p(s)ds
J[0,u] Ju
= $(2ix) - $(u) < (c - l)$(ix).
Hence 1 < p < c — 1. In particular, c > p + 1 > 2.
2) By the definition of p we know that for all s > 0 and u > 0
su<p(six)
Then for all t > 1
*(tix) ftuipisu) fl p
log-j-r^ = / , 'ds < -ds = plogt,
$(u) J\ $(su) Ji s
i.e., $(ftx) < *'$(u).
3) By Lebesgue's lemma we have (noting that sup{s : rf(s) < u} =
<p{u))
/ sdip(s) = / ip(s)ds = / ip{s)ds,
J[o,u] J{s'4>(s)<u} J[o,<p(u)]
\!>(<p(i/)) + $(u) = / i/)(s)ds + wp(u) - / sdip(s)
•/[O^(tx)] ^[0,u]
= utp(u) < p$(u).
Thus 3) follows. □
Making use of the above two lemmas, we can show the following
10.34 Lemma. Let $ be a moderate convex function on it+,<p be its
right derivative, £ and rj be a pair of non-negative r.v.. If
E[*(Q] < oo, E[*(Q] < E[M0\, (34-1)
then
«[*(0] < P^'EMri)}- (34-2)
Proof Let \& be the conjugate convex function of $, and i/j be its
right derivative. By the convexity of 9 and p > 1 we have
»(^) < -*M0).
\ O / 0

284 Chapter X Martingale Spaces H1 and BMD
By Lemmas 10.31 and 10.33 we have
< prsin) + -*fo>(0) < ppHv) + —*(0-
p p
From (34.1) we obtain
E[*(Q] < (fE[*{rj)\ + ^E[*(Z)Y
Thus (34.2) follows. □
Remark. If in the lemma $(t) = tp(l < p < oo), then directly from
Holder inequality we get
E[MQ] = Ebb?-1)] < P{E[e])^{E[rf])l.
Prom (34.1) we have
E[&] <jPE[rf\. (34.3)
It is finer than the corresponding (34.2) (in this case p = p).
The following lemma is called Garsia 's lemma,
10.35 Lemma. Let A = (At) be an adapted increasing process, £ and
7] be a pair of non-negative integrable r.v.. If £ > Aqo a.s. and one of the
following two conditions holds:
a) £ £ ^"oo and for any stopping time T
E[Z\FT] - AT-I[T>o] < E[r)\TT] a.s., (35.1)
b) £ € ^"oo, A is predictable, Aq = 0 and for any predictable time T
JE[£|FT\ - AT < E[r)\FT] a.s., (35.2)
then for all A > 0 we have
f (£ - X)dP < J r)dP. (35.3)
Furthermore, if $ is a non-negative monotone increasing convex
function on ii+ ratf/t $(0) = 0, then
E[9(Q] < E[W(0], (35.4)
where <p is the right derivative of $.
Proof. First of all, we point out that (35.3) and (35.4) are equivalent.
In fact, putting *(t) = (t- A)+ and (p(t) = 7[a,oo[(*) in (35-4) yields (35.3).
Conversely, integrating the two sides of (35.3) w.r.t. dip(\) yields (35.4).

§5. Burkholder-Davis-Gundy Inequality 285
Now we turn to show (35.3). Assume first a) holds. Put
T = inf{t : At > A}.
Then AT-I[t>o] < A. Since [£ > A] = [T < oo] U [T = oo,£ > A] e TT, by
(35.1) we have
/ (i-\)dP<\ (Z-AT_I[T>Q])dP< f rjdP.
•>K>A] J[Z>\] J[t>\]
Now assume b) holds. In this case, T = inf{t : At > A} is a predictable
time and T > 0. Let (Tn) be a sequence of predictable times foretelling
T. By (35.2)
E[^Tn] - ATn < E[r,\FTn] a.s..
Letting n —> oo, we obtain (Corollary 2.19 and Theorem 3.4.11))
E[£\TT] - AT_ < E[t]\Tt-} a.s.. (35.5)
Since [£ > A] = [T < oo] U [T = oo,£ > A] e TT- and AT- < A, (35.3)
follows from (35.5). □
By using of Garsia's lemma and Lemma 10.34, we obtain Burkholder-
Davis-Gundy inequality (or B-D-G inequality briefly) immediately.
10.36 Theorem. Let M be a local martingale and $ be a moderate
convex function on R+ such that $(M^) and $(\/[M]^) are integrable.
Then
p-(p+U(7 + 4V2)-*>E[$(y/[MT^)}
< E[*(M^)] < ^1(2v/6)^[^(V/[v/M]oc)], (36.1)
where p is defined by (32.1).
Proof. Set A = M*,£ = M^,t] = 2\/6>/[M]^- By Theorem 10.25
and Lemma 10.35 we have £?[$(£)] < E[w>(€)]- Then ty Lemma 10.34
we get
Em)]<pf+lE[*(r,)]. (36.2)
By Lemma 10.33 $(t?) < (2V/6)P^(>/[M]00). Thus the second inequality
of (36.1) follows from (36.2). The first inequality of (36.1) can be deduced
in the same manner. □
Remark. Let $(t) = tp(p > 1). The corresponding inequality (36.1)
is usually called Burkholder's inequality.

286 Chapter X Martingale Spaces Hl and BMO
§6. Martingale Space Hp, p > 1
10.37 Definition. Let M be a local martingale, 1 < p < oo. Put
Hp = {Me Moc : \\M\\hp < oo}.
Each element of Hp is called an W-martingale. Obviously, HP is a linear
space.
10.38 Theorem. Let 1 < p < oo. Put
Mp = {M e M : HMocllp < oo}.
Then Hp = MP, ||M||^p, H^f^ll? and ||M»||p are all equivalent norms.
Proof. Firstly, by Burkholder's inequality we know that ||M||^p and
IIM»IIp are two equivalent norms on Hp. Secondly, by Doob's inequality
we know that \\Mqq \\p and \\M^ \\p are two equivalent norms on MP. Hence
TiP = MP, \\M\\<hp, ||M^||P and ||Moo||P axe equivalent pairwise. □
10.39 Theorem. Let (p, q) be a pair of conjugate indices. Then
the dual space of Hp is W. Moreover, if M G Hp and N e Hq, then
K = MN-[M,N]eH1.
Proof. Since the dual space of LP is Lq and MP is isomorphic to LP,
by Theorem 10.38 we know that the dual space of Hp is Hq. Let M eHp
and N eHq. By Kunita-Watanabe inequality we have
E\ I \d[M,N]3\] < \\M\\W\\N\\W < oo.
Because K*x < M^N^ + f \d[M, N]s\ G L1, so K € H1. □
J[0,oo[
The next theorem is similar to Theorem 10.24. It gives a sufficient
condition for H. M e W.
10.40 Theorem. Let M be a local martingale, p > 1 and H be an
optional process such that
E[(L^[Mi')f]<o0-
Then
v
\\H. M||WP < Cp(e[(J0oo Hld[M)sy])1/P, (40.1)
where Cp is a constant depending on p only.

§7. John-Nirenberg Inequality 287
Proof. Denote L = H.M. For any bounded martingale N,K = LN —
H. [M, N] is a local martingale. By Kunita-Watanabe inequality
where q is the conjugate index of p (i.e. l/p+ 1/q = 1). By Theorem 10.24
we have E^} < oo, then AT^ < L^AT^ + / \Hs\\d[M, N]s\ <E L1 and
^[0,oo[
If € M. In particular,
1 J[0,oo[ J
£(^(|,0o,^[M1«)P/!])1/P||N|1'"
< CJN«,\),(e[( J Hldm,)"2})1*, (40.2)
where Cp is a constant depending on p only. Because the collection of all
bounded measurable functions is dense in Lq, from (40.2) we conclude
\\Lx\\p<Cp{E[{^Hld[M}s)Pl2])llv.
Whence (40.1) is deduced. □
§7. John-Nirenberg Inequality
The following lemma is due to Strook[l].
10.41 Lemma. Let X = (Xt) be an adapted cadlag process. Assume
that lim Xt = Xqq a. s. exists and is finite. If there is a non-negative
t—>oo
integrable r.v. £ such that for any stopping time T
EUXoo - XT-I[T>0]\\FT} < E[Z\fT} a.s., (41.1)
then for all X > 0, /i > 0 we have
liPiXlc > X + n) < 2 / idP. (41.2)
Proof. Let 0 < (if < //. Put
T = inf{t : \Xt\ > A}, S = inf{* : \Xt\ > A + //}.
Then T < S, XT-I[T>o] < A and
[X'oo > X + //] c [\XS\ > A + //] C [|Xr| > A] n [\XS - XT.I[T>0]\ > //].

288 Chapter X Martingale Spaces Hl and BMD
Because \XS - XT-I[t>o]\ < l^oo - XT-I[t>o]\ + l-^oo - Xs\, so
P(X^ > A + /0 < 1 / |X5 - XT_/[T>0]|dP
M •/ll*T|>A]
<^f/ IXoo-Xr-Z^oildP
M/L^[|A-T|>A] l '
+ / IXoc-XsldPl. (41.3)
Since [\XT\ > A] € ^r, by (41.1) we have
/ IXoo - XT-I[T>o]\dP < f t,dP<t t,dP.
J\\XT\>\) J[\Xt\>X]- J[X^>\]
Moreover, since lim X,„ i = Xc, by Fatou's lemma
7l-»00 (S+-)-
I \Xn-Xs\dP
[l*rl>A]
< Urn / IXoo-X.i JdP< / £dP.
[|ATr|>A] [|XT|>A]
Hence, by (41.3) we obtain
fi'P(X*00>\ + n,)<2 [ ZdP. (41.4)
Letting // | M in (41.4) yields (41.2). □
Remark. Let A G ^o- Imitating the proof of the theorem, we know
that for all A > 0, /x > 0
ImP([X^ >A + /i]HA)<2 / f dp. (41.5)
The next theorem is called John-Nirenberg inequality as usual.
10.42 Theorem. Assume thatX = (Xt) is an adapted cadlagprocess,
and lim Xt = X^ a.s. exists and is finite. If there is a constant c > 0
t—»oo
such that for any stopping time T
EftXco - Xt-IfxhWTt] < c a.s., (42.1)
then for 0 < A < — we have
E[eXX~\ < j-^, (42.2)
and for any stopping time T
JG[exp(A|Xoo - XT-I[t>0]\)\^t} < ^^^ *■'■■ (42-3)

§7. John-Nirenberg Inequality 289
Proof. By Theorem 10.41 we have
4cP(X£) > 4nc) < 2cP{X*x> > 4(n - l)c), n > 1.
Thus
P{X^ >4nc) <2_n<e~f.
When 0 < A < —,
8c
OO
^[e^-] < £ e4cA(n+1)P(4cn < X^ < 4c(n + 1))
n=0
< e4^ £ e-(^"4cA)n = e4c\^ _ e"(±-4cA)j-l ^ ^
n=0
a 1
Since e a < 1 = for 0 < a < -, by (42.4) we obtain
V^ 2
£[e«i|Se^_4arl<_j£_<r^_.
Let A e To and P(A) > 0. Making use of (41.4), by the same
argument we can obtain
Hence we have
E[exx~\f0}<YZ^\ a-S" (42-5)
Let T be a stopping time. Applying (42.5) to (.Fx+^^o-adapted
process (XT+t - XT-I[T>o))t>o yields
£ [exp (Asup \Xt - XT-I[t>o]l) \^t] < 1_8cX as-
In particular, (42.3) holds. □
The next theorem is a John-Nirenberg type inequality for BMO
martingales.
10.43 Theorem. Let M e BMO and ||M||hmc? = rn.
1) When A < -—, we have
E[eXM~] < r-|^. (43.1)
2) When A < —^, for any stopping time T we have
E[eMK[M]oo - [M]T_/[T>0])}|JTr] < YZTx^- (43-2)

290 Chapter IX Martingale Space H1 and BMO
Proof 1) Let T be a stopping time. By Jensen's inequality we have
EWM^ - MT_/[T>0]||^r] < (£[(Moo - MT_/[T>0])2|^r])1/2 < m.
Then (43.1) is deduced from Theorem 10.42.
2) Consider the increasing process A = [M]. Sicne M e BMO, for
any stopping time T we have E[Aoo — At^I[t>o]\^t] < m2 a.s.. By
Garsia's lemma (Lemma 10.35), E[A^] < E[m2(nA^'1)], n > 1. Thus
by induction we obtain
E[A^]<m2nn\, n = 0,l,2,---,
and
ElexpiXAoo)} < 1_\m2.
Then (43.2) can be shown in the same way as (42.3). □
Remark. If we do not use Garsia's lemma and apply directly Theorem
10.42 to [M], we can obtain the following weaker result: when A < -—^
for all stopping time T we have
EiexpiXWU - [M]r_/[T_>0])}|JTT] < ^^^
The following theorem gives another characterization for
HMO-martingales.
10.44 Theorem. Let M be a uniformly integrable martingale. Then
M £ BMO if and only if there is a constant c > 0 such that for any
stopping time T
JBllMoo-MT-Z^olll^rl^c a.s.. (44.1)
Proof. The necessity comes from Jensen's inequality. We want to show
the sufficiency. By Theorem 10.42, for A < — we have
8c
E[eMMMx - Mr_/[T>0]|)|JFT] < y^^-
Hence
EWn - MT.I[T>0])2\TT] < (1_g2cA)A2
and therefore M e BMO. D
a.s.

Problems and Complements 291
Problems and Complements
10.1 Let M be a local martingale and H be an optional process such
that H. M £ M2. Then for any stopping time T we have
E[((H. M)oo-(£T. M)t)2\Ft] < E\ [ H2sd{M\s\TT\ a.s..
10.2 Let M € BMO and H be an optional process with \H\ < 1. Then
\\H. MUbmo < V5\\M\\bmo.
10.3 n1'0 = H1 n Mfoc and W1,d = W1 nA*foc are all closed subspaces
ofW1.
10.4 Let M € BMO0. If AM > -1 + e, e e]0,1], then S(M) € M
10.5 Let M € Mfoc>0-
1) If ^[exp^AOoo}] < oo, then £(M) 6 M.
Li
2) If £[exp{ ^(M)^}] < oo, r > 1, then S(M) eHp,p= ——-.
2 2r — 1
10.6 Let M e M\oc and for any stopping time T, jE[|AMt|/[t<oo]] <
oo. Then
[M-*] = [[M]oo < oo] a.s..
10.7 Let A £ ^c with Aqq = oo and for any stopping time T
E[AAtI[t<oo]] < oo. Then
lim -sr- = 1, a.s..
10.8 Let M e Alloc- Let
and A be the compensator of B.
1) [Aoo < oo] C [M ->] a.s..
2) If for any stopping time T, E[\AMt\I[t<oo]] < °°>
[Aoo < oo] = [M —►] a.s..
3)K JB[i4oo] <oo,MGK
10.9 Let M be a martingale, and supjE[|M*|] < oo. Let TV 6 A^ioc,
t>o
and [N] < [M]. Then
P([N ->]) = 1-
10.10 Let X e <S, and p > 1. Set
||X||^=inf{|>/[Afl00+/ \dAa\
Ul v oo y[o,oo[
: X = M + A,
v

292 Chapter IX Martingale Space H1 and BMO
M € Alloc, A € V},
SP = {X € S : ||X||5p < oo}.
1) <SP is a vector space, and ■S? C «Sp.
2) W C «SP, and for M € Wp, ||M||W = ||M||5P.
3) For each X € ^
where Cp is a constant depending on p only.
4) Let X £ <SP, and X = M + A be its canonical decomposition. Then
||V/[^U+ / \dAs\\\ <2(1+P)\\X\\sp.
Whence Sp is a Banach space.
10.11 Let p > 1, q > 1, r > 1 and - + - = -. Suppose X e Sp and H
p q r
is a predictable process with ||^ff^,||9 < oo. Then H is X-integrable, and
\\H.X\\Hr < \\H^\\q\\X\\s,,
||(fr.jr)^||r < cr||^||9||x||5P,
where Cr is a constant, depending on r only.
10.12 Let X,XW e Sp. We say that (X^) converges prelocally in
Sp to X if there exists a sequence (2*) of stopping times with 2* f oo
such that for each k
Urn \\(xW-X)Tk-\\SP = 0.
n—»oo
The following two statements are equivalent:
1) Urn § 2-fc£[(X(n) - X)l A 1] = 0.
2) Prom each subsequence of (X^) one can extract a subsequence,
which converges prelocally in Sp to X (see Problem 8.20).

Chapter XI
The Characteristics of Semimartingales
In this chapter we first introduce random measures which axe the most
useful tools to investigate the jumps of semimartingales. Then by using
jump measures we establish the integral representation of
semimartingales, in connection with which the predictable characteristics of
semimartingales are introduced. It is interesting that the classical Levy-Ito
decomposition for processes with independent increments is just a
special form of the general representation of semimartingales. In the last
paragraph we study another simple but useful type of semimartingales—
step processes, which play an important role in applied probability and
statistics.
§1. Random Measures
11.1 Definition. Let (E,B(E)) be a measurable space. Define
(&,?) = (ft x R+ x E, T x B(R+) x B(E)\
6 = Ox B(E), V = Vx B(E).
0 (resp. V) is called optional (resp. predictable) a-field in ft. An O
(resp. ^-measurable function defined on ft is called an optional (resp.
predictable) function on ft.
(E,B(E)) is supposed to be a Lusin space, i.e., a Borel subspace of a
compact metric space with its Borel field. For example, (E,B(E)) may
be a discrete space, (R, B(R)) or the n-dimensional space (Rn,B(Rn)).
In our book we discuss mainly real stochastic processes. By convention,
(E,B(E)) is taken as (ii, B(R)), unless otherwise stated.
11.2 Lemma. Let W be an optional (resp. predictable) function on
17, (at) be an optional (resp. predictable) process, and T be a stopping

294 Chapter XI The Characteristics of Semimartingales
time. Then
W{u,T,aTM)/[T<oc]M € TT (resp. TT-)- (2.1)
Proof. It is easy to see that (2.1) holds when W(u, t, x) = /(cj, t)g(x),
f(w,t) e 0(resp.V) and g(x) e B(E). Then for any optional (resp.
predictable) W (2.1) follows by a monotone class argument. □
11.3 Definition. An extended real function /x defined on fix (Z3(i2+)x
B(E)) is called a random measure if
1) for all u) £ £2 /x(u;, •) is a cr-finite measure on B(R+) x B(E),
2) for all B G B(fl+) x B(E),ia(-,B) is a r.v. on (0,.F).
For any B e P, define
MM(B) = ^ [ / /g(w,«, s)M"i dt, dxj\.
J R+ xE
Then MM is a measure on ($7, j^), and is called the measure generated by
fjL. \i is said to be integrable if MM is a finite measure: MM($7) < oo. \i is
said to be optionally (resp. predictable) a-integrable, if the restriction of
MM on 0 (resp. P) is a cr-finite measure.
Clearly, the concept of random measure is a generalization of the
concept of increasing process. Let A = (At(u>)) be an increasing process.
Take E = {x0}: a set of one point, B(E) = {0, E}. Then
li(u,dt,dx) = dAt(u>)6X0(dx)
is a random measure, and
fjL([0,t]xE) = At.
It should be stressed that in general for a random measure /x, /x([0,£] x
B) may be equal to infinity identically for all t > 0 and B £ B(E).
If W eT+, then
v(u, B) = f W(u, t, x)ia(lj, dt, dx), B e B(R+) x B(E), (3.1)
Jb
is still a random measure. (3.1) is also denoted by u = W.fi or dv = Wd\i.
Let W e T. If for every t > 0
/ \W\dfi < oo,
J[Q,t]xE
we define W * /x = (W * fit) as follows:
W*fj,t= f Wdn, t > 0.
J[0,t]xE
Obviously, W * /x is a process with finite variation.

§1. Random Measures 295
A random measure /x is said to be optional (resp. predictable), if for
any optional (resp. predictable) function W such that W * /x exists, W * /x
is an optional (resp. predictable) process.
It is evident that if for every t > 0,1 */x* < oo, then /x is optional (resp.
predictable) if and only if for every B G B(E), 1# * /x = (/x([0, £] x 2?))*>o
is optional (resp. predictable). The following result is can also be easily
obtained.
11.4 Lemma. Let /x be an optional (resp. predictable) random
measure, and W be an optional (resp. predictable) non-negative real function.
Then v = VT./x is an optional (resp. predicatable) random measure.
11.5 Theorem. Let /x and v be two optional (resp. predictable) and
optionally (resp. predictably) a-integrable random measures. If the
restrictions of Mp and Mv on O (resp. V) are identical, then /x = v, i.e.,
\i and v are indistinguishable:
P({u>: 3B e B(R+) x B(E) such that n(u>,B) ^ v(u,B)}) = 0.
Proof. Let An e O (resp. V) such that An | ft and M^(An) =
Mv(An) < oo for each n. Let V be a countable 7r-class such that cr(V) =
B(E). For every n and D eV define
U = (IX JD)*/x, V = (IxID)*v.
Then both U and V are optional (resp. predictable) integrable increasing
processes. For any non-negative optional (resp. predictable) process H
we have
E[J HtdUt] = M^I^JdH] = MV\I~JDH] = E[j HtdVt].
Hence U and V are indistinguishable. Therefore
P({u : f I7 dfji= f I2 dvVt> 0, VL> e V\) = 1.
J[0,t]xD An J[0,t]xD An
By the uniqueness of the extention of measures we obtain
P({u : Llzjl* = L1!^ V^ e B(H+) x B(E)}) = L (51)
Letting n —> oo in (5.1) leads to /x = v. □
11.6 Theorem. Let m be a measure on (ft, T) such that its restriction
on O (resp. V) is a-finite. There exists an optional (resp. predictable)
random measure /x such that m = Af ^ if and only if
i) for any evanescent set N C ft x f2+, m(N x E) = 0,

296 Chapter XI The Characteristics of Semimartingales
ii) for any A € O (resp. V) with m(A) < oo and bounded measurable
process X
m(XIx) = m{°XI~) (resp. m{XI~) = m^XI^)).
In this case, the optional (resp. predictable) random measure /i is uniquely
determined.
Proof. We deal with the optional case only. Necessity, i) is trivial:
M^(N x E) = 0 for any evanescent set N. Noting that Y = I~* p
is an optional integrable increasing process, for any bounded measurable
process X we have
m{XI~) =Mtl(XIx) = E[J XtdYt]=E[J °XtdYt]
= M^XI-) = m(°XI~).
Sufficiency. At first, we assume m is a finite measure. Then m can be
decomposed as follows:
m(duj, dt, dx) = n(o;, £, dx)m(du>, eft, E), (6.1)
where for every B G B(E),n(uj,t,B) is the Radon-Nikodym derivative
of m(dw,dt,B) w.r.t. m(dw,dt,E) on 0, and at the same time it is an
optional process (Theorem 5.14); for all (u;,£),n(u;,£, •) is a probability
measure op B(E).
Applying Theorems 5.11 and 5.13 to m(du;, dt, E), we know that there
is an optional integrable increasing process A = (At) such that for any
non-negative measurable process X
m(X) = E[f XtdAt\. (6.2)
Put
/i(cj, dt, dx) = n(u, £, dx)dAt(u).
It is not difficult to check that /x is an optional integrable random measure.
By (6.1) and (6.2), for any B e B(E) and bounded measurable process X
we have
m(XIB) = m(°XIB) = / °Xt(u))n(u, t, B)m(dw, dt, E)
JtoxR+
= E[J 0Xt(u)n(u,t,B)dAt(u)]=E[j Xt(u)n(u,t,B)dAt(u)]
= E\ f Xt(u)ii(u,dt,dx)} = M„(XIB).
lJR+xB J
Then by the uniqueness of the extension of measures we obtain m = M^
on T.

§1. Random Measures
297
Now assume that m is cr-finite on O. There is a sequence (An) of
disjoint sets in O such that Q = \JAn and m(An) < oo for each n.
71
Applying the result shown above to finite measure m(WI~ ), there is an
optional integrable random measure /xn such that m(WIj ) = Mnn(W)
for any VT € ^+. Put
oo
n=l *n
Then it is easy to see that /x is an optional random measure and m = JVf fJL.
The unqiueness of /x follows from Theorem 11.5. □
11.7 Definition. Let /x be a random measure. If there exists a
predictable random measure v satisfying
1) v is predictably cr-integrable,
2) the restrictions of MM and Mu on V are identical,
then we say that /x /ias dwa/ predictable projection or compensator v, and
i/ is the dwa/ predictable projection or compensator of /x. Of course, the
dual predictable projection of a random measure (if exists) is uniquely
determined by Theorem 11.6. The dual predictable projection of /x is
denoted also by /xp or /x.
Remark. We may define the dual optional projection of a random
measure. But we do not need it in our book.
11.8 Theorem. A random measure /x has dual predictable projection
if and only if it is predictably a-integrable.
Proof. The necessity is trivial. Only the sufficiency is required to be
proved.
For any bounded non-negative measurable process X and bounded
non-negative Z3(2?)-measurable function h put
m(Xh) = M^VXK).
Since MM is cr-finite on?, m can be uniquely extended to a measure on
T. Obviously, m and M^ have the same restriction on?. It is easy
to see that m satisfies the requirements in Theorem 11.6. Hence there
is a predictable random measure v such that m = Mu. Therefore v is
predictably cr-integrable, and v = /x. □
11.9 Theorem. Suppose that random measure /x has dual predictable
projection. Let W G T* such that v = W. /x is a predictably a-integrable
random measure. Then v has dual predictable projection:
v = U.ji,

298 Chapter XI The Characteristics of Semimartingales
where U = M^[W\P}.
Proof. By the assumption Mu is cr-integrable on V. This means that
under MM W is ^-integrable w.r.t. V. Thus U = M^[W\P] is finite. Let
H be a predictable function on Q such that M„(\H\) = M^(\HW\) < oo.
Then
M„{H) = Mp(HW) = Mp(HU) = M~(HU) = MV~(H).
Hence V = U.Ji. □
11.10 Corollary. Suppose that random measure /z has dual predictable
projection Jiy and W € T such that X = W * fi is a process with locally
integrable variation. Then X has dual predictable projection: X = U *Ji,
whereU = M^[W\V\.
11.11 Theorem. Suppose that random measure /z has dual predictable
projection Ji, W 6 V*, and T is a predictable time. Then
Je
W(T,x)t({T},dx)I[T<oo]
= JE?[^W(T,x)/i({r},dx)Jrr<oo]|^r-] a.s.. (11.1)
Proof. Let An € V such that An | £2, M^An) < oo and W is bounded
on An for each n. In this case, X^ = (W/r )*// is an integrable
increasing process and has dual predictable projection X^ = (WIr )*/J.
Hence AXp/[T<oo] = £[AXp/[T<00]|.Fr_] a.s., i.e.,
/ W(T,x)IXn(T,x)fi({TUx)I[T<oo]
J E
= f;[^(T,x)/-JT,x)/i({T},dx)/[T<TO]|^ a.s.. (11.2)
Letting n -> oo in (11.2) yields (11.1). □
11.12 Defintion. A random measure /z is called an integer-valued
random measure if
1) /i takes on values in {0,1,2, • • •, +oo},
2) foram>0,/z({*} x E) < 1,
3) /i is optional and optionally cr-integrable.
11.13 Theorem, /z is an integer-valued random measure if and only
if
H(dt,dx) = ZS(3A)(dt,dx)ID(s), (13.1)

§1. Random Measures 299
where D is a thin set (D is called the support of /z), /3 = (fa) is an optional
process.
Proof Sufficiency. It suffices to justify that /z is optional and optionally
(T-integrable. Let (Tn) be a sequence of stopping times with disjoint graphs
such that D = UPn]> and W 6 O such that W * /z makes sense. Then
71
W*fi = EW(Tn,0Tn)IlTn>ool.
n
By Lemma 11.2 W * /i is optional. Hence // is optional. On the other
hand, putting An = ( \J [Tk] U Dc) x E, we have An e O, An | ft and
AfM(i4n) < n, i.e., /z is optionally cr-integrable.
Necessity.^ Denote D = {(u,t) : n({t} x E) = 1}. £> is a thin set.
In fact, let An e O such that An ] Q and M^(An) < oo for each n.
Then B^ = It * U is an optional integrable increasing process, and
D = U[A#(n) # 0]- Assume D = \J[Tn], where (Tn) is a sequence of
71 71
stopping times with disjoint graphs.
Since B(E) is countably generated and each one-point set in E is
measurable, if Tn(u>) < oo, there exists a real number /3rn(u,)(u;) such that
M^{(rnM,/5rn(u,)H)» = l- Put
P = ZPTJlTn].
n
Then (13.1) holds. Finally, it remains to prove that /3 is optional. In fact,
since for each B e B(E),
[0rn <E £, Tn < oo] = H{Tn} x B) = 1, Tn < oo],
and /z({Tn} x B)I\Tn<00] is the jump size of optional process I[rn]xB * M
at time Tn, /3rn/[T„<oo] € ^Tn- Therefore, /? is optional. □
11.14 Theorem. Suppose that integer-valued random measure \i with
support D has the dual predictable projection v. Put
a = (at), at = u({t} x E), t> 0. (14.1)
J = [a>0], (14.2)
K = [a = 1]. (14.3)
77ien a is a predictable thin set, 0 < a < 1, J is the predictable support of
D, and K is the largest predictable set contained in D(up to an evanescent
set).
Proof. Let An e V such that An | £2 and MM(j4n) < oo for each
n. Then B<n> = Jr * /x is an optional integrable increasing process, and
An
has dual predictable projection JB(n) = I~ * v. Thus a = lim AZ?(n) is
A.n 71

300 Chapter XI The Characteristics of Semimartingales
predictable. Clearly, D = \J[AB^ ^ 0] and the predictable support of D
n
is
U[AB(") ^ 0] = [a > 0] = J.
71
Hence a is a thin process. For any predictable time T
aTI[T<oo] = #[Mm x E)I[T<oo]\TT-\ a.s.. (14.4)
But 0 < //({t} x B) < 1. By the section theorem we have 0 < a < 1.
Now assume that T is a predictable time such that [T] C K = [a = 1].
By (14.4) we have
E[fi({T} x £)/[T<oo]] = £[aT/[r<oo]] = P(T < oo).
Since 0 < fi({T} x £')/[T<00] < 1, we conclude that
/x({T} x£) = l a.s. on [T < oo].
Hence [T] C D, and consequently, K C D.
On the other hand, if if is a predictable subset of Z?, and there is
a predictable time T such that [T\ C H \ if, P(T < oo) > 0, again
by (14.4) we know ar^[T<oo] = I[T«x>] as> i-e> CH C if, contradicting
[T] C if \ if. It must be if C if, i.e., if is the largest predictable set
contained in D. □
11.15 Theorem. Let X = (Xt) be an adapted cadlag process, and
D = [AX^o], Then
fi(dt,dx) = £ 6(sAXa)(dt,dx)ID(s)
3>0
is an integer-valued random measure and has dual predictable projection
v. fi is called the jump measure of X, and v the Levy system of X.
Proof It is well-known that D = [AX ^ 0] is a thin set and AX
is an optional process. By Theorem 11.13 /x is an integer-valued random
measure. It is required to prove that MM is cr-finite on V.
For n > 1 put
Tnfi = 0, Tn,m = inf it > Tn,m_x : ± < |AXt| < —^-}, m > 1.
Then Antm = [0,TB,m] x (f-,-^-1 U {0}) e ?, U An,m = Q and
M-n n — 1-1 / n,m
MM(An,m) < m. D
In the remainders of this paragraph we suppose /x is a given integer-
valued random measure having the dual predictable projection i/, and
/i({0} x E) = 0. We continue to use the notations defined in (13.1) and
(14.1)-(14.3).

§1. Random Measures 301
Our goal is to define the stochastic integral of a predictable function
W w.r.t. fi- v. In fact, if W * /z e A\oc, then
M = W * fi — W *v
is a local martingale with locally integrable variation and Mo = 0, because
W * v is the dual predictable projection of W * /x. It is natural to define M
as the stochastic integral of W w.r.t. p — v, and denote M = W*(ii — v).
Besides,
AMt = I W(t,x)ti({t},dx) - / W(t,x)v({t},dx), t > 0.
JE JE
Hence it is natural to give the following definition of stochastic integrals
w.r.t. random measures in general.
11.16 Definition. Denote
Vt{dx) = u({t},dx), t>0.
If W is a predictable function and for all t > 0, / \W(t,x)\Vt(dx) < oo,
JE
denote
Wt= J W{t,x)Vt(dt), *>0,
JE
Wt= I W{t,x)ii({t},dx)- f W(t,x)v({t}.dx)
JE JE
= W(t,(3t)ID(t)-Wu t>0.
Clearly, W = (Wt) and W = (Wt) axe all thin processes, and W is
predictable. By Theorem 11.11, we have ?(W) = 0. Put
Q(^) = {weP:Vt>of \W(t,x)\Vt(dx) < oo and y/x(W)2 <E ^c}.
Then by Theorem 7.42 for every W £ G(ii) there exists a unique purely
discontinuous local martingale M such that AM = W. M is called the
stochastic integral of W w.r.t fjb — i/, and denoted by W * (/i — v), or
Mt= I W(s, x)(/x(ds, dx) - v(ds, dx)), t > 0.
J[0,t]xE
It is easy to see
[M] = E(AM)2 = £(W)2.
Obviously, ifWeV and W * fi e A\oc, then W e £(/x) and W * (/i -
iz) = W' * /z — W * i/, as pointed out above.

302 Chapter XI The Characteristics of Semimartingales
11.17 Theorem. Let W e <?(//) and M = W *(fi-v). There exists
V e £(/i) such that M = V * (// — u) and
[a = l]c[V = 0]. (17.1)
Such aV is unique up to an M^-null set.
Proof. By the definition of stochastic integrals, on ft we have
AM = W - W M^-a.e..
Consequently, for any version U of M^AM^P] U = W — W M^-a.e.,
i.e., Ip^w_fy * Moo = 0 a.s., and hence I^w_^ * "oo = 0 a.s.. Put
U_
V = U + Z 7[o<1].
Since U = W — Wa = W(l — a), we have
V = W - W + WI[a<1] = W- WI[a=1], M„-a.e., (17.2)
V = W-WI[a=l]a = WI[a<1]. (17.3)
Noting that [a = 1] C D, we obtain
Vt = W(t,pt)ID(t) - WtI[at=1]ID(t) - WtI[at<l]
= W(t,pt)ID(t)-Wt = Wt, t>0.
Hence V € £(//) and M = V * (/x - v). (17.1) foUows from (17.3).
On the other hand, the definition of V does not depend on W. If
[a=l]c[W = 0],
then by (17.2) we have
V = W M^-a.e..
The uniqueness of V follows. □
11.18 Lemma. Let H be a predictable process. Then
X(HIJDc) e Aioc <=* £(#/j(l - a)) <E A{oc. (18.1)
In this case, £(i/Jj(l —a)) is the dual predictable projection o/E(i//j£>c).
Proof. Let J = UPn]> where (Tn) is a sequence of predictable times
71
with disjoint graphs. Then we have
E
E\Hs\IjDc(s)]=ZE[\HTJIDc(Tn)I[Tn<^
= EE[\HTn\(l - aTn)I[Tn<oo]] = e[e\Hs\Ij(s)(1 - a.)], (18.2)

§1. Random Measures 303
because aTJ[Tn<oo] = E[ID(Tn)I[Tn<oo]\FTn_] a.s. and HTn e TTn-. XT
is a stopping time such that
E\ E \H3\Ijd<s)] < oo or E\ £ \H3\Ij(s)(1 - as)] < oo,
Ls<T J Ls<T J
by replacing \H\ with i/7[o,ri m (18-2) one arrives at the conclusion of the
lemma. D
11.19 Theorem. Let W € V and for all t > 0, / \W(t,x)\ut(dx) <
Je
Put
\W-W\2 _/ W2
A =
4=^*i/ + E( ^(l-a)V
W\ Vl+ W\ >
\ + \w-w\ vi + |w|
B = (\W- W\%w_iym + \W- W\Im_frl>b]) * "
+nw2i[^b] + \w\i[w>b]), b>o.
Then
W € G(a) ^AeA+c^Be A+,. (19.1)
Proof. Since VF is a thin process, we have
W2
s<t 1 + \WS\
= \wM.m.)-w.r x
»/[0,*]x£ 1 +
w.»>-****.*)+ !:-!£-/„.(.)
/ W2 \
By Corollary 11.10 and Lemma 11.18, £ ( ^J G A?oc *=> A e A?oc,
and in this case A is the dual predictable projection of £ f ^=^).
By similar computation we have
E(W J[\w\<b] + l^l^l^^t]) = S((W J[|1V|<6)
and
S^ V|<6] + l^l7[|H>|>6]) € ^oc *=* B € .4+,
In this case, B is the dual predictable projection of ^KW2I\\w\<b\ +
l^l7#|>6])- Then (19-1) follows from Lemma 7.41. D

304 Chapter XI The Characteristics of Semimartingales
11.20 Corollary. Q{ti) is a vector space, and for any W\, W2 € Q((i)
and real numbers a, b we have
(aWi + bW2) *(//-!/) = a(Wi * (M - v)) + b(W2 * (// - v)).
11.21 Theorem. 1) Following Gi(fJ-), (?2(m) are subspaces of Q{n):
Qfa) = {W € V : V* > 0 / \W(t,x)\Vt(dx) < 00 and Y.(\W\) € A&,
Je
Q2(n) = {W € V : Vt > 0 / \W(t,x)\Vt(dx) < 00 and £(W2) € AL}.
Je
2) We Gi(n) ^ \W-W\*v+Z(\W\(l-a)) € .4+c *=> W*{»-v) €
3) W € &(//) *=* \W - W\2 * v + Z(W2(1 - a)) € A&. <=► W * (/* -
i^) £ Mfoc. In this case, we have
(W*(v- u)) = \W - W\2 * v + £(W2(1 -a)) = W2*v- £(W2),
Me /as^ equality holds only if W2 * v G ^[^c.
Proo/. 1) is easy: £ (J^U) < £(|W|) and £ (^) < E(^2)-
The proofs 2) and 3) are similar to that of Theorem 11.19. In fact, we
have£(|W|) = |W-W|*m + £(|W|/dc) and £(W2) = |W-W|2*// +
£(w2/zjc). n
11.22 Theorem. For every W € <?(//) Mere exisf U 6 £i(m) a™*
V € ftj(/i) sucft Ma* W = t/ + V.
Proo/. Put M = W *(ti-u) and A = E(AM/[|AM|>i]). We have
already known A € A\oc (Lemma 7.16). Put
u = (w-w)i[]w_^>l] + wi[^>l],
V = (W - w)I[\W_w\<i\ + WI[\w\<iy
Then W = U + V. Since
AAt = AMtIftAM,\>i]
= jE(W(t,x) - Wt)Im_^>1]ti({t},dx) - Wtl^^l - /*({*} x E))
= JEU(t,xM{t},dx)-wti^t>lV
the predictable projection of / U(t,x)fji({t},dx) = AAt + WtI,.^. ,, is
Ut = J U(t,x)ut(dx) = AAt + Wtl0tl>iy

§2. The Integral Representation of Semimartingales
305
Hence U = A(A - A), and U e Gi(fi) (t/ * (/x r v) = A - A). Now that
V = W - U e G(fi) and A(V * (/x - u)) = V is bounded by 4(\V\ <
2), v*(ti-v)eM2{£zndVeg2(ti). □
11.23 Theorem. Let W e </(/x) and M = W * (/i - v), H be a
predictable process. Then H is integrable w.r.t M if and only if HW G
C/(/i). In this case, we have
H.M = (HW) * (/i - u). (23.1)
Proof. Since
H2.[M] = E(if2AM2) = E(tf2W2) = E(W)2,
The fact that H is integrable w.r.t. M, i.e., y/H2.[M] £ A^ is equivalent
to HW e G(n). In this case, we have A(H. M) = HAM = HW = (HW).
Hence (23.1) holds. □
§2. The Integral Representation of Semimartingales
11.24 Theorem. Let X be a special semimartingale, and
X = M + A
be its canonical decomposition, where M is a local martingale and A is a
predictable process with finite variation and Aq = 0. Let /z be the jump
measure of X, and v be its dual predictable projection. Then
Md = x*(/x-i/). (24.1)
Proof. For any predictable time T we have
AATI[T«x>] = E[AXTI[T<oo]\^T-]
= E[JEXfi({T},dx)I[T<oo]\TT-] = JEXv({T},dx)I[T<oo] a.s..
Hence, AA is indistinguishable from ( / xv({t},dx)Y and
/ xfi({t},dx) - f xv({t},dx) = AXt - AAt = AMt = AM?, t > 0.
Je Je
By the defintion of stochastic integrals, Md is just x*(fi — u). □
11.25 Theorem.Letf X be a semimartingale, /z be its jump measure,
and v be the dual predictable projection ofji. Then
X = X0 + a + Xc + (x/[|*|<i]) * (A* - ") + (*J[M>il) * M, (25.1)

306 Chapter XI The Characteristics of Semimartingales
where Xc is the continuous martingale part of X, a is a predictable process
with finite variation and olq = 0. Moreover, we have
v({0} xE) = v(R+ x {0}) = 0, (25.2)
(x2Al)^ueA^ (25.3)
Aa=(f xvt(dxj). (25.4)
KJ\x\<l '
Proof. Noting that (xl^yy) * /i = Y,(AXI[\AX\>i]), we know
Y = X - XQ - (xl[lx>l]) * /x, t > 0, (25.5)
is a special semimaxtingale (|Ay| < 1). Denote its canonical
decomposition by
Y = M + a, (25.6)
where M is a local martingale with Mq = 0, and a is a predictable process
with finite variation and ao = 0. Clearly,
MC = YC = Xc, (25.7)
the jump measure of Y is 7[|x|<i]. /i, and its dual predictable projection is
/[|x|<i]- ^- By Theorem 11.24 we have
Md = (x/[w<1])*(M-i/). (25.8)
Then (25.1) follows from (25.5)-(25.8).
(25.2) is trivial. (25.4) comes from (24.2). Only (25.3) is required to
be proved. Since
(x2I[\x\<i]) * V = SfiAlf/y^i^]) = E(AF)2
and |AF| < 1, (z21[\x\<i]) * M £ A^c an<^ lts ^ua^ predictable projection
(z2%e|<i]) * v e Afoc. On the other hand, I[\x\>i] * M £ Aoc ** evident,
hence /[|z|>i] * v € A^oc. In a word, (x2 A 1) * v e -4j£c. □
(25.1) is called the integral representation of semimartingale X.
Denote (3 = (Xc). The triplet (a, (3, v) is called the predictable
characteristics (or predictable triplet, or local characteristics) of semimartingale X.
Predictable triplet is an important tool to investigate semimartingales,
though a semimaxtingale or its law can not be uniquely determined by its
predictable characteristics in general.
11.26 Corollary. A semimartingale X is a special semimartingale
if and only if
(l*|J[|x|>i]) * M = S(|AX|7[|AX|>1]) € A+c.

§2. The Integral Representation of Semimartingales 307
In this case, the canonical decomposition of X is
X = (XQ + Xc + x * (/i - v)) + (a + (x/[W>1]) * i/), (26.1)
where \i is the jump measure of X, v is the dual predictable projection of
M-
Proof. By the integral representation (25.1) we know that X is a
special semimartingale if and only if a + (xi^^) */x £ ^ioc, or, equivalently,
(x^[|x|>i]) * M € A\oc, because a G ^4ioc- But (x/[|x|>i]) * v is the dual
predictable projection of (xli^x\>x\) * /x. (26.1) follows directly from (25.1).
D
11.27 Corollary. Let X be a semimartingale having integral
representation (25.1), and f be a bounded C2-function on R+. Then the canonical
decomposition of special semimartingale f(X) is
f(X) = M + A, (27.1)
M = f(X0) + f'(X.).Xc + [f(X. + x) - /(*-)] * (/i - i/), (27.2)
A = f'{X-).a + \f'\X-).0
+[/(*_ + x) - f(X.) - */'(X-)JIW<i]] * v. (27.3)
In particular, the special semimartingale Y = exuX (u € R) has the
following canonical decomposition:
Y = Y0 + (Y-).N + (Y-).H, (27.4)
N = iuXc + (eiux - 1) * (ji - u), (27.5)
u2
H = iua- —0 + (etux - 1 - iuxl^^]) * v. (27.6)
Proof. By making use of Ito formula, the integral representation (25.1)
and Theorem 11.23 we have
f(X) = f(X0) + f(X-).(X - X0) + \f"{X.).(Xc)
+ Uf(X)-f(X.)-f'(X.)AX)
= f(X0) + f'(X-).a + f'(X-).Xc + (xf'(X.)I[lxm) *(n-u)
+ (x//(X_)/[W>1])*/x + i/"(X_).^
+ (/(X_ + x) - /(*_) - xf'(X-)) * v
= f(X0) + f'(X-).Xc + (x/'(X_)/[,,!<!,) *{n-u) + /'(*_). a

308 Chapter XI The Characteristics of Semimartingales
+ ±f'(X-).0 + (f(X-+x)-f(X-)
-x/'(X-)/[|x|<i])*/i.
Since / is bounded and f(X-)yf(X-) is locally bounded, (f(X- + x) -
f(X-) - xf'(X-)I[\x\<i]) * /i is a purely discontinuous process with finite
variation and locally bounded jumps, and consequently, belongs to A^
and has dual predictable projection (f(X-+x)—f(X-)— x/'pf-.)/^^])*
v. Now it is straightforward to obtain the canonical decompositions (27.1),
(27.2) and (27.3).
Applying (27.1), (27.2) and (27.3) to the case of f(x) = eiux gives
(27.4), (27.5) and (27.6). □
11.28 Corollary. Let X be a semimartingale with predictable
characteristics (a,/3,v). Then X is stochastically continuous if and only if
for every t > 0
!/({«} x E) = 0 a.s..
In this case, a is also stochastically continuous.
Proof For every t>0we have
i/({t} x E) = 0, a.s. <=► E[u({t] xB)]=0« P(AXt ^ 0) = 0,
because
v{{t) xE) = P[AXt ± 0|JFt_].
The stochastic continuity of X means just that P(AXt ^ 0) = 0 for all
t>0.
If X is stochastically continuous, so is a by (25.4). □
11.29 Lemma. Let X be an adapted cadlag process. If for some real
u ^ 0, eluX is a semimartingale, then X itself is a semimartingale.
Proof. Let g be a C2-function on the complex plane such that
g(e*>) = y, if|y|<£.
Put To = 0, and
Tn = inf{* > rn_x : \Xt - It,., I > ^7}, n > 1.
Then Tn f 00, and for each n > 1
g(e^Xt-XT^) = u(Xt - Xrn_,), Tn_! < t < Tn.
Now it is not difficult to verify directly that for each n > 1
XTn _ XT„_, = lg(Yn) + ^^ _ ^^ _ i5(^-XVl)))/[rB|a)|]

§2. The Integral Representation of Semimartingales 309
where is a semimartingale. Hence for each n >
1, XTn - XTn~l is a semimartingale. Then X e S\oc = S. □
11.30 Theorem. Suppose X is an adapted cadlag process. Let a be
a predictable process with finite variation and ao = 0, (3 be an adapted
continuou increasinjg process with (3q = 0, and v be a predictable random
measure such that
i) for each t > 0, (x2 A 1) * ut < oo,
ii) 0 < a < 1, a = (v({t} x E))y
iii) u({0} xE) = v(R+ x {0}) = 0,
iv) Aa= ( [ xv({t},dx)).
W[|x|<l] '
u2
Denote ku(x) = etux — 1 - iuxl[\x\<i] and H(u) = iua —— /3 + ku(x) * v.
Then the following statements are equivalent:
1) X is a semimartingale with predictable characteristics (a,/3,v),
2) For any bounded function f e C2(R), f(X) - f(X0) - f'(X-).a -
\f"{X-).f3 - [f(X. + x)- f(X-) - xfiX-Vmq] * v e Moc,o,
3) For any real u, eiuX - eiuX° - eiuX~. H(u) e M{oCj0.
Proof 1)=>2)=>3) have already shown in Corollary 11.27. It remains
to show 3)=>1).
First of all, eluX is a semimartingale. Then by Lemma 11.29 X itself
is a semimartingale. Let (5, /?, v) be the predictable characteristics of X.
Put
H(u) = iua — -u2/3 + ku(x) * v.
Z*
Then eiuX - eiuX° - eiuX~. H(u) <E Moc,o, eiuX~. (H(u) - H(u)) <E MloCj0
and H(u) — H(u) G -Mioc,o- But H(u) — H(u) is a predictable process
with finite variation. Therefore, for each ix, H(u) and H(u) are
indistinguishable. Because H(u) and H(u) axe continuous in ix, we have
P({uj : V(*, u) Ht(u) = Ht(u)}) = 1. (30.1)
By direct computation we know
Ht(u) -If' Ht(u + rv)dr = V-(3t + [ eiux(l - S^^)u(ds,dx)
2 7-i 6 J[o,*]x£ v vx /
and it is the characteristic function of the following measure
v f / sinuxx
—Pt6o(dx) + j [1 )v(ds,dx).
Noting that v(R+ x {0}) = 0, by (30.1) we know that (3 and p,v and v
are indistinguishable respectively. Finally, so are a and 5. □

310 Chapter XI The Characteristics of Semimartingales
11.31 Theorem. Let X be a semimartingale with predictable
characteristics (a,/3, v). Then the following statements are equivalent:
1) There exists a sequence (Tn) of stopping times such that Tn ] oo
and
E\ sup \Xt - XQ\2] < oo (31.1)
lt<Tn J
(in this case, semimartingale X is said to be locally square integrable).
2) For every t > 0
x2 * vt < oo. (31.2)
3) X = Xq + M + A, where M is a locally square integrable martingale
with Mq = 0, and A is a predictable process with finite variation and
Aq = 0.
If it is the case, we have
(M) = 0 + x2 * v - £(AA)2. (31.3)
Proof. 2)=>3). Since for each t > 0, J[|z|>i] * "t < oo, by (31.2) we
have |£|/[|x|>i] * ut < oo for each t > 0. By Corollary 11.26 X is a special
semimartingale. Let X = Xq + M + A be its canonical decomposition,
where M G M\oc$ and A e A\oc^ is predictable. Prom the proof of
Theorem 11.24 we know AAt = ( f xv({t), dx)). Then £ (AA3)2 < x2 *
JE s<t
uu t > 0. By Theorem 11.21 x*(/x-i/) 6 M^Q and M = Xc+x*(fi-u) e
M2oc0. At the same time,
(M) = (Xc) + (x*(fi-v))=!3 + x2*v- £(AA)2.
3)=>1). Since M e jVf?oc, A = A_ + AA, A- is locally bounded,
E(AA)2 is a predictable increasing process and Ti(AA)2 € .Aj£c, there
exists a sequence (Tn) of stopping times such that MTn € M2, ATn~ is
bounded, and JE? f £ (AAS)2] < oo. Then (31.1) holds.
ls<Tn
1)=>2). Let Sn = inf{t > 0 : £ (AXS)2 > n} A Tn. Then 5n | oo, and
for each n
E\ £ (AXs)21 <n + 4f;[sup|X,-X0|2]<oo,
L3<Sn J L*<Tn
i.e., x2 * fit = £ (AX,)2 is a locally integrable increasing process. But its
3<t
dual predictable projection is x2 * i/$. Hence (31.2) holds. □
11.32 Corollary. Let M be a locally square integrable margingale with
predictable characteristics (a, /?, i/) and Mo = 0. T/ien
(M) = f3 + x2* v.

§3. Levy Processes
311
11.33 Theorem. Let f be a non-random cadlag function on i2+.
Then f is a semimartingale if and only if f is a function with finite
variation, i.e., f has finite variation on every finite interval.
Proof. The sufficiency is trivial. Only the necessity is required to be
proved. Assume that / is a semimartingale. Let / = /o + M + A be
its canonical decomposition, and (Tn) be a localizing sequence such that
MTn E Mq and ATn € Aq. Denote by Fn(t) the distribution function of
Tn. Then we have
fo + E[AtATn} = E[ftATn] = J ftAsFn(ds) = ftFn(]t,oo}) + J f3Fn(ds)
[O.oo] [0,t]
and
/,F»(]t, oo]) = /0 + E[AtATn] - J fsFn(ds)
is a function with finite variation. For every to > 0,
Fn(]t0, OO]) = P(Tn > t0) -> 1, OS U -> OO.
We may take n large enough such that Fn(]£o, oo]) > 0. Then / has finite
variation on [0, to], because Fn(]t, oo]) > Fn(]to, oo]) > 0. Thereofre / is a
function with finite variation. □
§3. Levy Processes
Henceforth we always consider Levy processes (i.e., stochastically
continuous processes with independent increments) as cadlag processes, owing
to Theorem 2.68. Recall that for a Levy process X = (Xt)
tpt(u) = E[eiu(Xt-Xo>>], ueR,
is continuous and never vanishes,
eiu(Xt-X0)
Zt(u) = ——, t > 0,
is a martingale.
11.34 Theorem. Let X be a Levy process. If X is a semimartingale,
then for all u 6 R <Pt(u) is a function with finite variation. Conversely,
if for some u ^ 0, ipt(u) is a function with finite variation, then X is a
semimartingale.
Proof. Without loss of generality, we may assume Xq = 0. If X G <S,
then for all u e R,eiuX e S. Since Zt(u) ^ 0 and Zt-(u) ^ 0, <pt(u) =

312
Chapter XI The Characteristics of Semimartingales
piuXt
e S (refer to Problem 9.16). By Theorem 11.33 ip{u) is a function
with finite variation.
Conversely, if for some u ^ 0, y>t(u) is a function with finite variation,
then eiuXt = Zt(u)ipt{u) e S. By Lemma 11.29 we have X eS. D
11.35 Corollary. Let X be a Levy process. There exists a continuous
(non-random) function f such that X — f is a semimartingale.
Proof. Take ft = aig(E[el(Xt~x°)]) to be a continuous function with
/o = 0. Since ipt(u) ^ 0, this is possible. X — f remains to be a Levy
process. But
£[ei(AWt-*o)] = |JE[g*(*-*o)]|
is a monotone decreasing function. Hence, X — f £ S. □
11.36 Theorem. Let X be a stochastically continuous
semimartingale. Then X is a Levy process if and only if its predictable triplet (a, /?, u)
is non-random. In this case, we have
i) a is a continuous function with finite variation and ao = 0,
ii) (3 is a continuous monotone increasing function with /3o = 0,
iii) v is a a-finite measure on (R+ x E,B(R+) x B(E)) and for all
t>0
v({t) xE) = i/(R+ x {0}) = 0, (x2 A 1) * ut < oo.
In particular, X is quasi-left-continuous.
Proof. Necessity. We suppose Xo = 0 for simplicity. Applying the
formula of integration by parts to Y = eluX yields
Yt = Zt<pt = 1 + / VsdZs + f Zs-d<ps
Jo Jo
= 1 + / VsdZs + [ Ys-—dips. (36.1)
Jo JO <Pa
Comparing (36.1) with (27.4) and noting |YL| = 1, we have
Ht(u) = f -±-dvs{u), t > 0, (36.2)
JO <PsW
i.e., for each u,H(u) is indistinguishable from a non-random continuous
function. But H(u) is also continuous in u, so for almost all uj (36.2) holds
for all t £ i£+ and u € R. We have already seen in the proof of Theorem
11.30 that (a, /?, v), the predictable triplet of X, is completely determined
by {H(u),u £ R}. Hence (a,/3, v) is non-random. Conditions i), ii) and
iii) follow from Theorem 11.30 and Corollary 11.28 immediately.

§3. Levy Processes 313
Sufficiency. It suffices to show for all u G R and 0 < s < t,
E[Ju(Xi-Xa)\Fs\ = E[Ju(Xt-Xa)],
i.e., for any A € Fs with P(A) > 0,
E[IAeiu^Xt^x^] = P{A)E[J<Xt~x°\ (36.3)
Taking 5 to be a new origin and applying Corollary 11.27 to Yt =
eiu(Xt-xa)^ £ > 5j we have
Since
sup
a<r<t
Yt = 1 + / Yr-dNr + f Yr-dHr, t > s.
Js Js
f Yr-dNr\ = sup \?t - 1 - / Yr-dHr\ < 2 + / \dHr\
Js s<r<t ' Js ' Js
and / \dHr\ is non-random, ( / Yr-dNrJ is a martingale (Theorem
7.12).*Then
E[IA J'Yr-dNr] = 0,
E[IAYt] = E[IA] + E[IA J' YT-dHr) = £?[/,*] + J' E[IAYr-]dHr.
Putting ft = p. . , we have
ft = * + J fr-dHr,
and by the exponential formula ft = eHt~H' does not depend on A. Thus
E[IAYt] E[IaYt]
P(A) P(Q)
This is just (36.2). □
E[Yt\.
11.37 Corollary. Let X be a Levy process with Xq = 0. If X is
a semimartingale, then its law is uniquely determined by its predictable
triplet (a,f3,u).
Proof. We have shown above that
ipt{u) = exp{iuat--u2(3t+ f {eiux-l-iuxl[lx{<1])v{ds,dx)}, (37.1)
^ J[0,t]xE

314
Chapter XI The Characteristics of Semimartingales
Thus for all 0 < s < t
E[eiu(xt-xa)] = Vb^u) = My)
= exp{iu(at - as) - ^(A - P*)
I (eiux - 1 - uixJ[W<1])i/(dr, dx)\.
J]s,t]xE u i- J )
+
The law of a process with independent increments is determined by its
initial law (i.e., the law of Xq) and the distributions of all its increments.
In our case, Xq = 0. By (37.1) the law of X is uniquely determined by
(a, 0,1/). □
11.38 Theorem. Let X be a process with Xq = 0. Then X is a
normal Levy process if and only if the following conditions are satisfied:
i) There is a continuous (non-random) function f such that Y = X — f
is a continuous local martingale,
ii) (Y) is non-random.
Proof Sufficiency. Evidently, we may choose /o = Yq = 0. The
predictable triplet of Y is (0, (F),0). It is non-random. By Theorem
11.36 Y is a Levy process. By (37.1)
2
£;[eiu^-y')]=exp{-y(A-/3s)}, 0 < a < t, (38.1)
where /? = (Y). Hence Y is a normal Levy process. Obviously, so is
X = Y + /.
Necessity. Since X is a normal process,
u2
<pt(u) = E[eluX<] = exp{iuft - —fit], t > 0,
where ft = E[Xt], (3t = D[Xt]. By the stochastic continuity of X, f
and /? axe continuous functions. By the independence of increments /? is
monotone increasing. Evidently, Y = X — / is still a Levy process, and
(38.1) remains true. Since E[Yt] = 0, Y is a martingale (Theorem 2.69).
From (38.1) we see that the predictable triplet of Y is (0,/?,0). Hence Y
is a continuous martingale and (Y) = (3 is non-random. □
11.39 Corollary. A process X = (Xt) with Xq = 0 is a standard
Wiener process if and only if the following conditions are satisfied:
i) (Xt) is a continuous local martingale,
ii) (X2 — t) is a local martingale.
Proof. It suffices to notice that Condition ii) is equivalent to (X)t = t

§3. Levy Processes
315
and (38.1) is reduced to
E[ei^Xt-x^]=exp{-^-(t-s)}, 0<s<t. D
Corollary 11.39 is well known as the martingale characterization for
Wiener process. It is also called Levy's theorem. As its first application, we
will show an important relation between Wiener process and continuous
local martingales below.
11.40 Lemma. Let M be a continuous local martingale. Then for
almost all u,M.(u) and (M).(u) have the same constancy intervals, i.e.,
for any a < b if M(u) is constant on [a, 6], so is (M).(uj) and vice versa.
Proof. For every rational r > 0 put
Tr = inf{t > r : (M)t ? (M)r}, Sr = ini{t >r:Mt^ MT).
Since (/]rTrj.M) = /jrTrj.(M) = 0,/jj.j^.M = 0. Similarly, we have
7ir5rj.(M) = {I]rSryM) = 0. Then it is not difficult to see that for
almost all u for each r, Tr(cj) = St(lj),M.(u)) and (M).(u) are constant
on [r, Tr(cj)], consequently, M.(u) and (M).(u>) have the same constancy
intervals. □
11.41 Theorem. Let M be a continuous local martingale with Mq = 0
and (M) oo — OO. Put
rt = inf{5 : (M)3 > t}, Nt = Mn, Gt = Tn, t > 0.
Then (Nt) is a standard Wiener process w.r.t. (Gt) and (Mt) is
indistinguishable from (N(M)t).
Proof. In fact, (rt) is the change of time associated with (M) (Theorem
3.48). Since (M)oc = oo, each rt is finite. Besides, we have t^ = oo and
Goo = ^oo- Since (M) is continuous, we have (Lemma 1.37) (M)Tt =t,t>
0. By Theorem 7.32 for each t, (M3T')5>o € M2 and (M32Art-(M)sArt)s>0 €
M. By Doob's stopping theorem, for all 0 < s < t
E[Nt\Gs] = E[MTt\TTa] = MTa=Ns a.s.,
E[N? - AGs] = E[M* - (M)Tt\T3] = Ml - (M)Ta = N* - a, a.s.,
i.e., (Nt) and (N? — t) axe (^)-martingales. Evidently, (Nt) is right-
continuous and (Nt~) = (Mn_). Since (M)rt_ = t = (M)Tt, by Lemma
11.40 (Nt~) is indistinguishable from (MTt) = (Nt). Hence (Nt) is
continuous. Then by Corollary 11.39 (Nt) is a standard Wiener process w.r.t.
(Gt). Since (M)T{M)t = (M)t, again by Lemma 11.40 (Mt) is
indistinguishable from (MT{M)t) = (N{M)t). □

316 Chapter XI The Characteristics of Semimartingales
11.42 Theorem. Let X be an adapted point process, i.e.,
oo
* = E Jfrnioo[,
n=l
where (Tn) is an increasing sequence of stopping times such that Tn ] oo
and for each n > 0, Tn < oo =► Tn < Tn+i(To = 0). Then the follomng
two statements are equivalent:
1) X is a Levy process and for all 0 < s < t, Xt — X3 has a Poisson
law (Such a process is called an inhomogeneous Poisson process unless it
is a homogeneous Poisson process),
2) There is a continuous increasing function At such that X — A is a
local martingale, null at 0.
Proof. 1)=>2). Denote At = E[Xt]. Since for all s < t, Xt - Xs > 0,
A, = E[Xt] = E[XS] + E[Xt - X3] > E[XS] = A3.
Hence A is monotone increasing. By the stochastic continuity of X,
c-(At-A.) = p(Xt _xs = 0)->l ast-s-^O,
i.e., A is continuous. By Theorem 2.69 X — A G .Mioc,o-
2)=^1). The jump measure of X is
ji([0,t]xJ8) = At«i(J8), BeB(E).
It is easy to see that the predictable triplet of X is (A, 0, u) and it is
non-random. Thus X is a Levy process by Corollary 11.28 and Theorem
11.36. Furthermore, by (37.1)
(ft(u) = exp{iuAt + / (elux - 1 - iixx/r|x|<1i)c/As<5i(c/x)}
J[0,t]xE u i- j j
= exp{A,(e--l)}.
Hence for all s < t, Xt — X3 has a Poisson law with parameter At — A3.
□
Theorem 11.42 is well known as the martingale characterization for
Poisson process. It is also called Watanabe's theorem. Similar to Theorem
11.41, a point process differs from a Poisson process by a change of time.
Even the proof is similar. It is left to readers as an exercise (Problem
11.11).
Now we come back to the general discussion on Levy processes.
11.43 Theorem. Let X^\ • • •, X^ be both Levy processes and semi-
martingales, null at 0. If[X^\X^] = 0, j'^ k,j,k = 1, • • • ,n, then
X^l\ • • •, X^n' are independent.

§3. Levy Processes 317
Proof. First we deal with the case of n = 2. By the assumption
bX^AXW = 0 and ((X(1))c, (X™)c) = 0. Let Z<*> = * x, ife =
1,2. It is easy to see AZ^AZ™ = 0 and
v(l) v(2)
«*">•. <**>■> - (%!")(%-) •«*«*><. <*(2>><>=».
Thus [Z(1), ZW] = 1, Z(J)Z(2) is a martingale. Hence we obtain
[c*(«iXt(1)-^*t(a))] = E[Su*xll)]E[e^xl\ (43.1)
For any n,m > 1, 0 = *o < *i < • • • < *n, 0 = so < si < • • • <
sm, Uj e R, j = l,-..,n, vk e R, k = l,---,m, applying (43.1) to
E ttj(x£> - Xgj ) and £ Vk(xHlk - X$8h_x) and letting t - oo,
;=1 fc=l
we obtain
E[exp{i t «i(41) " *£i) + ' £ V"(X« ~ X^}]
= E[exp{i jtmixS* - 4-.)}]^[«p{i£1,'*(jr2)" *2-i)}]'
i.e., X^ and X(2) are independent.
Making use of the above argument, by induction, we arrive at the
conclusion for general n. □
«
Remark. The converse of Theorem 11.43 is also true. We leave it to
readers as an exercise.
11.44 Lemma. Let (£n) be a sequence of r.v. converging in probability
to a r.v. f. If for each n, £n has a Poisson law with parameter An, then
Xn —> A (A may be 0 or +oo), and f has a Poisson law with parameter
X (when X = 0 or +oo, this means £ is a.s. identical with 0 or +oo
respectively).
Proof. By the assumption for u € R
£[ei<«]=exp{An(eiu-l)}.
If there is a subsequence Xnk —> A and A is finite, then £ has a Poisson law
with parameter A. If there is a subsequence Xnk —► +oo, for any positive
integer I
P(£ < 0 < limP(U < /) = lip S -^e-A"* = 0,
k k jf=o J.
i.e., P(£ = +oo) = 1. In a word, it must be An —> A. At the same time, £
has a Poisson law with parameter A. □

318 Chapter XI The Characteristics of Semimartingales
11.45 Theorem. Let X be a Levy process. Then
Xt = X0 + Xt + ( xdfi+ f xd(fjL - v), (45.1)
J[0,t]x[\x\>l] J[0,t]x[\x\<l]
where 1) X is a continuous normal process with independent increments
and Xq = 0;
2) /i is the jump measure of X having the following properties:
i) for any B € B(i2+) x B(E), fi(B) has a Poisson law,
ii) for any n > 1 and disjoint sets B\,-,Bn e B(R+) x B(E), /x(#i)>
•••,/i(Bn) are independent, moreover, if Bj c]s,oo[xE,j = l,--,n, for
some s > 0, then (/z(Z?i), • • • ,/z(JBn)) is independent of !FS;
3) v = E[fi], the dual predictable projection of //, is a a-finite measure
on B(R+) x B(E) and for each t > 0, i/(il+ x {0}) = u({i] x E) = 0,
(x2 A 1) * vt < oo;
4) Xo,X and /z are independent
In addition, we have
<pt(u) = exp{iuft - -u2(3t + (elux - 1 - ii/xJ^i^]) * i/*j, (45.2)
where ft = E[Xt] and fit = D[Xt] are continuous, (it is monotone
increasing, /o = Po = 0.
Proof By Corollary 11.35 there is a continuous function g such that
X — Xo — g € Sq. Levy process X — Xq — g is independent of Xq and
has the same jump measure as X. Hence we may suppose X G «So- By
Theorem 11.36 the predictable triplet of X (a, /?, u) is non-random.
Xt = at + XI + / xdfi + / xd(n - u).
*/[0,t]x[|x|>l] •/[0,t]x[|rr|<l]
Putting X = a + Xc, we obtain (45.1).
X is a continuous semimartingale and its predictable triplet (a,/?,0)
is non-random. By Theorem 11.36 X is a Levy process, and by Theorem
11.38 X is a normal process. 1) is established.
For any B e B(R+) x B(E)
Efr(B)] = M„(IS) = Mv(Is) = "(*)•
Then 3) follows from Theorem 11.36.
Let B e B(R+) x B(E) and i/(B) < oo. Put
Y = Is * /x, A = Is*v.
Then V is a point process and A is a continuous monotone increasing
function such that Y - A is a local martingale. By Theorem 11.42 Y is
a Poisson process. For each t > 0, Yt has a Poisson law with parameter

§3. L6vy Processes 319
A*. By Lemma 11.44 /x(B) = Um Yt has a Poisson law. If v(B) = oo, by
a-finiteness of v and Lemma 11.44 fi{B) still has a Poisson law.
Let jBx, • • •, Bn e B(R+) x B(E) be disjoint. We want to show that
M^i)> "miH(Bn) and X axe independent. Obviously, we may suppose
i/(Bi) < oo, • • •, u(Bn) < oo. Put yW) = Ig * /i, j = 1, • • • ,n. We have
known that y^'\ j = 1,• • •, n, axe Poisson processes. Besides, Ay^Ay^)
= 0 when j ^ k. Hence we have
[yU\yW] = o, i?fefcj [yO),x] = o, j,fc = i,...,n.
By Theorem 11.43 y(1\ • • •, y(n> and X axe independent. Since fi(Bj) =
lim Yi\j = 1, • • • ,n, we arrive at the required assertion. Furthermore,
t—+00
if Bj c]s, oofxJEJ,,/ = 1, • • •, n, for some 5 > 0, then fi(Bj) = lim (Y^ -
Ys),j = 1, • • •, n. Thus (/x(Bi), • • •, fJ>(Bn)) is independent of ^"3. Now
2) and 4) axe established.
Finally, (45.2) follows from (45.1) and (37.1). □
(45.1) is the famous Levy-Ito decomposition of Levy processes. We
also call (/, /?, u) in (45.2) the characteristics of Levy process X. And
the law of Levy process X is uniquely determined by its initial law and
its characteristics. The next two theorems illustrate the applications of
Levy-Ito decomposition.
11.46 Theorem. Let X be a Levy process and Xq = 0.
1) If X is a semimariingale and jE[|JQ|] < oo, t > 0, then X is a
special semimariingale.
2) If X is a special semimariingale, then E[\Xt\] < oo,£ > 0.
3) // X is a local martingale, then X is a martingale.
Proof. 1) Since (Xt—E[Xt]) is a martingale, E[Xt] is a semimartingale.
Hence E[Xt] is a function with finite variation, and Xt = (Xt — E[Xt\) +
E[Xt] is a special semimartingale.
2) By Corollary 11.26 (^i])*^ A>c- Thus (xl[\x\>i])*v € A>c-
Hence, Y = (*/[N>i]) * (n - v) € Wioc and Ys = (YsM)t>0 € W for all
s > 0. By Theorem 11.21.3)
((x/[W<i]) * (A* ~ ")) = (x2/[l*|<i]) * "•
Hence, Z = (a?/[|*|<i]) *(**-«') ^ -^L axid Z* 6 -M2 for all 5 > 0.
Similarly, since (Xc) = /?, (X^At)t>0 e X2 for all 5 > 0. But we have
X, = X\ + Yt + Zt + at + (xl[]x>1]) * i*, (46.1)
so for all 5 >0,JE[|Xa|] < oo.

320 Chapter XI The Characteristics of Semimartingales
3) In this case we have a+(x/[|x|>i])*i/ = 0 in (46.1) by the uniqueness
of the canonical decomposition. Then for all s > 0, Xs G A4, i.e., X is a
martingale. □
11.47 Theorem. Let X be a Levy process and AX be bounded. Then
for all p> 0 and 0 < s < t
E[\Xt-Xs\P]<oo.
Proof. Without loss of generality, we may suppose Xq = 0 and |AX| <
1. It suffices to show £[|Xt|p] < oo for all t > 0. We have
(ft(u) = exp{iuat - -u2/3t + / (elux — 1 — iux)du)}.
2 y[o,t]x[w<i]
For any positive integer m
E[\Xt\2m] < oo *=> ^(2m)(0) exists and is finite
d?m / \
~T~2^[[I[\x\<i](e%ux — 1 — iux)] *vt) exists and is finite
/ x2mdu < oo.
•/[<Mlx[|x|<ll
/ x2mdu < f x2dv < oo.
«/[0,t]x[|x|<l) J[0tt]x[\x\<l]
Thus £[|X*|2m] < oo. D
§4. Step Processes
11.48 Definition A process X is called a step process1^ if all its
trajectories are cadlag step function having at most a finite number of jumps
in every finite interval, i.e., X can be expressed as:
X = X0 + £ £n/[rn)oo[- (48.1)
71=1
where 1) Tn f oo; 2) for each n > 0,T*n < oo => Tn < Tn+i(T0 = 0 by
convention); 3) for each n > l,£n ^ 0 «<=> Tn < oo. In fact, for n > 1, Tn
is the n-th jump time of X:
Tn = inf{t > Tn_x : Xt ± XTn_x}, n > 1,
and £n = AXTnI[Tn<oo] is the n-th jump size of X.
But
^ Usually, a step process is also called a jump process. But in this book the term
"jump process" is reserved for the jump process of a cadlag process.

§4. Step Processes 321
It is easy to see that step process X is adapted if and only if each Tn
is a stopping time and £n G Trn •
Recall that the natural filtration F°(X) = (?t(X)) is defined as
J?(X) = a{Xs, s<t} = <r{XsAU s > 0}, * > 0.
For each n > 0 we have
7?(X) n[Tn<t< rn+i] = (r{xsATn, s > o} n [Tn < t < rn+1],
because on [Tn < t < Tn+i], XsAt = XsATn for all s > 0. Obviously,
<r{XaATn,s > 0} = *{*o,Ti, • • •,Tnj&, • • •,£„}, n > 1.
Denote Qn = a{X0,Tu ■ ■ • ,Tn,fi, • • ■ ,£»}." > 1, and GQ = a{X0}. Then
j?(X) = u fe n [r» < t < rn+i]), t > o.
n=o v 7
Clearly, we have
Qn n [rn+i = oo] = Qoo n [Tn+i = 00], n > 0.
Hence, F°(X) is a filtration of discrete type discussed in Chapter V §5.
We will utilize all results there. For example, by Theorem 5.56 we know
that for all F°(X)-stopping time T
J^{X) = a{XsAT, s > 0}.
In the remainders of this paragraph, the step process X is given and the
reference filtration F = (Ft) is taken to be the complete natural filtration
FP(X) of X. F°(X) = (??(X)) is simply denoted by F° = (7?). The
jump measure of X is denoted by /x:
00
fi(dt,dx)= £ 6iTn£n)(dt,dx)I[Tn<oo].
71=1
Obviously,
X = Xo + x * /x.
11.49 Theorem. For each n > 0 let Gn(dt,dx) be the conditional
distribution of (Tn+i,£n+i) w.r.t Trn:
Gn(dt,dx) = P[Tn+x e dt, £n+i e dx\FTn]i
and
Hn(dt) = Gn(dt,E) = PpTn+i € dt\TTn}.
Then
"(*&)=!ofM7|^-«' (491)
is the dual predictable projection of /x.

322 Chapter XI The Characteristics of Semimartingaies
Proof. First of all, we point out that (49.1) is meaningful. In fact,
let 5n+1 = inf{t : Hn([t, oo]) = 0}. Then Sn+1 <E TTn, Hn(]Sn^uoo]) =
0, Hn([t, oo]) > 0 for t < 5n+i, and
P(Tn+X > Sn+i) = E[P[Tn^ > Sn+ll^Tj] = E[Hn(]Sn+U Oo])] = 0.
Thus Tn+! < 5n+i a.s.. When t < Tn+1, a.s. either t < 5n+i, fln([*» oo]) >
0, or t = 5n+i. In the latter case, if ffn([5n+1, oo]) = 0, i.e., JJn({5n+1}) =
0, then Gnlt'S'n+i}, dx) = 0. Therefore i/ is well defined. By Theorem 5.55
2) v is a predictably random measure.
In order to show u = /x it suffices to prove that for any JB G B(E), Iq*v
is the dual predictable projection of Ib * M, owing to the monotone class
argument. Obviously Is* V € >tj^c, since /# * /ixn = /z([0,Tn] x B) < n.
Hence, it suffices to prove for any stopping time T and n > 0
£[/x([0, T A Tn+1] x J3)] = B[i/([0, T A Tn+1] x J8)].
But we have
/i([0,TATn+1] x B) = t I[Tk<T\n{\Tk,Tk+l AT] x 5)
fc=0
and the same expression for v. Now it suffices to prove
£[/[rn<T]M]rn,:rn+1 at] x b)] = £[/!Tn<^i/(]Tn,Tn+1AT] x b% n > o.
(49.2)
By Theorem 5.54 there is Rn € ^r„ such that r A rn+i = Rn A Tn+i.
Then
„nr r l * m - fTn+lAT G^dt^ - fTn+lARn Gn(dt,B)
E{T[Tn^nu{\Tn,Tn^ AT] x B)}
-FlT f\ fTn+iARnGn(ds,B){ U
- E{l[Tn^ J^ Hn(dt) J^ ^^jy}
= EWn<r\jTn Hnidt)^ I[a<^I[a<t]hI^}
= E{I[Tn<T\ jH I^^Gnids, B)}.

§4. Step Processes 323
On the other hand,
E{I[Tn<T]ti{]Tn>Tn+1 AT] x B)}
= E{I[Tn<nti(]Tn,Tn+1 ARn]x B)}
= EiI[Tn<T]I[Rn>Tn+l4n+i€B]}
= E{I[Tn<T]P[Rn > Tn+1,£n+1 e B\TTn}}
= E{I[Tn<n I[Rn>a]Gn{ds,B)}.
Thus (49.2) follows. □
11.50 Remarks. 1) (49.1) can be deduced from Theorem 5.69. Due
to its importance, we give a detailed proof here.
2) In (49.1) one may take Gn(dt,dx) as the conditional distribution
of (Tn+i,fn+i) w.r.t. Tj. , i.e., we may consider the dual predictable
projection u as an F°-predictable random measure.
3) For any B e B(E),Gn(dt,B) < Hn{dt\ then
Gn{u), dt, B) = Qn(u>, t, B)Hn(u, dt),
where the Radon-Nikodym derivative Qn(u,t,B) of Gn(u,dt,B) w.r.t.
Hn{uj, dt) can be taken such that for fixed (cj, t), Qn(u>, t, •) is a probability
measure on B(E), and for fixed B e S(£),Qn(-, •, JB) is TTn x B(fi+)-
measurable, i.e., Qn(u, t, dx) is a transition probability measure from (ft x
R+,FTn x B(R+)) to (E,B(E)). In fact, on {Tn+i < oo} we have
Qn(Tn+udx) = P[£n+i € dx\fTn+1-\ a.s.. (50.1)
To see this, take B e B(E), D e B(R+) and C e ?Tn,
P([Tn^ € Afn+i € B] n C) = f Gn(D x B)dP
Jc
= l (/ Qn^B)Hn(dt))dP
= / «[Q„(TM.llB)/[rn+l6D1|JTn]rfP
Qn(Tn+1,B)dP.
= /
/[Tn+iG^]nC
By Corollary 5.57 ^rn+i- = ^r„ V <r{Tn+i}, thus (50.1) follows.
Put
oo
Q(t,dx)= £ QnM*)/[rn<t<rn+1],
71=0
oo fj (dt)

324 Chapter XI The Characteristics of Semimartingales
Then
i/(dt, dx) = Q(t, dx)A(dt), (50.2)
where A* = A([0,*]) = ^([0,£] x E) is the dual predictable projection
oo
of counting process /z([0, t] x E) = £ I[Tn<t]i Q(v,t,dx) is a transition
71=1 ~
probability measure from (fi x R+,V) to (E, B(E)). The expression (50.2)
has clear probabilistic meaning and is constantly used.
4) If A(dt) -C dt (Lebesgue measure), there is a non-negative
predictable process (A*) such that
A(rft) = \tdt.
oo
(At) is called the intensity of the counting process £ J[t„,oo[- I*1 this case?
71=1
v(dt,dx) = \tQ(t,dx)dt.
\(t,dx) = XtQ(t,dx) is also called the intensity of the step process X.
11.51 Example. Let X be a regular temporally homogeneous
Markov chain with state space Z and Q = (qij) be its density martix:
0 < q% = -qu = £ Qij < °°-
Let (rij) be the transition matrix of its jump chain:
Tij = <
(1-*«) — ■ if«>0,
Qi
[ 6ij, if qi =0.
On [Tn < oo] we have for j ^ 0
G»(*,{i}) = «awe-«»w(*-r">/lr-t00lrjraw^Tii+idtl
and for t > Tn
By (49.1) for j # 0,
OO
»'(*,{i})= E 9XrnrXrn,A-Tn+^]rn,Tn+1l(^)
71=0
= qxt-,Xt-+jdt. (51.1)
Let ix(j) be a function defined on Z such that for all i
E«jMi)l <°°
Denote
(0")(O = £«ju(j).

§4. Step Processes 325
By direct computation we have
u(Xt) - u(X0) = / [u(X3- + x)- u(Xs-)]ii(ds,dx),
J[0,t]xE
because on [Tn+i < oo]
u{XTn+l) - u(XTn) = / [u(X3- +x) -u(X3-)]ii(ds,dx).
J]Tn,Tn+1]xE
On the other hand, by (51.1)
/ [u(X3- + x) - u(Xs-)]v(ds,dx)
J[0,t)xE
= I £ MXS- + j) - u(X3-)]qXs_}xa-+jds
Jo j^o
= / [ E qxa-jU(j) + qXa_}XaMXs-)\ds
JO lj*Xa- J
= / (Qu)(Xs-)ds = f (Qu)(Xa)ds.
Jo Jo
Hence we get the following well-known result: the process
u(Xt) - u(XQ) - f\Qu)(Xa)ds = ( [u{Xs- + x) - u(Xa-)]d(n - v)
Jo J[0,t]xE
is a local martingale.
11.52 Definition. Let H be a probability measure on ]0, oo]. Define
tH = w£{t:H([t, oo]) = 0],
[0,te], if tH < oo and H({tH}) > 0,
*<*> ■-■■ othe^se,
= f [0,«ff],
~ 1 [0,*ff[,
„ , x f* H(ds)
Jo
t>0.
H([s,oo]y
Evidently, Fn(t) is monotone increasing on R+, F#(0) = 0. If t < in,
then
Fn(t) = I
Jo
« £T(cfa) „ #([0,*])
< TT/, '' < oo,
H{[s,oo])-H ([«,«>])
AF m_ *({*» r1
because i/(]£, oo]) > 0. By Doleans-Dade exponential formula
H(]t, oo]) = e"W n (1 " AFff(«)), « € [0, t*[.

326 Chapter XI The Characteristics of Semimartingales
Hence
r i-e-wn(i-AFfl(»)), t<tH,
H([0,t}) = \ >* (52.1)
I 1, t > tH,
where Ffj(t) is the continuous part of Fn(t).
UH({tH})>0,
FH{tH) - amy < °°-
If tn < oo at the same time, then
AP ,, x H({tH}) T
&FH {tH) = -T7TT. iT = 1,
H{[tH,oo\)
FH(t) = FH(tH), t > tH.
If #({*«}) = (), by (52.1)
0 = H({t„}) = e-FC^-) n (l-*FH(s)).
s<tH
Then either FcH(tH-) = ooor ll<t {l-*FH{s)) = 0, i.e., £ AFH(s) =
3<tH
co. In a word, Fff (£#-) = oo, and Fnit) = oo for t > £//.
11.53 Lemma. Let H and H' be two probability measures on ]0, oo].
IfFH{t) = FH'(t) for t € *(#) n $(H'), then H = H'.
Proof. If tH < tH>, then FH(t) = FH'(t) for t < tH- In this case,
FH(tH-) = Fw(tH-) < oo, H({tH}) > 0 and tH € *(#). Hence
FH(t) = F#/(*) for t < tH. Since tH < oo, AFH.(tH) = &FH{tH) = 1 and
if'(]£//, oo]) = 0. This contradicts tfi' > ty. Thus tu < t^' is not true.
By symmetry it must be tu = *//'• Then by (52.1) H = H'. □
11.54 Theorem. Let P' be another probability on J^, such that
i) Plto = PIjj,
ii) under P1, v (taken as F°-predictable) remains to be the dual
predictable projection of /z. Then
PV = P\?o .
I-' OO '"' OO
Proof. By induction it only needs to show if P'\j*> = P\j* , then
P'\#> = p\^T • Since ^n+i = ^ V <7{rn+i,f»+i}, if suffices to
show under both P and P' we have a.s.
Gn{dt,dx) = P[Tn+1 € ctt,£n+i € d*|J$J
= P'[Tn+1 e rft, £n+1 € dx|*?J = G'n(dt, dx). (54.1)

§4. Step Processes 327
Since F$n+l n [Tn = oo] = ^ n [Tn = oo], it remains to show that (54.1)
holds on Qn = [Tn < oo]. First we will show on Qn
Hn(]Tn,t}) = Gn(]Tn,t] xE) = G'n(]Tn,t] x E) = H'n(]Tn>t]).
We adopt the following abbreviated notations:
H(]0,t]) = Hn(]Tn,Tn + t}), H'(]0,t]) = H'n(]Tn,Tn +t]),
Fit) = f H{ds) F'(t) = I* H'{ds)
r{t) J0 H({s,oo}Y *{t) Jo H'([s,oo]Y
T = inf{t : H([t, oo]) = 0}, T = ini{t: H'([t, oo]) = 0},
$ = $(#), *' = *(fT).
By Theorem 5.55.2) there is a process B € J^ xB(R+) such that \*v = B
on ]Tn,Tn+1]. Denote C(t) = BTn+t - BTn,t> 0. Then
p(nn n [F(t) = c(t),t < rn+1 - rn]) = P(ft„).
Noting that F(t) and C(t) axe ^j. -measurable, for any fixed t > 0 we
have
0 = P([F(t) # C(*),« < Tn+1 - Tn]fin) = JE?[/[F(t¥C(t)]ff([«,oo])/nB].
But H([t, oo]) > 0 for t € $, so
P([F(«) # c(«)] n [«€ *] n n„) = o. (54.2)
Since T is also T^ -measurable, by the same argument we have
P([F{T) ^ c(T)] n [T e *] n nn) = o. (54.3)
Denote 4 = [F(t) = C(t), Vt G *] n fin. Then Ae^n and
^nCl{( U [F{r)*C{r),re*])U[F{T)*C(T),Te*]}nn.
By (54.2) and (54.3) we have P(A) = P(f2n) and P'(A) = P(A) =
P(fin) = P'(nn).
Denote A' = [F'(t) = C(t),Vt e $'] n fin. By the same argument
we have P(A) = P'(A') = P'(ftn) = P(«»), and P(AA!) = P\AA') =
P/(fin) = P(ftn). On AA' F(t) = F'(t) for all * e $ n $'. By Lemma
11.53 H = H', i.e., Hn = H'n.
For any B G #(#) the result established above can be applied to
Gn{\Tn,t) x B) and G'n(]Tn,t] x £). Since B(£?) is countably generated,
it is not difficult to see that (54.1) holds on £2n under both P and P'.
D
Obviously, a step process X is a semimartingale, and its predictable
triplet is ((x/^i^!]) * i/, 0, i/). Theorem 11.54 means that the law of a step
process is uniquely determined by its initial law and Levy system.

328
Chapter XI The Characteristics of Semimartingales
oo
11.55 Definition. Let N = £ ^[Tn,oo[ be a point process, i.e.,
71=1
(Tn)n>i is an increasing sequence of r.v. such that Tn | oo and for each
n > 0, Tn < oo => Tn < Tn+i (T0 = 0). Let (£n)n>i be another sequence
of r.v.. (Tn,fn)n>i is called a marked point process (or multivariate point
process). In fact, a marked point process is not a process in ordinary sense.
Only when for each n > 1, Tn = oo «<=> £n = 0, (Tn, £n)n>i and the step
oo
process X = Yl £n^[Tn,oo[ can be determined by each other.
71=1
For a marked point process (Tn,£)n>i we still define
_ oo
*? = U (Gn n [Tn < * < rn+i]), t > o,
71=0
Gn = a{Tu •, Tn, £i, • • •, £n}, n > 1, (<?0 is arbitrary),
and call F° = (rf) its natural filtration. Similarly by (48.1) we define its
jump measure /z. It is not hard to see that the two main theorems 11.49
and 11.54 remain true for a marked point process, even the initial a-field
Go may be arbitrary and need not be trivial.
Problems and Complements
11.1 Let /i and v be two optional (resp. predictable) and optionally
(resp. predictably) a-integrable random measures.
1) The following statements are equivalent:
i) P({w : m(w, ■) <^i ■)}) = !.
ii) Mp < Mv on T,
iii) M^ < Mv on 6 (resp. V),
iv) n = W.i>, where VT £ (5+ (resp. P+).
2) The following statements axe equivalent:
i)P({w:^(u;,Oii/(a;,.)}) = l,
ii) M^lMv on (^(resp. P),
iii) M^LMv on j^.
11.2 Let /i be an integer-valued random measure and v be its
compensator. If /i is quasi-left-continuous, i.e., the support of /x is totally
inaccessible, then W h-* W * (/z — i/) is an isometric mapping from L2(J7, P, M„)
to A42*d.
11.3 Let X be a semimartingale with predictable triplet (a, /?, i/).
Then X G Moc (resP- Moc) if and only if dxl%r|>i]) * ^ € ^c (resp.
x2 * i/ £ ^c) and a = -(x/[W>1]) * ia

Problems and Complements 329
11.4 Let X be a semimartingale with predictable triplet (a, /?, u).
Then l)IeV« (|x| Al)*^^ and /? = 0;2)lGV+^
(|x| A 1) * 1/ € Ate /? = °> K-R+ *]oo, 0]) = 0 and ac > (xJ[|x|<i]/[O=o]) *i/,
where ac is the continuous part of a; 3) I E *Aioc ^=^ |a;| * i/ £ -Aj£c and
0 = 0.
11.5 Let X be a semimartingale. Then
1) D = {t: P(AXt 7^ 0) > 0} is at most a countable set.
2) X = X1 + X", where X' is a stochastically continuous
semimartingale, and X'l = Yl AXS (the series is absolutely convergent in pro-
0<s<t,seD
bability).
11.6 Let X be a Levy process, /z be its jump measure and v be its Levy
system. Let /(£, x) be a Borel function on R+ xE. If V£ > 0 I[f^o]*ut < °°
or V£ > 0 |/| * vt < oo, then F = / * /x is also a Levy process, and
E[eiuY<] = exp( / (c*/(*.*> - i),/(<fa, dx)).
W[0,t]x£ J
11.7 Let X be a Levy process with characteristics (/,/?, i/).
1) X e V+ if and only if /? = 0, v(R+x] - oo,0]) = 0 and V* >
0 {xI[o<x<i]) * "i < °°i ft = ft ~ (^[o<x<i]) * ft is monotone increasing.
In this case,
(pt(u) = exp{iftu + / (eiux - l)di/}.
•/[0,*]x[x>0]
2) X e V if and only if /? = 0,V* > 0 Oxl/fi^i]) * ut < oo and
ft = ft — (xl[\x\<i]) * ^ is a function with finite variation. In this case,
<pt{v) = exp{iftu + f (eiux - l)di/}.
J[0,t]xE
Moreover, Yt = \dX3\,t > 0, is also a Levy process, and
■/[CM]
E[ju(Yt-Y0)] = exp|i|x /* |rf£| + / (ju\x\ _ 1)rfl/\
1 JO J[0,t|x£ J
11.8 Let X be a Levy process and v be its Levy system. Then X is a
step process if and only if Vt v([0, t] x E) < oo and
<pt(u) = exp{ / (eiux - l)di/}.
11.9 Suppose X is a continuous Levy process and Xq = 0. Then X is
a normal process.
11.10 Let X be a Levy process and Xq = 0. Then X is a Poisson
process if and only if X is a point process.

330
Chapter XI The Characteristics of Semimartingales
11.11 Let X be a point process and A be its compensator with A^ =
oo. Let (rt) be the change of time associated with A. Then (XTt) is a
Poisson process with parameter 1 w.r.t. (.?>*)•
11.12 Let /i be an integer-valued random measure. Let m be a a-finite
measure on (R+ x E, B(E)) such that Vt > 0, m({t} x E) = 0. If m is the
compensator of /z, then
i) m = E[fi]^
ii) for each B e B(R+) x B(E),ii(B) has a Poisson law,
iii) for any disjoint J§i,---,Bn € B(R+) x B(E) and Bi c]s,oo[x£,
i = 1, • • •, n, for some 5, /i(JBi), • • •, fJ>(Bn) and Ts are independent.
11.13 Let B = (Bt) be a standard Brownian motion. Set
Ft = Ft\/(r{B1}, 0<*< 1,
Wt = Bt- f Bl~Bsds, 0<t<l.
Then (Wt)o<t<i is an (^)o<*<i-Brownian motion, and
rt aw
Bt = tBx + (l - *) / —L, o < t < 1.
70 1 - 5
(Note that (£* — tB\)o<t<i is a Brownian bridge.)
11.14 An adapted continuous d-dimensional process W = (W1,--,
Wd) is a d-dimensional standard Wiener process if and only if (Wl, W^)t =
6ijt,t > 0.
oo
11.15 Let X = ^ I[rnioo[ be a point process. Then X is a Poisson
71=1
process if and only if there is a continuous monotone increasing function
A* with Ao = 0 such that for all n > 0
P[rn+1 € dt\TTn]I[Tn<oo] = e-(A<-ATn)l[Tn<t]dAt.
11.16 Let (fn)n>i be an i.i.d. sequence of real r.v. with common
distribution function F, and N = (Nt) be a homogeneous Poisson process
which is independent of (fn)n>i- Find the Levy system for the following
processes:
l)Xt = Zi + --- + tNt (Xt = 0 when Nt = 0), t > 0,
2) Xt = max(0,£i, • • • ^Nt)(Xt = 0 when Nt = 0), t> 0.
oo
11.17 Let X = ^2 Jfrn,oo[ be a point process. Assume Ti, T2 — T\,
71=1
• • •, Tn+i — Tn, • • • axe i.i.d. real r.v. with common distribution function
F (F(0) = 0), i.e., X is a renewal process. Find the Levy system of X.
11.18 Let X be a step process. Let f(t,x) be a Borel function on
il+ x R and continuously differentiate in t. Then
/(«, Xt) = /(0, X0) + f |-/(a, Xa)ds + E [f(s, X.) - f(s, X.-)].
J0 OS 0<s<t

is is a
Problems and Complements 331
11.19 Let X be a regular temporally homogeneous Markov chain with
state space Z. Let Q = (q^) be the density maxtrix of X. Assume
f(i,j) to be a function on Z x Z such that for all i,f(i,i) = 0 and
E«vI/(m)I < oo. Then £ f(Xs-,Xs)-[tZqx8jf(XsJ)ds
j 0<s<t JO j
local martingale.
11.20 Let X be a step process, and v be its Levy system. Then X is
a Markov process if and only if v has the following form:
v(dt,dx) = Q(t,Xt-,Xt-+dx)A(Xt-,dt),
where 1) Q(t, x, dy) is a transition probabiUty measure from R+ xflto
#withQ(*,x,{x}) = 0;
2) A(x, di) is a cr-finite transition measure from R to R+ with A(x, {t})
< 1, and there exist two sequences (/n) and (gn) of Borel functions on R
such that for each x £ H, R+ can be expressed as a union of disjoint
intervals:
oo
n=l
and for all * €]/n(x), 5n(a;)[
A(x,]/„(x),«[) < oo, A(x,{t}) < 1.

Chapter XII
Changes of Measures
In this chapter we will discuss the well-known Girsanov's theorems
which describe how to transform semimaxtingales and stochastic integrals
under the change of measures. We will also give some applications of
Girsanov's theorems, including the characterization for semimartingales.
§1. Local Absolute Continuity
In the first three paragraphs of this chapter we make the following
basic assumptions. On the basic space (Jl,^70) we axe given:
i) a right-continuous filtration F° = (yrt)t>o with T^ = T°,
ii) two probability measures P and P'.
Set
P=\{P + P').
Evidently, we have P « P and P; « P on f°. Define
F = (F°)? : ?t = (F?f = J* V A\ t > 0,
where Af is the a-field generated by all P-null sets of J7^ = (F°)p.
Henceforth, we take F = (F°)p to be the reference filtration. Since we will deal
with the two measures P and P' at the same time, such a selection is
natural and reasonable. Thus stopping times, optional processes, local
martingales, semimaxtingales, • • •, mean always F-stopping times, F-optional
processes, F-local martingales, F-semimartingales, •••, respectively,
unless otherwise stated. For any stopping time T we denote by Pt and PtT
the restriction of P and P' on Tt respectively. E, E' and E denote the
mathematical expectations under P, P1 and P respectively.

§1. Local Absolute Continuity
333
These basic assumptions and notations will be not repeated later. But
we emphasize that one should be careful with null sets and evanescent sets.
Usually, null sets and evanescent sets mean P-null sets and P-evanescent
sets respectively, unless clarity dictates otherwise.
12.1 Definition. We say that Pi is locally absolutely continuous
w.r.t. P, and denote it by P,<£P, if for all* > 0 P'
equivalently, P[ <& Pt.
*> « P
12.2 Theorem. Suppose that there is a sequence (Tn) of stopping
times such that Tn | oo and for each n, P'Tn <& Pxn. Then for every
stopping time T with P'{T < oo) = 1 we have P'T -C Pt- In particular,
. loc
P'<P.
Proof Let A e TT and P(A) = 0. Since A[T < Tn] e TTATn and
P(A[T < Tn]) < P{A) = 0, we have P'(A[T < Tn]) = 0. Then
P\A) = P\A[T > Tn]) < P'(T > Tn).
But limsupP'(T > Tn) < P\T = oo) = 0. Whence P'(A) = 0. D
71—>00
. loc
12.3 Lemma. Suppose P <P. Let S and T be two stopping times.
Then
1) P(S < T) = 0 => P'(S < T) = 0, in particular, P(S < oo) = 0 =>
P*(S < oo) = 0,
2) P(S = T < oo) = 0 =* P'(S = T < oo) = 0.
Proof. 1) Noting that V£ > 0 [5 < T, S < t] e Tt, we have
P(S < T) = 0 => Mt > 0 P(S < T, S < t) = 0
=> V* > 0 P'(S < T, S < *) = 0 => P\S <T) = 0.
Proof 2) is similar. □
. loc
12.4 Theorem. Suppose P «P. T/ien there exists a unique adapted
non-negative cadlag process Z = (Zt)t>o satisfying
1) Zoo = lim %t exists P-a.s., and
t—►oo
P(Z0O = oo) = P(sup Zt = oo) = 0, (4.1)
t>o
P'^ = 0) = P'(inf Zt = 0) = 0, (4.2)

334 Chapter XII Changes of Measures
2) for every stopping time T, whenever P'T -C Pt> we have
in particular, under P, (Zt) is a martingale. Z = (Zt) is called the density
process of P1 w.r.t. P.
Proof. Let (Yt) and (Y[) be the cadlag modifications of yE 7"=n«^i )
and (e\—— \Tt J respectively. Set
r = inf{*:y* = 0} and r' = ini{t: Y{ = 0}.
Recall that for t < r (resp. t < r') Yt > 0 and Yt- > 0 (resp. Y( > 0 and
y/_ > 0), for t > r (resp. t > rf) Yt = 0 (resp. Y( = 0) (Theorem 2.62).
Thus
P(r < oo) = / YTdP = 0, Pi(r' < oo) = / Y^dP = 0.
J[r<oo] J[t'<o6\
By Lemma 12.3, P'(r < oo) = 0. Therefore P(r < oo) = 0, i.e., we may
consider r = oo. Now define
Zt = ^r, t>0.
Apparently, Z = (Zt)t>o is an adapted non-negative cadlag process. As
t —► oo,
Y'
Zt-^ P-a.s..
dP' „ dP
Since Y' = —=-, Voo = —^,
°° dP dP
P'(YL = 0) = / CdP = 0, PCYoo = 0) = / YocdP = 0,
•/[V4=o] •/[yoo=o]
we have P(Y4 = ^oo = 0) = 0, i.e., Z& = Y^/Voo makes sense. At the
same time we obtain
P{Z^ = oo) = P(Y^ > 0, Yn = 0) = 0,
p'fZoo = o) = p'(y4 = o) = o.
On the other hand, because Zt is bounded on every finite interval, sup Zt =
t
oo <$=> Zoo = oo. On [t' < oo] we have inf Zt = Z^ = 0, and on
[r1 = oo] we have V* > 0, Zt > 0, V* > 0, Zt- > 0, and therefore,
inf Zt = 0 «<=> Zoo = 0. (4.1) and (4.2) are established.

§1. Local Absolute Continuity 335
Let T be a stopping time such that P'T <C Pt- Then Pt ~ Pt and
= ^JL/^J: = Y^/Yt = Zt P.a.s..
dPrl dPT
dP'T dP'r
dPT dPTl dPT
In particular, for all t > 0
HP'
Z« = ^ P«. (4.4)
By (4.4) and right-continuity, Z = (Zt) is uniquely determined. □
Remark. If P' -C P, it is not necessary to introduce P. It can be
replaced by P simply. The reference filtration F = (Tt) can be taken to
be {F°)p = ((^)p), and the density process Z = (Zt) is just the cadlag
modification of (E —— yFt J.
Prom the above proof one can see immediately the following
12.5 Corollary. If P'^P, then B = [0] U [Z_ > 0] is a predictable
set of interval type, where Z is the density process. In fact, we have
IB = IfI[o}ri + IfcI[o}r]i (51)
R = ini{t : Zt = 0}, F = {u> : 0 < R(u>) < oo, ZR({Jj)_(u) = 0} (5.2)
or
n
Rn = inf (t: Zt < -}, n > 1. (5.3)
We will continue to use all notations in Corollary 12.5 in this chapter.
. loc
12.6 Theorem. Suppose P <P. Let T be a stopping time. Then
^(R < oo) = 0,
2)J"(T = oo) = 1 «=► P(T >R) = 1.
Proof. 1) follows from the proof of Theorem 12.4, since R = r'. We
continue to use the notations there.
0 = P'(T < oo) = / Y^dP <^> YrI,T<oo] = 0 P-a.s.
J[T«x]
•*=► P(T >R) = l.
Obviously, P'(T = oo) = 1 =► P(T > R) = 1. Conversely, by Lemma 12.3
P(T > R) = 1 => P(T <R) = 0*=> P'(T < fl) = 0 =► P'{T > R) = 1.
But P'{R = oo) = 1, so P'(T = oo) = 1. □

336
Chapter XII Changes of Measures
. loc
12.7 Corollary. Suppose P' <P. Let X be an optional process.
Then X is Pf-evanescent if and only if XIiqjh is P-evanescent.
Proof. Let T be the debut of [X ^ 0]. Then
X is P'-evanescent <=> P'(T = oo) = 1
<=> P(T > R) = 1 <=> Xlyojn is P-evanescent. D
12.8 Lemma. Suppose P' <P. Let X be an (F°)p-adapted process,
whose trajectories P-a.s. are cadlag (resp. continuous). Then there exists
an F-adapted cadlag (resp. continuous) process X such that X is P-
indistinguishable from X.
Proof. For each r e Q+ choose Yr e ^? so that P(YT = Xr) = 1.
Write At = {u: there exists a cadlag function / on il+ such that for all
r G [0,t] n Q+ Yr(u) = f(r)}, t > 0. Then At e J^ (its proof is put to
the end). Put
S(u>) = mf{t:u><?At}.
Since A^ is monotone increasing in t, for all t > 0
[S <t]cA$C[S< t],
[S<t]D f)Ai£??+=??-
Hence S is an F°-stopping time. By the assumption P(S = oo) = 1. By
Lemma 12.3 P'(S = oo) = 1. Therefore P(S < oo) = 0, [5 < oo] € TQ.
Set
f lim Yr(u), if S(u) = oo,
{ 0, if 5(w) < oo.
Then X is F-adapted and cadlag, and P-indistingusihable from X.
If the trajectories of X P-a.s. are continuous, put
T(u) = inf {* : AXt ^ 0}.
Then T is an F-stopping time and P(T < oo) = 0. Similarly, we have
P'(T < oo) = 0, P{T < oo) = 0 and [r < oo] € Tq. Put
T(o/) = oo,
vi \ f *(w)' if:
{ o, if:
X't(u) = .
■ T(u) < oo.
Then X' is F-adapted, continuous, and P-indistinguishable from X.

§1. Local Absolute Continuity 337
Finally, we show At € Jf. For any positive integer I define by
induction
f FoM, if iimyr(w) = y0("),
T,fo(a/) = 0, Z,,oM={ ri°
I 0, otherwise,
?Wi(u,) = inf {r € Q+ : r > r,in(cj), \Yr(u>) - Zl§n(u>)\ > ^} A t,
Zi,n+lM = S
lim yrM, if TJ n+iM < £ and
riT|f„+i(u/)
Urn Yr(u) = YTl^lH(^\
' «Mf,fl+l
+oo, otherwise.
We want to show
oo oo oo
At = n u n pu =«]. (8.i)
1=1k=ln=k
Evidently, the set on the right-hand side of (8.1) belongs to J^. Let
u G At. Assume I is fixed. If T^n(u) < t, then Z^n(u)) < oo and
Tltn(u>) < ritn+i(o;). If for all n > 1, Tln(u) < t, then r,,n(u;) t 5 < t,
|£in+i(^) _ ^n(^)| > ^7? and liml^.(a;) does not exist. This contra-
diets u) e At. Hence u> e U£Li f|£UPu = *]• Conversely, let u> G
flSi U£i nSS=fc[TifW = t]. For Z > l define ft = min{n : Ti%n{u) = t] and
{Z|in((j), 5 € [Tu(a;),Tu+1(a;)[, n < fc - 1,
Ft(u;), se[t,oo[,t€Q+,
0, *€[*,oo[,t£Q+.
Then /j is cadlag on R+, and it is straightforward to verify
sup i/^-r^i^i
r€[0,t]nQt Z
sup sup \fi(s) - /j+m(*)| < rpj.
m>l a>0 *
Hence (fi)i>i uniformly converges to /, / is cadlag on R+ and for all
r € [0, t] D Q+ /(r) = Yr(w), i.e. u64 □
. loc
12.9 Theorem. Suppose P <P. Let X 6e an F-adapted process
whose trajectories P-a.s. are cadlag (resp. continuous, resp. right-
l) if Ti%n+i(u) £ Q+, it is required only that the limit exists.

338
Chapter XII Changes of Measures
continuous increasing, resp. right-continuous and with finite variation).
Then the trajectories of X P'-a.s. are the same as P-a.s..
Proof. By Lemma 12.8 there is an adapted cadlag (resp. continuous)
process X such that [X ^ X] is P-evanescent. By Corollary 12.7 [X ^
X] is P'-evanescent, i.e., the trajectories of X P'-a.s. are cadlag (resp.
continuous).
If the trajectories of X P-a.s. axe increasing, for all s < t, P(XS <
Xt) = 1, and consequently P'(XS < Xt) = 1. It is already known that
the trajectories of X P'-a.s. axe cadlag. Therefore the trajectories of X
P'-a.s. are increasing.
If the trajectories of X P-a.s. are functions with finite variation, then
P(T < oo) = 0, where T = inf{t : Jj0 ti \dXs\ = oo} is a stopping time.
Similarly, P'(T < oo) = 0, i.e., the trajectories of X P'-a.s. are functions
with finite vaxiation. □
§2. Girsanov's Theorems for Local Martingales and
Semimartingales
. loc
In this paragraph we always suppose P <£iP and Z = (Zt) is the
density process of P' w.r.t. P.
The concepts of local maxtingale, semimaxtingale, • • • are dependent
on measures. Therefore, we use notation M\oc(P) for the class of all
processes which are local martingales under P. According to Lemma 12.8
and Theorem 12.9, for every X G A4ioc(P) there exists an F-adapted
cadlag process X, P-indistinguishable from X. Hence we consider each
process of M\oc(P) to be P-adapted and cadlag. The situations for other
classes of processes axe similar, and we do not repeat the formulations.
12.10 Lemma. Let S and T be two stopping times such that P's <
Ps and P'T < PT Let r.v. ^fT and £'[|£|] < oo. Then
E[ZZt\Fs]I[s<t\ = ZsE'[t\rs]Iis<T\ P-a.s.. (10.1)
Proof Let D e Fs. Then D[S <T]e FSat and
E[ID[s<T\iZT] = E[ID{S<T[Z] = E'{ID[s<T]E'[t\Ts]}
= E[ID[3<nZsE?[li\Fs\Y
Thus (10.1) is deduced. □

§2. Girsanov's Theorems for Local Martingales and Semimartingales 339
12.11 Lemma. Let X be an adapted cadlag process and (Tn) be an
increasing sequence of stopping times such that limTn > R P-a.s. and
71
for each n,XTn e M\oc(P). Then X e (M\oc(P))B. (The predictable
set of interval type B and stopping time R are defined in (5.1) and (5.2)
respectively.)
Proof. We may suppose P( limTn = Rj = 1. Otherwise, Tn can be
replaced by Tn A R. Let T be a stopping time such that [0,T] C B. We
want to prove XT e M\oc(P). Put
T'n = (Tn)lTn<T], D = f][Tn<T].
n
It is not difficult to verify: i) Tn |; ii) on D for all n > 1, T'n = Tn < T < R,
and therefore T = R; iii) on Dc for n large enough Tn>T and T'n = oo.
Hence (T^ A n) P-a.s. foretells Rd, and Rp is predictable.
Since Z € M\oc(P),
Zrd-I[rd<0o] = E[ZRdI[Rd<oo]\^Rd-] = 0. P-a.s..
Thus for P-almost all u e [RD < oo] [0,T(cj)] C [0,R(u)[. But in the
above we know that on D we have T = R. Hence it must be P(Rd <
oo) = 0, and therefore P(T^ | oo) = 1. On the other hand,
(lY-^)TGMoc(P).
So XT e Xioc(P). a
Remark. In fact, the lemma is concerned only with the measure P
and it holds not only for the class of local martingales but also for any
class of processes whenever it is stable under the localization.
12.12 Theorem. Let X be an adapted cadlag process. Then X G
Mloc^) if and only if XZ <E (MXoc(P))B.
Proof. Since only local martingales are concerned, we may assume
X0 = 0.
Necessity. Let X € M\oc(Pf). There exists an increasing sequence
(Tn) of stopping times such that Tn | oo P'-a.s., for each n, XTn € M(Pf)
and P*Tn <& PTn- The last requirement can be satisfied by replacing Tn
with Tn A n, if necessary. By lemma 12.10
E[(XZ)Tn\?t]IHTn] = (XZ)tI[t<Tn] P-a.s..
But (XZ)TnI[Tn<t] € ft- In a word,
E{(XZ)Tn\Tt] = (XZ)tATn P-a.s.,

340 Chapter XII Changes of Measures
i.e., (XZ)Tn G M(P). By Theorem 12.6.2) P(limTn > R) = 1. Then by
Lemm 12.11 XZ G {M^P))3.
Sufficiency. Let XZ G (M\oc(P))B'• There exists an increasing
sequence (Tn) of stopping times such that Tn | R P-a.s., for each n, (XZ)7"
G M(P) and P^n <C PTn- On [* < Tn] we have
(XZ)tATn = E[(IZ)Tn \?t] = ZtE'[XTn \Tt] P-a.s.,
E'[XTn\Ft] = XtATn, P'-a.s.,
because under P', Z never vanishes. But Xrn J[Tn<$] G ^i, so
£'[XrJ.Ft] = X(A:rn, P'-a.s.,
i.e., Xr« G A4(P')- Since P'(Tn | oo) = 1, X G MXoc{P'). D
12.13 Theorem. //A" € Moc,o(P) and lx,z) € (A,c(P))B, then
±-.{X,Z)€Aioc(P'),
X' = X- -±-.(X, Z) G Moco(P'),
where [X, Z] and (X, Z) are defined under P.
Proof. Write C = ^-.(X,Z). By Theorem 12.9 (X,Z) G Aioc(P).
On ]0,Rn] we have Z_ > -, where /?„ is defined in (5.3). Thus C*" €
AlociP'). But P'(i?n t oo) = 1, so C G A>c(P')-
Since XZ - (X, Z) G (Moc,o(P))B, for each n, (XZ)^ - (X, Z)*" G
A1ioc,o(P)- On the other hand, by the formula of integration by parts and
Yceurp's lemma
(CZ)R» - (X, Z)*" = (CZ)R» - Z-.C*"
= C^.Z*" + [C'SZ*"] G Mioc(P),
Hence (XZ)11" - (CZ)*" = (X'Z)H- € Moc(P)- BY Theorem 12.12
X' € Moc(P'). □
Theorem 12.13 is called the Girsanov's theorem for local martingales.
12.14 Theorem. If X & S{P), then X G S(P'), and [X^P*) is
P'-indistinguishable from [X](P).
Proof. Decompose X as X = Xo + M + A, where M G A1ioc,o(P)»
|AM| < 1 and A G V(P). We have [M,Z](P) G A>c(P) (Problem 7.10).

§2. Girsanov's Theorems for Local Martingales and Semimartingales 341
By Theorem 12.13 M' = M - C € MiOCt0(P'), where C = —.(M,Z) €
V(P'). Then under P' X can be decomposed as X = X0 + M' + (C + A),
C + A€ V(P'). Thus X e S(P').
For all t > 0, P't < Pt. By the remark after Theorem 9.33, [X]t(P)
(resp. [A"]t(P')) is the limit in P (resp. P/) of sum of quadratic differences
of X on [0,t]. Hence
[X}t(P') = [X]t(P) P'-a.s..
By the right-continuity of trajectories (Theorem 12.9) [X](P') is P'-
indistinguishable from [X](P). D
12.15 Corollary. If X e McXocQ(P), then X' = X - ^-.{X,Z) <E
' Z/—
M^oiP1) and (X')(P/) is P1-indistinguishable from (X)(P).
Proof. It follows immediately from Theorems 12.14 and 12.9. □
12.16 Corollary. If X € Mfoc(P) and[X,Z](P) € (A>c(P))B, then
X' = X-^-.(X,Z)&Mfoc(P').
Proof. It suffices to show that the P'-local martingale X' is purely
discontinuous. Write C = —.(X,Z). Under P'
//—
[X](P') = [X](P) = £(AX)2,
[Xf](P/) = [X](P') - 2[X,C](P') + [C](P')
= E(AX)2 - 2E(AXAC) + E(AC)2 = E(AX')2,
this implies X' is purely discontinuous under P'. □
12.17 Corollary. If X 6 S(P) and Xc is the continuous martingale
part of X under P, then (Xc)' = Xc - —— .(XC,Z) is the continuous
Li-
martingale part of X under P'.
Proof Decompose XasX = X0 + M + A, where M e M\ocp(P),
|AM| < 1 and A e V(P). Then Xc = Mc, (Mc)' = Mc - —.(MC,Z) e
Zj _
^bc,o(p/)- (Mdy = Md- j:-(Md>z) e MtiP'),
x = x0 + (Mcy + (Mdy + (a + J-.(m,z)).

342
Chapter XII Changes of Measures
Hence (Xc)' = (Mc)' is the continuous martingale part of X under P\
D
12.18 Theorem. Let X be an adapted cadlag process. Then
i)Xe viP1) <=^xe (V(P))B,
2) X € «S(P0 <^ X € (S(P))B,
3)Xe SP(P') *=*Xe (S(P))B and X + ^-.[X, Z] € (SP(P))B,
4) X€Moc(P') <=► X6 («S(P))B andX+J-. [X, Z] € (Moc(P))B,
5) X € ^oc(P') «=» X € (V(P))B andX + -±-. [X, Z] € (^ioc(P))B,
in £/iis case, #ie duaZ predictable projection of X under P' is
P'-indistinguishable from the dual predictable projection of X + -=-.[X, Z] under P.
Proof Without loss of generality, we may assume Xq = 0.
1) Let X € V(P'). Put Tn = inf{t : / \dX,\ > n} A R*. Then
Jo
P'(Tn T oo) = 1, P(Tn T R) = 1. If Tn = oo, then for all t > 0
/ |dX,| < n. If Tn < oo, / " |dXs| < n + \AXTn\. Hence Xr» € V(P),
and therefore X € (V(P))fi.
Let X € (V(P))B (resp. («S(P))B). Then for each n,X^ € V(P)
(resp. S(P)), X*" € V(P') (resp. S(P')). But P'^ | oo) = 1. So
X e V(P') (resp. 5(P')).
2) Let X e Moc(P')- Then for each n,^)^ € Mioc(P). Choose
F(x,y) € C2(ii2) such that F(x,y) = x/y for |y| > 1/n. Then
Y = F((XZ)Rn,Z)eS(P).
When t < Rn, we have \Zt\ > 1/n. Hence YIy0 #n[ = X/[o)^n[. Since
X*"=XI[0>Rnl + XRnI[Rn e «S(P).
This implies X € («S(P))B. We have established X € «S(P') =^Jf€
(<S(P))B. The converse implication has already been shown above in 1).
3) Let X € «SP(P'). Then under P' X = JV+A, where iV 6 MOC,0(P)
and A € V0(P') is predictable. Put Tn = inf {t: / |dA3| > n} Ai?n. Then
P(Tn }R) = 1, XT« € «S0(P), (NZ)T» € A^loc,o(P) and AT" € V0(P).
By the formula of integration by parts,
(NZ)Tn + (AZ)Tn = (XZ)Tn = X-.ZT" + Z_.XTn + [Xr",Z].

§2. Girsanov's Theorems for Local Martingales and Semimartingales 343
Since ATn is predictable and (AZ)Tn - Z-.ATn e M\OCi0(P), Z-.XTn +
[XTn,Z] - Z_.AT- e -Mioco(P). Since Z_ > - on ]0,Tn], we obtain
' n
XT» + j-.[XT«,Z] - AT» € Mloc>0(P), i.e., XT« + j-.[XT",Z] € SP(P).
Hence X + ^=-.[X, Z] € («SP(P))B by Lemma 12.11 and its remark.
Z—
Now let X € (5(P))B and X + -£-\X, Z] € {SP{P))B. Then X*" €
Z—
S{P), and y(n> = X*" + J-.[X*», Z] € SP(P). Let y<B> = Af<n> + A^

jibe the canonical decomposition of Y^n\ where M^ G Mioc,o(P) and
^W 6 V(P') is predictable. Since (i4<n+1))fl» = A<-n\ then
n=l
and is predictable (Jfo = 0). In fact, X + -=-.[X, Z]-A€ (Mioc(P))B.
Put N = X - A. Then N1^ = X*" - A*" = M'n» - ^-.[X*", Z],
Z—
(NZ)R* = AL.Z*- + Z-.N** + [N^^Z]
= N-.ZR« + Z-.MM - [X*~, Z] + [N**, Z]
= N-.Z*" + Z-.MM - [A**,Z] € Miocfi{P).
Hence N € Miocfl(P'), X = N + A is the canonical decomposition of X
under P'. This means X € «SP(P').
4) follows from proof 3) with A = 0.
5) Let X € Ax^i5')- Then X € V(P') n «SP(P'). By 1) and 3) we
have X € (V(P))S and X + J-. [X, Z] € («SP(P))B. Hence
Z—
x + ^-.[x,z]e(Aloc(P))B.
Conversely, let X € (V(P))B and X + ^-. [X, Z] € (A>c(P))B- Then by
Z—
1) and 3) X € V(P') and X € «SP(P'). Hence X € AkC-P')- The last
assertion follows from proof 3), in which A is both the dual predictable
projection of X under P' and the dual predictable projection of X +
-=-.[X, Z] on B under P. □

344 Chapter XII Changes of Measures
12.19 Corollary. If A € (Aoc,o(p))B> then
A € Aloc(P') «=» [A,Z] € (A>c(P))B.
/n this case, under P' we have
A*p' = A">p + ±-.(A,Z),
where AP'P and AP'P are the dual predictable projections of A under P
(on B) and P1 respectively.
Proof. This is a consequence of Theorem 12.18.5). Under the
assumption, A € A*(i") <=> A + -^-.[A,Z\ € (Aioc(P))B <^ f~\A,Z\ G
(A>c(P))B <=► [A,Z] € Mioc(P))B, and
A*p' = {A+ j-.[A, Z]fP = A*p + -±-.(A, Z). O
Theorems 12.14 and 12.18 axe Girsanov's theorems for semimartin-
gales. The next theorem is another Girsanov's theorem for local
martingales.
12.20 Theorem. Let X e M\oc,o(P)' Then
1) X e S(P') andM = X- 1 [X,Z] + Y e Mloc,0(Pf), where Y is
Z
the dual predictable projection ofY = AXfi/[fi<00]/[floo[ under P.
2) X £ Sp(P') <=$> [X, Z] £ (A\oc(P))B, and in this case the canonical
decomposition of X under P' is
X=(X-±.{X,Z)) + -±-.(X,Z).
3) X € ■Mioco(P') <^ [X,Z] € (Mioc,o(P))B.
Proo/. 1) Since / |dYa| < yJ[X)t, Y € Aloc(P). Because A =
. [X, Z] is P'-indistinguishable from —.[X,Z], it suffices to show
Z Z
for each n, (MZ)^ € A^ioc,o(P), where M = X - A + Y.
{ZA)R" = A-.Z*" + Z.A*",
Z.A*» = /'J[Z>01.[X*»,Z] = [X^,Z]R-.
Thus (ZA)*- - [X^,Z\R- € ^Iioc,o(P). We have also (ZY)*» -Z-.Y*"
= Y.Z*» € Xioc,o(P) and (ZX)^ - [X*», Z] € Moc,o(P). Hence
{MZ)*" - {[X**,Z] - [X*»,Z]R~ + Z.Y^} e Moco(i'). (20.1)

§2. Girsanov's Theorems for Local Martingales and Semimartingales 345
Noting ZRI[R<oo] = 0, we have
[X^, Z] - [XR",Z)R- + Z-Y**
= AZRAXRuI[R<00]IlRooi + Z/*_ AYj^" J[fl<00j/[Ri00[
= AZRAXRI[Rn=R<00]IyRool - AZRAXRI[Rn=R<00]IlRt0oi = 0.
By (20.1) we know (AfZ)*» € A4ioc,o(P)-
2) Let X € SP{P'). By Theorem 12.18.3) X + J-.[X,Z] € («SP(P))B,
Z—
and for each n, A"*" + -^-.[X*", Z] € «SP(P). Because X«" 6 Moc,o(P),
Z —
we have ^-.[X*»,Z] € A>c(P) and [X*«,Z] € A>C(P). Thus [X,Z] €
Z—
(Aoc(P))B- The another half is just Theorem 12.14.
3) By Theorem 12.18.4),
X € A4loc,o(P') *=*► X + ^-.[X, Z] € (Moc,o(P))B
<=> -±-.[X,z}e(Mlocfi(P))B^[x,Z}e(Mlocfi(P))B. n
12.21 Theorem. Suppose X € A*ioc,o(P) and [-X", £] G ^loc(P)- £e<
H be a predictable process such that under P, H.X exists (i.e., y/H2.[X]
€ ^ioc(P)) and [H.X,Z] € Aioc(P). Set X' = X - ^-.(X,Z). Tften
Z—
under Pf, H.X' exists and
H.X' = H.X - ^-.(H.X, Z). (21.1)
Z—
Proof. Write M = H.X, M' = M - -£-.(M,Z). We deal with the
continuous and purely discontinuous cases separately.
Firstly, assume X 6 A*foc,0(P)- By CoroUary 11.15 X' € Mfoc0(P')
and (X')(P') is P'-indistinguishable from (X)(P). Since >/#*-W €
■Aloc(P), y/H2.(X') € A>c(P') (CoroUary 12.19). So under P', tf.X'
exists. On the other hand, under P' we have
(M',H.X') = H.(M',X') = H.(M,X) = (M,H.X) = tf2.(X),
<M') = (M) = tf2.(X>,
(#.X') = H2.{X') = H2.(X).
Thus (M' - H.X') = 0, AT' = #.X'. (Evidently, in the above (X), (M)
and (M, X) are defined under P.)

346 Chapter XII Changes of Measures
Secondly, assume X 6 M^C{P). By Corollary 12.16 X' € Mf^F).
At the same time, we have M € M^C{P) and M' € A^foc(P')- Since
AX' = AX-^-A(X,Z),
AM' = HAX - ^-A(X, Z) = HAX',
Z—
under P', H.X1 exists and H.X' = M1. D
12.22 Theorem. Suppose X e S(P). Let H be a predictable process
such that under P, H.X exists (denoted by H-X). Then under P',H.X
pi pi . p
exists (denoted by H • X), and H • X is P -indistinguishable from H-X.
Proof. We may assume Xq = 0. Let X = M + A be an
.//-decomposition of X under P, where M G M\oc$(P) and A € V(P). We may
assume that AM and HAM axe bounded. Otherwise, we may set
C = E(AM/[|AM|>1 or |/jam|>i])> N = C - C
(C is the compensator of C under P), and replace M and Aby M — N
and AT + A respectively. Thus [M, Z], [F^M, Z] e A\oc(P). Put M' =
M - -j-.(M, Z), A' = A + -j-.(M, Z). Then by Theorem 12.21
Z— Z—
Hp'm' = HPM - ^-.(HPM, Z) = HPM - ^-.(M,Z).
Z— Z—
On the other hand, H.A € V(P) *=> H.A 6 V(P'), and (HPM, Z) 6
V(P) =► ^-.(M.Z) = -^-.(HPM,Z) € V(P'). Thus
#. A' = H. A + ^-. {M, Z) € V(P').
Z—
Therefore, under P', X = M1 + A1 is an if -decomposition of X, and
HPX = Hp-M' + H.A' = HPM + H.A = HPX. D
Theorems 12.21 and 12.22 axe the Girsanov's theorems for stochastic
integrals. As a simple appUcation of Theorem 12.22, we show the local
property of stochastic integrals in the next theorem, which is concerned
only with the measure P.
12.23 Theorem. Assume that X and Y are semimartingales, H and
K are predictable processes such that H. X and K. Y exist. Let A G T such
that on A, X and Y are indistinguishable, H and K are indistinguishable.
Then on A, H.X and K. Y are indistinguishable as well

§3. Girsanov's Theorems for Random Measures
347
Proof. We may assume P(A) > 0. Put P'(-) = „?*'• Then
P < P, X (resp. H) is P'-indistinguishable from Y (resp. K).
Thus HP'X = KP'Y. By Theorem 12.22 ff.A" (resp. K.Y) P-
pi p/
indistinguishable from H • X (resp. K • Y)
Hence H.X is P'-indistinguishable from K.Y, i.e., on A, H.X and K.Y
are P-indistinguishable. □
§3. Girsanov's Theorems for Random Measures
In this paragraph we still suppose P CP. At the same time, we
suppose /i is an integer-valued random measure:
fj,(dt,dx) = £ 6{sM(dt,dx)ID,
S>0
where (3 = {fit) is an optional process, D c]0, oo[ is the support of \x.
Under P (resp P') the measure generated by /z is denoted by M^ (resp.
M'). We suppose M^ is cr-finite on 'P. The dual predictable projection
of n under P is denoted by v.
12.24 Lemma. J/iV = (Nt) e Sp and E[\N0\] < oo, then N and AN
are cr-integrable w.r.t. V under M^. (Note that here only the measure P
is concerned.)
Proof. Let An G V such that An | fi and M^(An) < oo. Then
CW = /r */x€.4+. Put
T0(n) = O,!^ = inf{t > T%\ : ACln) ± 0},m > 1.
We have P( lim T™ = oo) = 1. By the assumption there is a sequence
(S&°)m>i of stopping times such that S&° < I&0, P( lim S& = oo) = 1
— to—>oo
and AT5m is of class (D). Then
7ft
)|<OC.
Hence N is cr-integrable w.r.t. V under MM. Since N- is locally bounded,
AN is also cr-integrable w.r.t. V under M M. □
12.25 Theorem. M' and M'v are a-finite on V, and M'^ < M'u
onV.

348 Chapter XII Changes of Measures
Proof. Take An e V such that M^(An) < oo and An | 11 For all
t > 0 and A e V by Theorem 5.32
M'^AAn(lO,t}xE)) = E'V^iit] = JBIZtU^*^)] = ^[(Z/^ )*//,].
Letting t —► oo and n —> oo yields
M'l£IA) = Mll{ZIA), AeV. (25.1)
By Lemma 12.24, Z is cr-integrable w.r.t. V under M^. Hence M^ is
cr-finite on P. Denote by u' the dual predictable projection of fi under P*'.
Under P, JB^71) = I-r * i/ is the dual predictable projection of I~ * //.
Put T„m = inf{* : 5,(n) > m}. Then P( lim Tnm = oo) = 1, and
therefore P'( lim Tnm = oo) = 1. Since Afl<w> < 1,
m—>oo '
MU^([0,Tn,m] x £)) = JG'pgj < m + 1.
Hence M'u is cr-finite on V.
Now we may assume Mf^(An) = M/l//(^47l) < oo and M'u(An) < oo as
well. For all t > 0 and A 6 V by Theorem 5.33
AfUAMp), t] x £)) = B[Zt(/^n * i/f)] = ^[(Z-/^) * ut]
Letting t —► oo and n —► oo yields
M'Ma) = Mii(Z-Ia), AeV. (25.2)
Under P;, C(n) = Jj * i/ is the dual predictable projection of I~ */i.
Put 5nm = inf{* : c[n) > m\ A m. Then P'( lim Snm = oo) = 1 and
M „>(An(lO, S„,m] xE))<m+l. For aUAGP
M'^AAnftZ- = 0][0,5„,ml x E)} = Mul[AAn([Z- = 0][0,S„,m] x £)]
= ^[^n.m(/^n[z_=0] * */)s„.J = E[{Z-IAA-n[z=Q]) * ^n,J = 0,
and therefore M^(A[Z- = 0]) = 0. Combining with (25.1), we obtain
M^Ia) = M^IA[Z_>0]) = M„(ZJA[Z_>01). (25.3)
Then M^ < Af (, on P follows from (25.2) and (25.3). □
12.26 Theorem. There exists a non-negative predictable function Y
on fi such that

§3. Girsanov's Theorems for Random Measures
349
1) i/ = Y.v is the dual predictable projection of // under P', and for
P1-almost allu for allt>0
0<v'(u,{t}xE)<l, (26.1)
i/(w, {t} x E) = 1 => i/(w, {«} xB) = l, (26.2)
2) Mp[Z\V] = Z-Y.
Proof. By Theorem 12.25 denote by Y the Radon-Nikodym derivative
of MJ, w.r.t. Af'„ on P. Then for all A € P
M ^(7,0 = M'V{YIA) = M'yMa).
This means v' = Y.v is the dual predictable projection of /z under P'.
Denote at = u({t} x £) and a!t = i/({*} x E). Put
{y, if a < 1, a' < 1, or a = a; = 1,
1, otherwise.
In order to prove (26.1) and (26.2), it suffices to prove that Y' and Y are
P/-indistinguishable. We have already known that [a' > 1] (resp. [a > 1])
is P'-evanescent (resp. P-evanescent). Therefore it is only required to
prove that [a = 1 ^ a'\ is P'-evanescent. Suppose there is a predictable
time T > 0 such that [T] C [a = 1 ^ a'] and P'(T < oo) > 0. By Theorem
11.14 [T] \ D is P-evanescent, i.e., P(/x({r} x £*) ^ 1, T < oo) = 0.
For all * > 0 we have P(fj,({T} x E) ^ 1, T < t) = 0, and therefore
PV({T} x£)/l,T<t) = 0, P'Mm x £?) # 1, T < oo) = 0,
^/[t<oc] = «V({T} x E)I[T<oo]\TT-\ = I[T<oo], ^-a.s.,
i.e., a'T = 1 P'-a.s. on [T < oo]. This contradicts [T] C [a' ^ 1]. Hence
[a = 1 7^ a'] is P'-evanescent.
By (25.1) and (25.2) for all A e V
M^ZJYIa) = M'^YIa) = M'AIa) = M^/a) = M^Z/^).
Thus 2) is estabished. D
Remark. In fact, if Y is a non-negative predictable function on SI
such that M^[Z\V] = Z-Y, then Y. u is the compensator of /z under P',
since for all A 6 V
M'^Ia) = M^ZIa) = M^Z.YIa) = MUYIa) = M'Ym(Ia).

350 Chapter XII Changes of Measures
12.27 Lemma. Suppose M = W * (// - u) (W € Q{y)), N € A4k>c,o,
V = M^ANIV] and [M, N] € A>c- Then
(M,N) = (VW)*v. (27.1)
(Note that here only the measure P is concerned.)
Proof. Write H = (M, N). For any predictable time T > 0
A#r/[r<oo] = E[A[M,N]TI[T<oo]\fT-] = E[AMTANTI[T<oo]\TT_\
= E[(W(T,0t)Id(T) - WT)ANTI[T<oo]\FT_]
= E[ANtW(T,Pt)Id(T)\FT-].
Observe that AMtANtI[t<<x>) and WtANtI[t<<x>] are c-integrable w.r.t.
TT-, then so is ANTW(T,f3r)ID(T). Furthermore, if E[\ANTW(T,fo)
Id(T)\] < oo, then for any bounded predictable process X
E[ANT W(T,0t)Id(T)Xt] = e[ f ANWXImd^
= M^ANWXIp^) = M„{VWXIm) = Mv{VWXIm)
= E[(VW)TXTI[T<oo]}.
Hence for any predictable time T
A#r/[r<oo] = E[ANTW(T, 0t)Id(T)\FT-} = (VW)TI[T<0o] a.s..
Therefore AH = {VW), and the purely discontinuous part of H is
Hd = £(A#) = E(VW) = (VWIj) * v, (27.2)
where J = [a > 0] is the predictable support of D.
Let K be a predictable process such that K.[M, N] € A. Noting that
J is a thin set and IjcHd = 0 (by (27.2)), we have
E[ J" KtdH<\ =E[J°° KtIjc{t)dH$] =E[T KtIjc(t)dHt]
= E[J°°KtIjc(t)d[M,N]t] =E[Z KtAMtANtIjc(t)]
= E\ £ KtW(t,pt)ANtIjcD(t)\ = M^ANWKIjc)
Lt>o J
= M^VWKIjc) = M„(VWKIjc) = e[[°° Ktd((VWIjc) * i*)].
Hence the continuous part of H is
Hc = (VWIjc)*v. (27.3)
Then (27.1) Mows from (27.2) and (27.3). □

§3. Girsanov's Theorems for Random Measures 351
Remark. If in the lemma the assumption [M, N] e A\oc is replaced
by [M, N] £ Afoc, where A is a predictable set of interval type, then the
dual predictable projection of [M, N] on A is (M, TV) = (VWIa) * ^-
12.28 Theorem. Suppose M = W * (/x - v)(W G G(v,P)) and
[M, Z] € (AXoc(P))B. Put A = {W(Y - 1)) * v. Then
1) A is P'-indistinguishable from -—.{M,Z)y
2) W € 0(/i, -P') and
W^'(/x - i/) = M - -J-.(Af, Z) = M - A (28.1)
Proo/. Denote M' = M - —-.(M,Z). By Corollary 12.16 M' €
Zj —
Mfoc(P/). On the other hand, by Theorem 12.26 M„[AZ\V] = Z_(y~l).
By Lemma 12.27 (M,Z) = (Z-W(Y - 1)) * u. Thus under P', ,4 =
—.(M, Z). Furthermore,
LI —
AMl=W(t,(3t)ID(t)- f W(t,x)u({t},dx)
JE
- [ W(t,x)[Y(t,x)-l]v({t},dx)
Je
= W(t,0t)ID(t)- f W(t,x)t/({t},dx).
Je
Hence W € G(», P') and M' = W%(p - v'). D
12.29 Theorem. IfV zs P'-indistinguishable from v, then
1) Af „[AZ|P] = 0,
2) G(ijl,P) C G{ii,P'), and for all W € G(ji,P) W**(ji - u') is P7-
indistinguishable from W*(fi — v).
Proof. We may take Y = 1. 1) follows immediately from Theorem
12.26. Let W € G{n,P). By Theorem 11.19
A = J^L .„ + E (-^t)d - «)) 6 ^C(P).
1 + |W-W| Vl^T ' toc
Since under P', i/ = v and a' = a, W may be used for P' as well. Under
P1, A is still a predictable process with locally integrable variation. Again
by Theorem 11.19 WeG(fi,P').

352
Chapter XII Changes of Measures
Denote M = W*(p - v) and M' = WP* (/x - u'). Then X = M -
M' e S(P/). But AX is P'-indistinguishable from zero, sole ^(P7)
and M e <SP(P')- BY Theorem 12.20.2) [M,Z\ € (A>c(^))B. Then by
Theorem 12.28 under P' we have M = M'. □
12.30 Theorem. Le* X € 5(P) and
X = X0 + a + Xc + (xl[lx>l]) *n + (x/[|x|<i]) *(**-«')
be its integral representation under P, where ji is the jump measure of X
and (a,/3, u) is the predictable triplet of X under P. Then under P' the
integral representation of X is
X = XQ + a' + (Xcy + (x/[N>i]) * /x + (x/[w<i]) * (/a - v'\
where (Xc)f = Xc - —.(Xc, Z), (a', /?', v') is the predictable triplet of X
under P/ which satisfies the following conditions:
\)a' = a + j-.(Xc, Z) + ((Y - l)sJN<i]) * u,
ii)/?' = /?,
iii) v1 — Y. v, where Y is a non-negative predictable function
determined as in Theorem 12.26.
P
Proof. Write W = x/[|x|<i] and M = W*(n — v). Since \W\ < 1 and
|AM| < 2, we have [M, Z] e A\oc{P). By Theorem 12.28
W^ifi - u') = W*(n -u)- (W(Y - 1)) * v.
By Corollary 12.15 (Xc)' € McXoc^P') and /?' = ((Xc)')(Pf) = (XC)(P)
= (3,vf = Y.u follows from Theorem 12.26. After rearrangement we obtain
immediately the integral representation of X under P' and the expression
for a1. □
12.31 Theorem. Let X e S(p), (a,/?,i/) and (a',/3',1/') be the
predictable characteristics of X under P and P' respectively. Then (a, /?, v)
and (a/, /3', v') are P'-indistinguishable if and only if the following
conditions are satisfied:
i) M„[AZ\V] = 0,
ii) (Xc, Z) is P-indistinguishable from zero.
Proof. Under P',/?' = (3 holds always. By Theorem 11.26 and its
remark we know that the condition i) is equaivalent to Y = 1, i.e., i/ =
v. Now by Theorem 12.30 and Corollary 12.7, a' = a under P/ <=>

§4. The Characterization for Semimartingales 353
^-.(Xc, Z) = 0 under P' <^> (Xc, Z) = 0 under P' *=> (Xc, Z)I[QjR[ =
0 under P «<=> (Xc, Z) = 0 under P. The last equivalence takes place,
since Z = ZR and (Xc, Z) is continuous. □
§4. The Characterization for Semimartingales
In this paragraph we suppose as usual that (fi,^7, P) is a complete
probability space equipped with a filtration F — (Tt)t>o satisfying the
usual conditions. Denote by L1 and L°° the spaces of all integrable r.v.
and all bounded r.v. respectively. If G and H axe subsets of L1, denote
G — H = {x — y : x € G,y e H}, and denote by G the closure of G in L1.
12.32 Theorem. Let K be a convex set in Ll and 0 e K. Then the
following three statements are equivalent:
1) For allf] e (Ll)+\{0}, there exists c> 0 such that cq £ K -(L°°)+.
2) For all A € !F with P(A) > 0 Mere existe c > 0 swc/t that cIA £
K-{L°°)+.
3) JYiere exzste a( G L°° swc/i that ( > 0 a.s. and sup2£[C£] < °o>
Sex
where {Ll)+ and {L°°)+ are the sets of all non-negative elements of Ll
and L°° respectively.
Proof. 1) => 2) is trivial. 2) => 3). Let A e T and P(A) > 0. By
the assumption there exists c > 0 such that cIa & K — (L°°)+. Since
K — (I/00)"1" is convex and L°° is the dual space of L1, by Ascoli-Mazur
theorem there exists 0 G Z/°° such that
sup E[0(£ - 7])} < cE[0IA}. (32.1)
Putting f = 0 and t] = a0~, a > 0, in (32.1) yields
aE[(0-)2} <cE[0IA}. (32.2)
Since (32.2) holds for all a > 0, we have 0" = 0, a.s., i.e., 0 e (L°°)+.
g
Besides, it is obvious that P{0 > 0) > 0. Replacing 0 by -=j^r, if necessary,
E[0\
we may assume E[0] = 1. Then by (32.1) we have
sup E[0£] < c.

354
Chapter XII Changes of Measures
Put H = {0 e (L°°)+: E[0] = 1 and sap E[0(\ < oo}. We have shown
H is non-empty. Put C = {[0 = 0] : 0 e H}. We want to prove that C is
closed under countable intersections. Let (0n) C H and Cn = sup <E[0nf],
dn = ||0n||£oo. Choose a sequence (6n) of strictly positive reals such that
£ bn = 1, £ Cnbn < OO, £ Mn < OO.
n n n
Let 0 = Y. bnOn. Evidently, 0 £ if and [0 = 0] = f)[0n = 0]. This means C
n n
is closed under countable intersections. Thus there exists £ £ if such that
P([C = 0])=mfP(P = 0]). (32.3)
It remains to prove C > 0, a.s.. If P([C = 0]) > 0. Write A = [C = 0]. By
the result obtained in the above, there exists 0 £ H such that (32.1) holds.
Especially, we have E[0I[c=Q]] > 0. This implies P([0 > 0] n [C = 0]) > 0.
Hence P([0 = 0] n [C = 0]) < P([C = 0]). However, [0 = 0] n [C = 0] e C.
It contradicts (32.3).
3) => 1). Suppose 1) is not true. Then there exists 77 6 {Ll)+ \ {0}
such that for all c > 0 cq € K — (L°°)+. For each n there exists fn €
K, 7}n e (L°°)+ and 6n e L1 such that nrj = £n - r/n - <5n and ||<5n||Li < -.
n
We have £n > nrj + 6n, and for any strictly positive r.v. £
sup £[C£] > supJE[C6J = +00.
This contradicts 3). Thus 1) holds. □
12.33 Theorem. Let K be a convex set in L1. If for any sequence
(£n) C K we have — £[" —>0X\ then there exists C £ L°° such that C > 0
n
a.s. and sup !£[££] < 00.
Proof. First of all, we may assume 0 € K. Otherwise, we may take an
arbitrary 77 € K and replace K by {x-rj : x e K}. Suppose the assertion
1) in Theorem 12.32 does not hold. From proof 3) => 1) in Theorem 12.32
we know that there exist 7? € (L1)"1" \ {0}, (fn) C K and (6n) C Ll such
that for each n, H^nllz,1 < ~ and — > V + — • This contradicts —£+ ->0.
nun n
Therefore the assertion 1) in Theorem 12.32 is true, and we arrive at the
required conclusion by Theorem 12.32. □
J) It is easy to see that this condition is equivalent to the following one: for any
given e > 0 there exists c > 0 such that for all £ 6 K, P(£ > c) < e.

§4. The Characterization for Semimartingales
355
12.34 Definition. Denote by H the collection of all bounded
predictable processes of the following form:
n-i
H = £ &/|t<,t<+1|,
where 0 = t0 < t\ < • • • < tn < oo, & e Tti, |&| < 1, i = 0,1, • • •, n - 1.
Let X be a process. For every H G W define a process J(X, i/) as follows:
J(X,H)t = J:Zi(XtiM-Xti+lM), t>0.
1=0
Obivously, for every t the mapping (X, if) i—> J(X, H)t is bilinear.
Moreover, if X is a semimartingale, then J(X, H) = H.X.
The following theorem characterizes semimartingales.
12.35 Theorem. Let X be an adapted cadlag process. In order for X
to be a semimartingale it is necessary and sufficient that for every sequence
(#<">) C H and all t > 0, - J(X, H^)t 4 0.
n
Proof Necessity, let X eS, (H^) C W and * > 0. Since \^-H^
1/n, by Theorem 9.30
in
1-J(X,H^)t=(^.x)f,0.
n \ n 't
Sufficiency. We want to show for all t > 0, X1 = (XsAt)s>o 6 S.
Since X is cadlag, XI = sup|X,| < oo. We may assume 2£[Xt*] < oo.
3<t
Otherwise, we may replace P by an equivalent probability measure (the
assumption remains true in this case). Put
K={J(X,H)t:HeH}.
Then K is a convex set in L1. By the assumption, for every sequence
1 P
(£n) C K we have — £n —> 0. By Theorem 12.32 there exists a strictly
n
positive bounded r.v. £ such that E[Q = 1 and sup £?[££] < oo. Set
dP' = QdP. Then P' is a probability measure, equivalent to P. Since
C is bounded, E'[Xf] = E[CX£] < oo. We will prove that X* is a quasi-
martingale under P'. Let r : 0 = to < t\ < • • ■ < tn = t be a finite
partition of [0, t]. Put
#(T) = E^iW,], fc = sgn(£;'[X(j+1 - JfcJ^]).
t=0

356
Chapter XII Changes of Measures
Then #<T> € H and
E'[J(X,H(%] = E'[nt\i(Xti+1 - Xti)}
1 1=0 J
71—1 n rfl— 1
= E'[t tiE'[Xti+1 - Xti\Tti]] =E'[f: \E'[Xtl+1 - Xti\rti\
1 1=0 J L 1=0
Hence we have
Var(X',P') = suptfpEV^-M -Xufc]] + E'[\Xt\)
T L t=0 J
= suP£;,[j(A',JffW)t] + f;/[|xt|]
r
= supE[(J(X,HW)t] + £[CI*t|] < oo.
r
This means under Pf, Xf is a quasmaxtingale. Therefore, for all t > 0
X* G 5, and hence X G 5. D
12.36 Theorem. Let G = (Gt) be a filtration satisfying the usual con-
ditions such that for all t > 0, Gt C Tt. Suppose X is an F-semimartin-
gale and G-adapted. Then X is a G-semimartingale, [X\(F) and [X](G]
are indistinguishable.
Proof. Put
«(G) = { E Wl^J : |6|<i,< = o,-,n-l,n>l J"
Then H{G) C H. For every (#(n)) C H(G) C W, by Theorem 12.35 we
have for all t > 0
±J(X, #<»>)*£(),
n
because X is an F-semimartingale. Again by Theorem 12.35, X is also
a G-semimaxtingale. Since [X]t is the limit in probability of sums of
quadxatic differences of X on [0, £], it does not depend on filtrations. 0
12.37 Theorem. Let G = (Gt) be a filtration satisfying the usual
conditions such that for all t > 0, Gt C Tt- Suppose X is an F-semimarting'
ale and G-adapted. Let H be a G-predictable process such that H is in-
tegrable w.r.t. X and F (the integral is denoted by H-X), then H is also
integrable w.r.t. X and G (the integral is denote by H • X), H • X and
H?X are indistinguishable.
Proof. First, we show the integrability of H w.r.t. X and G. Set
A = T,(&x I[\ax\>i or \hax\>i]), Z = X - A.

§4. The Characterization for Semimartingales
357
By replacing P with an equivalent probability measure, if necessary, we
may suppose the two F-special sen
following canonical decompositions:
may suppose the two F-special semimartingales Z and H • Z have the
Z = N + B, HFZ = N' + B',
where N and N' are F-martingales, B and B' are F-predictable processes
with finite variation and Bq = B'0 = 0 such that for all t > 0
E\ I |dBs|] <oo,E[[ \dB's\] < oo (refer to Problem 12.12).
By Theorem 9.16 we have B' = H. B. Thus for all t > 0, e\ I \Hs\\dBafi
< oo. Let B be the dual predictable projection of B w.r.t. G. Then for
all t> 0, 17[ / |tfs||dBs|] < oo, i.e., H.B exists. By Theorem 5.30.2) for
s< t
E[Bt - Bt\Gs] = E[Ba - B3\G3] a.s.,
and therefore (noting that Z = N + B is G-adapted)
E[Nt + Bt- Bt\Qs\ = E[E[Nt\Ta)\Gs) + E[Bt - Bt\G3]
= E[Ns + Bs-Bs\gs} = Ns + Bs-Bs, a.s..
Hence N+B-B is a G-martingale. Since >Jh2.[N + B-B] < ^H2.[Z]+
E(|ff AB\) and \HAZ\ < 1, ^H2.[N + B - B] is locally integrable w.r.t.
G, i.e., H -(N + B — B) exists. Therefore, H is integrable w.r.t. X and
G, and H?X = H?(N + B-B) + H.(A + B), where the second term on
the right-hand side is a Stieltjes integral.
Now we show that H • X and If^X are indistinguishable. By Theorem
9.30 and its remark we may assume that H is bounded. Put C = {[(U]:
A € £o} U {]T,oo|[: T is a G-stopping time }. Obviously, for all A G C,
IA - X and I a • X are indistinguishable. Then by the monotone class
argument we arrive at the desired conclusion. □
12.38 Definition. Assume on the basic space (fi,^0) a
right-continuous filtration F° = (Jrt)t>o with J1^ = T® and a process X = (Xt)t>o
are given. A probability measure P on (fi,^70) is called a semimartingale
(resp. martingale) measure w.r.t. (X, F°) if under P, X is an (F°)p-
semimartingale (resp. local martingale). A semimartingale (resp.
martingale) measure w.r.t. (X, F°) is also called a solution of semimartingale

358
Chapter XII Changes of Measures
(resp. martingale) problem (X,F°). The collection of all solutions of
semimaxtingale (resp. martingale) problem (X, F°) is
denoted by TS(X,F°) (resp. Tm(X,F0)). The definition of semi-
martingale (resp. martingale) problem can be extended naturally to the
case of a family {X°, 0 e E} of processes.
For example, suppose on a probability space (fi,^7, P) a continuous
process W = (Wt) with Wo = 0 is given such that W is a standard Wiener
process w.r.t. its natural filtration. Set
*?= n*W-,r<*},*>0, ?<> = ?l,
s>t
X} = WU X? = W?-t, t>0.
Then by Corollaries 11.39 and 11.37 P is the unique solution of martingale
problem ({X\X2},F°).
Furthermore, we assume
i) F is a probability distribution on R,
ii) a and f3 are two processes with c*o = /?o = 0,
iii) v is a random measure with ^({0} x E) = v(R+ x {0}) = 0.
A probability measure P on(fi,^r0) is called a solution of semimartin-
gale problem (X, F°; F, a, /?, v) if under P, X is an (F°)p-semimartingale
with F as its initial law (i.e. the law of Xq) and (a, /?, i/) as its predictable
characteristics. The collection of all solutions of semimaxtingale problem
(X, F°; F, a, /?, v) is denoted by T3(X, F°; F, a, 0, i/).
12.39 Theorem. Let (fi,^70), F° and X be given as in Definition
12.38. // (Pn) C r3(X, F°) and P' is a probability measure on (f2,^°)
such that P' is absolutely continuous w.r.t. £n Pn, then P' G T3(X, F°).
In particularf rs(X,F°) is convex.
Proof. Put
-vo-/V*r 0 = *0<*i <-"<<n<oo,^e J?., 1
H -\i^l^il: |6|<if i = 0,...,n-l, n>l J*
and P = J2n ^nPn, where An > 0, n > 1, and £n An = 1.
For every (#<*>) C W° and t > 0, by Theorem 12.35 for all n
lJ(X,H(k)))t%0, as k^oo.
It is easy to see that we have
0, as k —> oo.

Problems and Complements
359
However, P' <S P. So we have
as k —► oo,
as well. Again by Theorem 12.35 we know P' € Ta(X, F°).
If Pi, P2 € ra(X, F°), and P = aPi + (1 - a)P2, 0 < a < 1, then
P<Pi + P2, and therefore P € T3(X, F°). D
12.40 Theorem. Le< (fi,^0), P°, X, F, a, 0, v be given as in
Definiton 12.38. Then rs(X,F°;F,a,P,i>) is convex.
Proof. Let Pu P2 € TS(X, F°; F, a, /?, i/), and P = aPi + (1 - a)P2,
0 < a < 1. First of all, F is still the law of Xo under P. Let /z be the
jump measure of X. Then M^(P) = aM^(Pi) + (1 — a)MAi(P2), and
on V, Af „(P) = aM„(Pi) + (1 - a)MI/(P2) = AfM(P). This means u
is still the dual predictable projection of /x under P. Let (a', /?', i/) be the
predictable triplet of X under P. By Theorem 12.30 under both Pi and
P2 /? is indistinguishable from /?', and therefore /? is indistinguishable from
/?' under P. By Theorem 12.29 (^[|x|<i]) * (m- *0 *s Pi-indistinguishable
P P2
from (x/[|x|<1])*(/x - v), (xl[\x\<\\) * (A* — v) IS P2-indistinguishable from
P P
(x/[|x|<ij)*(Ai - v). Therefore X - X0 - a - (xl{\x]>1]) * /i - (^[|x|<i])*
(// — u) is a continuous local martingale under both Pi and P2, and it is
a continuous local martingale under P, i.e., a! is indistinguishable from
a under P. In a word, (a, /?, v) is the predictable triplet of X under P.
Hence P € r,(X,F°,F,a,/?,i/). □
Problems and Complements
12.1 Let f be a r.v. defined on a probability space (ft, T, P). There
dP'
exists another probability measure P' on (ft, T) such that P' ~ P, ——-
is bounded, and for all n, 2£'[|£|n] < 00.
12.2 Let (£n) be a sequence of r.v. defined on a probability space
(ft,T, P). Suppose limnm_oo E[\£n - £m| A 1] = 0. There exists another
dP'
probability measure P' on (ft, T) such that P1 ~ P, —— is bounded, and
for all p > 1 Km-oo E'[\tn - Zm\P] = 0.
12.3 Let X e M\oc(P), A = Y>(AXI[lAXl>1]), A be the dual pre-
. loc
dictable projection of A under P. Assume P <£iP and Z is the density

360
Chapter XII Changes of Measures
proess of P' w.r.t. P. Put Y = X - -£-. (X - A, Z) - A + A. Then
y € M]oc(P'). Furthermore, X G <SP(P') if and only if A e A>c(P')-
12.4 Let M,X e M\oc(P). Assume that (X,M) exists and £(M)
is a non-negative uniformly integrable martingale under P. Set dP1 —
Z{M)OQdP. Then X - (X, M) <E .Mioc(P')-
12.5 Let W be a standard Wiener process and H be an adapted
measurable process. If E[exp(± J0°° H2sds)] < oo and dP' = £ (H.WOoodP,
then (W^ — /q Hsds) is a standard Wiener process under P'.
12.6 Assume that P <giP and Z is the density process of P w.r.t.
P. Let A e V(P), [;4,Z] € (^ioc(P))B. Then A <E A>c(P') *=> A €
(A>c(P))B. In this case, A(P') = A(P) + £-.(A, Z).
12.7 Assume that P' < Pand Z is the density process of P' w.r.t. P.
Let A e V0(P) and C = Z.A Then A e A>c(P') ^Ce (A>c(P))B.
In this case, i(P') = £-.C(P).
12.8 Let i G f, X G 5 and i/ G L(X). If for all u e j4,X.M is
a function with finite variation and Y = XIa, then the Stieltjes integral
H.Y exists, and on A, H.X is indistinguishable from if.F.
12.9 Let AeT, and 1,7^5.
1) If on A, X — Y is a process with finite variation and Xq = Yq, then
on A, Xc and Yc are indistinguishable.
2) If on A, X — Y is a continuous process with finite variation and
Xq = lo, then on A, [X] and [Y] axe indistinguishable.
12.10 Let X = (Xt) be an adapted cadlag process and for all t > 0,
E[\Xt\) < oo. Put Ka = {J(X,H)a: H = YTiZo &Wil>° = *° < *i
< * * * < t>n — a* & € ^ is bounded, i = 0, • • •, n — 1, n > 1}, a > 0. Then
the following three statements axe equivalent:
dP7
1) There exists a probability measure P' ~ P such that —— € L00
and under P',Xa = (XtAa)t>o is a martingale.
2) For all A e ^ with P(j4) > 0, IA <£ Ka - (L°°)+.
3)(L1)+nKa-(L~)+ = {0}.
12.11 Let X = (Xf) be an adapted continuous process and for all
t > 0, JE7[|^Ct|] < oo. ifa, a > 0, is defined as in the previous problem.
, dP'
In order that there exist a probability P ~ P such that —— G L°° and
under P', Xa = (Xt/\a) is a martingale, it is necessary and suflScient that
(L1)+nFa = {0}.

Problems and Complements
361
12.12 Let X G S. Then there exists a probability measure P' such
dP'
that P' ~ P, -r= G L°°, X G SP(P') and the canonical decomposition
(XMT
X = M + A of X under P' satisfies the following conditions: for all t > 0
M1 = (MsM)s>o € M2(P') and A* = (AsM)s>0 € .4(P').
12.13 Let X be an adapted cadlag process. Set
f , 0 = To < Ti < • • • < Tn, Tu 1 < i < n,
K= <K = J2 £iIlTi,Ti+i[ ■ are stopping times, & € FTi,
[ i=° K<|<1, t = 0,---,n-l, n>l
(£}=ofe(*r>+1- - xTi-)\ + k{xt - xTi-),
J°(X,K)t = l Ti<t<Ti+1,i = 0,---,n-l,
[ZUMXTJ+1-- XTj.), Tn<t.
Then the following statements are equivalent:
1) X is a semimartingale, and for all £ > 0, £ |AXa| < oc a.s..
s<t
2) For all (#<»>) C £ and t > 0 ± J°(X, K^)t 4 0,
3) There exists an adapted increasing process A such that for any
stopping time T and K G /C F[(J°(X, AT)r-)2] < F[Ar_(AT2.A)T_].
12.14 Let X be an adapted cadlag process. Suppose there exists a
sequence (Rn) of r.v. with Rn ] oo and a sequence (X^) of semimartingales
such that for each n Xt = X\n\ t < Rn. Then X is a semimartingale.
12.15 Let {A\,A2, • • •} be a countable partition of fi. Set Qt =
Tt V o{A\,A2, • • •}, £ > 0. Then an (^^-semimartingale is also a (Gt)-
8emimartingale.
12.16 On the basic space (fi,^0) a right-continuous filtration F° and
an F°-adapted process X are given. Let P be a probability measure on
(fl, J°). For A G T° with P(A) > 0 define PA(.) = P{>DA)/P{A). If
C = {A: PAe TS(X, F0)} is non-empty and B = ess supC, then B eC.
12.17 Let (fi,^0), F° and X be given as in the previous problem.
Then T(X,F°) = {P : P is a probability measure on (ft,-F°) and X G
A4(P)} is convex.
12.18 Let (fi,^"0), F° and X be given as in the previous problem.
Let C be an F°-predictable increasing process with bounded AC Then
T2m(X,F°;C) = {P : P is a probability on (ft,^°), * € M\oc(P) and
(X)(P) = C} is convex.

Chapter XIII
Predictable Representation Property
Predictable representation property means that every local martingale
can be represented as a stochastic integral of a predictable process. It is
meaningful and important not only theoretically, but also in applications,
such as filtering, control.
In this chapter, we are given a complete probability space (fi,^*, P)
endowed with a filtration F = (Jrt)t>o satisfying the usual conditions and
^ = ^00.
§1. The Strong Property of Predictable Representation
13.1 Definition. Let M = (Mt)t>o be a local martingale with
Mo = 0. Recall that Lm(M) is the collection of all predictable processes,
integrable w.r.t. M in the sense of local martingales. Write
C(M) = {H.M:He Lm(M)}, Cl{M) = C(M) n Hl.
If C(M) = -Mioc.cb we saY that M has the strong property of predictable
representation. Here we use the adjective "strong" because we will
introduce another weaker kind of predictable representation property in the
next paragraph.
13.2 Lemma. Assume M,L G -Mioc,o- U there exists a sequence (Tn)
of stopping times with Tn ] 00 such that for each n,LTn E C(M) then
LeC(M).
Proof Let LT" = H^'M, where H^ € Lm(M). Set if = § H^
n=l
J]Tn_i,T„]> (To = 0). It is easy to see that H € Lm(M) and L = H.M.
D

§1. The Strong Property of Predictable Representation 363
13.3 Lemma. Assume M G A4ioc,o- Then Cl(M) is a stable closed
subspace ofH1.
Proof. Obviously, Cl(M) is a stable vector space. It suffices to prove
the completeness of Cl(M). Let (H^n\ M)n>i be a fundamental sequence
of Cl(M) satisfying
< oc (i/(°) = 0).
71=0
So for all t > 0
/' g {H^ - Hin)\d[M]s
Jo n=o
< 2 (/Vin+1)-^in)lMM]a)1/2([M]t)1/2.
n=0 Wo '
Write A = [ £ |#<n+1> - ff(n>| < ool and tf = IA g (i/(n+1> - #<")).
L n=0 J n=0
Then A and H are predictable, and p( y°° 7^[M]S = o) = 1. Hence
e{(£ nnm,)m) < *[ £o (jf i»i"+» - H^dM,)1'2}
OO
= £ ||i/(n+1).M-tfHM||wi <oo,
n=0
i.e., H.M € £1(M). At the same time, as n —► oo
||tf. M - HH Af||Wi = II £ (#(As+1) - #(fe)). Af II
11 k=n "nl
< 2 ||(#(fe+1) - #(fc)). M||wi - 0,
k=n
so that //KM - H.M. D
13.4 Theorem. .Assume M G -Mioc,o- Tfeen tfie following statements
are equivalent:
1) C(M) = Alioc.O; *e-> A^ has the strong property of predictable rep-
resentation,
2) C\M) = Ml
3) -Mo° C £(M) (M°° is the collection of all bounded martingales).
Proof. Since M\OCjo = Wj^q, by Lemma 13.2 2)=^1) follows
immediately. 1)=>3) is trivial. Finally, we are to show 3)=>2). Let L G H\.
Because M™ is dense in U\ (Theorem 10.5), so there exists (N^) c M™
such that ||7V(n) - L||wi -> 0. However, N^ G Cl(M). By Lemma 13.3
we obtain L G Cl(M). □

364
Chapter XIII Predictable Representation Property
13.5 Theorem. Assume M G M\oc$. Then the following statements
are equivalent:
1) C(M) = M\OClo, i.e., M has the strong property of predictable rep-
resentation,
2) for all L G Moc,o, LM € Moc,o => L = 0,
3) for all N G M%>\ NM G -Mioc,o =► N = 0.
Proo/. 1)=>2). Let L,LM G -MiOc,0- By the strong property of
predictable representation we have L = H.M,H G Lm(M). So [L. M] = H\M\.
Since LM G A^ioc,o^ [L,M] G A^ioc,o^ ie> [L,M] G Wioc,o- Thus we know
the Stieltjes integral
(HIm<n]).[L,M] = (H2Im<n]).[M] G Moc.o.
However, (H2I\\H\<n})\M) G V4". Therefore, it must be zero. Letting
n -♦ oo, we find #2.[M] = 0, i.e., [L] = 0. Thus L = 0.
2)=>3) is trivial.
3)=>1). By Theorem 13.4 it suffices to show Cl(M) = H\. Let ip be
a bounded linear functional on Hq such that (f\Hi — 0. We are going to
show ip = 0. There exists TV G S.MC?o such that
<p(L) = E[[L,N}00], LeTil
(Theorem 10.21). Since for all L G Cl(M) £?[[£,,#]«,] = 0, for every
stopping time T
E[[L,N}T] = E[[LT,N}oo} = 0.
Thus [L,N] G A40- Because BMOioc = M^. and -Mioc.o = WJ^q. tnere
exists a sequence (Tn) of stopping times with Tn | oo such that for each
n, MT" G Wj and JVr" € .M§°. Noting MT" = /[0)Tnl. M € Cl(M),
we have [MT",iV] G M), [NTn,M] € jVf0 and MNT» € -Mioc.o- By the
assumption we obtain NTn = 0. So iV = 0, and therefore <p = 0. □
13.6 Corollary. .Assume M G A^ioc,o- TAen tf/ie following statements
are equivalent:
1) C(M) = Moc,o,
2) /or all L G A<ioc WtA L0 = 1, LM G M\oc => L = 1,
3) /or a// strictly positive L G A<ioc with Lq = 1, LM G A^ioc => L = 1.
Proof 1)=>2). Let L,LM G Moc and Lo = 1. Then N = L - 1 €
-Mioc.o and 7VM = LM — Me M\OCyo. By Theorem 13.5 we have N = 0,
i.e., L = 1.
2)=>3) is trivial.
3)=>1). Let TV G Alg0 and ^-^ € -Mioc,o- Let A; be a constant such that
N ' MTV
|7V| < k. SetL = 1+—. ThenL G MXoc,LM = M+—— e Mioc^o = 1

§1. The Strong Property of Predictable Representation 365
and L > 0. Thus L = 1, i.e., N = 0. By Theorem 13.5 this implies M has
the strong property of predictable representation. D
13.7 Theorem. Assume M G M\QC 0. Then the following statements
are equivalent:
1) £(M) = McXocfi,
2) Cl(M) = 7i,QC (W0,c is the subspace of all continuous HQ-martin-
gales),
3) M°°'c C C{M) (M°°'c is the space of all bounded continuous
martingales) ,
4) for all L G M^^ LM G Moc.o =* £ = 0,
5) /or a// TV G .M00''0, 7VM G Moc.o =► W = 0.
Proof 1)<=>2)4=>3) can be shown the same as Theorem 13.4.
1)=>4). Let L G Mfoc0 and LM G Moc.o- Then L = H. M,H e
Lm{M) and (L,M> = H.\m). However, (L,M) G Moc,o- It is both
purely discontinuous and continuous. It must be (L, M) = 0. Then
H2.(M) = H.(L, M) = 0, and hence L = H.M = 0.
4)=>5) is trivial.
5)=>2). Wq'C is a closed subspace of Wj (cf. Problem 10.3). Let (p be
a bounded linear functional on HqC such that <p\cl(M) = 0- We are going
to show (f = 0. (f can be extended to be a bounded linear functional on
Wo, and there is N G BMO0 such that
<p(L) = E[[L,N]00]i LeHl
Then
<p(L) = E[(L,Nc)00], LeH^. (7.1)
Hence for L G Cl{M) we have L7VC G -A/fioc.o- Take a sequence (Tn)
of stopping times with Tn ] oo such that for each n, MTn G H0,c and
(NC)T» G XS°'C. But MT» G £H^), so MT«7VC G Moc.o and 0 =
(MTn,7Vc) = (M,(7VC)T«). Thus M{Nc)Tn G Xioc,o- By the assumption
(ATc)Tn = o. Therefore iVc = 0. By (7.1) <p\Hi.e = 0. D
13.8 Lemma. Assume M G M\OCyo has the strong property of
predictable representation. Then for any stopping time T, MT has the strong
property of predictable representation w.r.t. (^at^o-
Proof. Write FT = (^iAr)t>o- It is not hard to justify that if L is a
uniformly integrable Fr-martingale, L is also a uniformly integrable F-
martingale and L = LT. Hence if L is an Fr-local martingale, then L is
also an F-local martingale and L = LT. And if L is an F-local martingale,
then LT is an FT-local martingale (see Theorem 3.53). Now let L and
LMT be FT-martingales with Lq = 0. It suffices to show L = 0. Observe

366
Chapter XIII Predictable Representation Property
that L(M - MT) = Lt(M - MT) is an F-local martingale (Theorem
7.38). As pointed out above, L and LMT are all F-local martingales. So
is LM. Since M has the strong property of predictable representation, we
have L = 0. □
13.9 Theorem. Assume M G A^ioc,o- Put
_ j , P' is a probability measure on T, 1
\ P' = P\r0 and M € Moc,o(i") J '
Then the following statements are equivalent
1) M has the strong property of predictable representation,
loc
2) p' <=r,p'<z:P=>p' = p,
3) i" 6/,,Jy -cP^i* = P,
4) P' £T,P' ~P^>P' = P,
dP'
5) P' € T,P' ~ P,— € L°° => P' = P.
(XMr
loc
Proof. 1) => 2). Let P' G 7\ P'< P, Z = (Zt) be the density process
of P' w.r.t. P, and R = inf{t : Zt = 0}. Because Z0 = 1, we have R > 0.
Since M G M\OCyo(Pf), there exists a sequence (Tn) of finite stopping
times such that Tn ] R P-a.s. and for each n, (MZ)Tn G -Mioc,o- Thus
ZTn and ZTnMTn are (^at„ )-local martingales. By Lemma 13.8 MTn has
the strong property of predictable representation w.r.t. {Ft/\Tn)' Then by
Corollary 13.6 we have ZTn = 1. This implies P' = P|^Tn,n > 1.
On [R < oo] we have Zr = 0. But Zrn = l,n > 1. This means
Tn < R, n > 1, and V^Tn = ^_. Thus P' = P|^H_. Especially, we have
P(R < oc) = P'f/J < oo) = 0,
i.e., P(Tn t oo) = 1. Hence Z = 1. This implies P' = P.
2)=>3)=> 4) => 5) is trivial.
5)=>1). Let AT g A4o° and NM G Moc,o- l4 suffices to prove N = 0.
/ TV \
We may suppose \N\ < 1. Set dP' = (l + -^)dP- For all A G ^b
/ N^dP = I N0dP = 0,
Ja Ja
1 dP' 3
so P' is a probability measure and P' = P\t0- Since - < —— < -,
we know P' ~ P. The density process is Zt = E —— LfJ = 1 + —,
NM
t > 0. Thus MZ = M + —— G MiOC)0, and M G M\oc,o{P'). By the
Zi
assumption we obtain P' = P, and therefore N^ = 0 and TV = 0. D

§1. The Strong Property of Predictable Representation 367
13.10 Definition. Let M G M\oc- Put
T(M) = {Pf : P' is a probability measure on T and M G M\oc(P')}.
Denote by Te(M) the set of extreme points of T(M), i.e., P' G re(M) <=>
P* G r{M) and if P1 = aPx + (1 - a)P2,Pi,P2 G T(M),0 < a < 1,
then P1 = P\ = P2. However, in general we do not know if T(M) is a
convex set.
13.11 Theorem. Assume M G M\oc$. Then the following
statements are equivalent:
1) M has the strong property of predictable representation, and Tq is
the trivial a-field M {i.e., the a-field generated by all P-null sets),
2) P G re(M).
Proof. 1)=*2). Let P = aPi + (1 - a)P2,Pi,P2 € r(M),0 < a < 1.
Because Pi <C P, so Pi = P\j?0. By Theorem 13.9 Pi = P, and hence
P2 = P. Thus P € Te(M).
2)=^-l). Assume i) £ is a bounded .Fo-measurable r.v. and E[£] = 0,
ii) N G .Mjf and NM € MloC]0. Set L = £ + N. Then L G .M00, and we
may suppose \L\ < 1. Define
dP1 = (l + If)dP, dP2=(l-!f)dP.
Since 22[Loo] = 0, Pi and P2 are probability measures on T. It is easy
to see that Pi ~ P2 ~ P and the density processes of Pi and P2 w.r.t.
Pare
respectively. Hence M^1' = M(l + i$) + \nM G .Mioc,o,MZ(2> =
M(l + «0 - ^M G -Mloc,o> and therefore M G A<ioc,o(-Pi) ^d M e
Zi Zi
Mhcfl(P2), i.e., Pi,P2 G r(M). But P G Te(M) and P = i(Px +P2).
It must be P = Pi = P2. Hence L^ = 0 and L = 0. This implies both
£ = L0 = 0 and N = L - £ = 0. 1) are established. D
loc
13.12 Theorem. Assume Pf«P, M G -Mioc,o(-P)> [M,Z] G
(•^loc(-P))5 an^ under P, M has the strong property of predictable
representation. Then under P', M' = M - — .(M, Z) G A<iocfo(^>/) ^a5
Zi—
the strong property of predictable representation as well. {Here we use the
notations in Chapter XII §1 and §2.)

368
Chapter XIII Predictable Representation Property
Proof. Assume N',N'M' € Mioc,o(P')- We want to prove under
P', N' = 0. Let (Tn) be a sequence of stopping times such that Tn |
R P-a.s. and for each n,(N'Z)T»,(N'M'Z)T" € M\OC,0(P), [M,Z)T» 6
Aloc{P) and Tn < Rn = iniit : Zt < -}. Write Y = (N'Z)T»,A =
n
—.(M, ZTn). By the formula of integration by parts,
Y(M')Tn = YMTn -YA = YMTn - (Y-).A - (A-).Y - [Y, A}.
Since Y(M')T», (A.). Y, [Y, A] € Mloc(P), we know [Y, MT«] - (Y-).A 6
Mioc(P) and {Y,MT") = (Y-).A = ^.{M,ZT»}. Noting Y = YT», we
obtain
(M,y-^.zT«) = o.
But Y - ^. ZT» e Mloc,o(P), so M(Y - ^. ZT») e Mloc,o(P). Since
M has the strong property of predictable representation,
Y-^.ZTn=0 or y = (y_).(-^-.ZT"). (12.1)
Because Y0 = 0, (12.1) has only null solution: Y = 0, i.e., (N'Z)Tn =
0. Since Tn < i?n, we have ^"/[o./ki > 0. Then N7[0tTn[ = 0 and
N'hoM = °- Hence under J>/^/ = o." D
§2. The Weak Property of Predictable Representation
13.13 Definition. Let X be a semimartingale, //,XC and (a,/3, u) be
its jump measure, continuous martingale part and predictable
characteristics respectively. Write
K(ri = {W.(»-v):Weg(ri}.
If -Mfoc 0 = C(XC) and .Mfoc = /C(/z), or equivalently
Mloc,0 = C(Xc)+JC(ri
(the right side is the linear sum of vector spaces), we say that X has the
weak property of predictable representation.
13.14 Theorem. Assume X G M\oc$ and X has the strong property
of predictable representation. Then X has the weak property of predictable
representation as well.

§2. The Weak Property of Predictable Representation 369
Proof. For all M G A^ioc,o we have M = H.X, where H G Lm(X).
Thus
Mc = H.XC, Md = H.Xd = (Hx) * (jx - i/)
by Theorems 9.3, 11.24 and 11.23. □
13.15 Lemma. .Assume X £ S. Let U G O suc/i £/ia£ M jJt/l'P] =
W exists fi.e., £/ is a-integrable w.r.t. V under M^). Then there exists
a sequence (Tn) of stopping times such that
i) D = [AX ^ 0] = \J[Tn] and [Tn] n [Tm] = 0 wAi/e n^m,
n
ii) /or eac/i n
W(Tn,AXrJJ[Tn<00,
= JB[C/(Tn,AXTn)/[Tn<oo]|^Tn_V<r{AXrn/[Tn<oo]}] a.s.. (15.1)
Proof. Choose (An) C P so that Q = \JAn,Ann Am == 0 when n ^ ra,
n
and for each n, MM(An) < oo, MM(|t7|/r ) < oo. Set 2?(n) = I~ */z, and
Tn,o = 0, Tn,m = inf{t > Tn,m_! : AB\n) ? 0}, m > 1.
Then (T,ntm)nm>i satisfies i). It is only required to show that T = Tn^m
satisfies ii).
Observe that f G Tt- V a{AXr/[r<oo]} <$=>> there exists V eV such
that f/[r<oo] = V(T, AXt)I[t<oo]- Furthermore, if £ is non-negative (resp.
bounded), V can be chosen to be non-negative (resp. bounded) also.
Let V G V be bounded. Set V = IxnIyrn,m-l,Tn,m]V- Since M^WV)
= M^UV), we obtain
E[W(T, AXT)V(T, AXT)I[T<oo]] = E[U(T,AXT)V(T,AXT)I[T<oo]],
E[W(T, AXT)^I[T<oo]] = E[U(T, AXr)£/[r<oo]],
where f G ^"r- V a{AIT/[T<00]} is bounded. Then (15.1) follows. D
13.16 Theorem. Assume X G S. Then the following statements are
equivalent:
l)Mfoc = tC(n),
2) 0 = VWa{AX},
3) for all M G Mfoc, M^[AM\V] = 0 =► M = 0,
4) /or a// M G .M00^ (£/ie space o/ all bounded purely discontinuous
local martingales), M^[AM\V] = 0 => M = 0,
5) i) if T is a totally inaccessible time, [T] C [AX ^ 0];
ii) for every stopping time T,Tt = Tt- V <t{AXtI[t<oo]}'
Proo/. 1)=>2). First of all, we observe that O = V V tf{-Mioc,o}- In
fact, for any stopping time T set A = J[r,oo[- Then M = A — A G A^ioc,o

370 Chapter XIII Predictable Representation Property
and A = M + A eW a{M\oc^}. Noting M = M- + AM, it is only
required to show AM eW a{AX} for all M G A^ioc,o-
By the assumption Md = W * (/z — v), where W G G{ii). Then
AM, = AM/ = W(t, AXt)ID -Wu D = [AX ^ 0].
It is easy to see W G V, D G a{AX), (W(t, AXt)) G V V a{AX}. Hence
AMg7>V<t{AX}.
2)=>3). Let M G Xfoc0 and MM[AM|P] = 0. Because AM G O =
V V a{AX}, there exists V eV such that
AMt = V{t,AXt), t>0.
Then
0 = M^AMV] = E\Z [AMtV(t, AXt)ID]\
lt>o J
= ,b{E[(am,)2/d]},
lt>o J
AMID = 0. (16.1)
Noting that M = 0 <*=* AM = 0 (since M G Alfoc0), by (16.1) it
remains to show AMfec = 0. Let T be a totally inaccessible time. Then
the predictable projection of /[xj/lk is zero because it is the jump process
of an adapted integrable increasing process having only totally inaccessible
jumps. On the other hand, since O n Dc = V fl Dc, there exists Y eV
such that
I[t]Idc = YIdc. (16.2)
Taking predictable projections on the two sides of (16.2), we obtain
y(l-a) = 0,
where at = v{{t} x E),t > 0. However [a = 1] C D, so Dc C [a < 1] C
[y = 0], ImIDc = YIDc = 0, i.e., [T] C £>. Thus
(AMIDc)TI[T<oo] = 0, a.s.. (16.3)
Let T be a predictable time and Z eV such that AMJ/jc = Z/^k- By
(16.1) we have
0 = E[AMTI[T<oo]\TT-] = E[{AMIDc)TI[T<oo]\fT_]
= ZT(1 -ar)^[r<oo].
Similarly, [{IDc)T = 1,T < oo] C [aT < 1,T < oo] C [ZT = 0,T < oo],
and hence (ZIdc)tI[t<oo] = 0> i-e•> (16.3) holds for any predictable time
T. Therefore (16.3) holds for any stopping time T, i.e., AMfec = 0.
3)=>4) is trivial.

§2. The Weak Property of Predictable Representation 371
4)=>5). Let T be a totally inaccessible time. Set S = ?|(jD)TjrT<ool=o]-
Then (Id)sI[s<oo] = 0- It suffices to show P(S < oo) = 0. Set A = I[s,oo[-
Then N = A — A € M^ . Since S is totally inaccessible, A is continuous
and AN = AA = I[S] For any W G P+
M^ANW] = E[W(S, AXs)(Id)sI[s<oo)] = 0,
i.e., My[AN\V] = 0. By the assumption TV = 0, A = A. So A is
continuous. It must be P(S < oo) = 0. i) is established.
Let T be a stopping time. In order to show ii) we may suppose T > 0,
since TT fl [T = 0] = TT- fl [T = 0] and T may be replaced by T[T>0]. Let
C € bTT,
^ = ^[T,oo[, */ = £ " Efcl^T- V *{A*T/[T<oc]}].
Since ^[r/l^.] = 0, we know A G .M°°'d and AA = r//^. For any
W G P+ we have W(T, AXT)(/D)T/[r<oo] G ^r- V a{AXTI[T<oo]} and
MM[AAW] = E[W(T, AXT)<n{ID)TI[T<oo]]
= E[W(T,AXT)(ID)TI[T<oo]E[V\fT- V a{AXTI[T<oo]}}] = 0,
i.e., MjJAAl'P] = 0. By the assumption, A = 0, i.e., 77/[t<oo] = 0 a.s..
But on [T = oo] Tt coincides with Tt- • Hence
£ = E[£\TT- V <t{AXtI[t<oo]}} a.s..
This means Tt = Tt- V (t{AXtI[t<o6\}'
5)=4>1). Let M G A<foc and £/ = MM[AM|P]. By Lemma 13.15 there
exists a sequence (Tn) of stopping times with disjoint graphs such that
D = \J[Tnj and for each n
n
U(Tn,AXTn)I[Tn<oo) = E[AMTnI[Tn<oo]\FTn-V0{AXTJ[Tn<oo]}}
= E[AMTJ{Tn<oo] \7rn] = AMTJ[Tn<oo] a.s..
This means
( / U(t, x)n({t},dx)) = (U(t, AXt)(ID)t) = AMID. (16.4)
is the predictable projection of AM Id-
Set
U_
1-a
W = U+r— /(a<i].
If AM = W, we know W G G(fi) and M == W * (/z — v), hence ii) is
established. We are going to show AM = W. Since £)c C [a < 1], by

372 Chapter XIII Predictable Representation Property
(16.4) we have
Wt = U(t,AXt)(ID)t + T^-I[at<i](lD)t -Ut- j^-I[at<i)<h
= AMt(lD)t ~ T-^—I[at<l](lD')t -Ut + UtI[at<l]
= AMt(ID)t - i^(lDc)t - UtI[at=l], (16.5)
Because p(AMId) = U, we have
UI[a=1] = *>(AMIDI[a=1]) = p(AM/[a=1]) = p(AM)/[a=1, = 0. (16.6)
By (16.5) and (16.6) it remains to show AMIqc = — Idc, ie., for
any stopping time T
UT
AMT{IDc)TI[T<oo] = ~1_a {Idc)tI[t<oo], a.s.. (16.7)
If T is a totally inaccessible time, [T] C D and (16.7) holds clearly.
Finally, it suffices to show that (16.7) holds for any predictable time T. In
this case, there exists £ G Tt- such that
AMt{Idc)tI[t<oo\ = £(J£>0r/[r<oo].
Since
UtI[t<oo] = E[AMt(Id)tI[t<oo]\Ft-) = -E[AMt(Idc)tI[t<oc]\^t-1
we have
—: (Id°)tI[T<oq] = ~, E[AMT{Idc )tI[T< ooll^T-]
1 — ax 1 — o>t
= i E[$t{IDc)TI^T<(X^\TT_\
1 — ax
= £{Idc)tI[t«x>]
= AMt{Idc)tI[t<oo], a.s.. D
13.17 Theorem. Assume X G S. Then the following statements are
equivalent:
1) X has the weak property of predictable representation,
2) for all M G M\OCy0, (Mc, Xc) = 0 and M^[AM\V] = 0 =* M = 0,
3) for all N G <M§°, (7VC, Xc) = 0 and M^[AN\V] = 0 => TV = 0.
Proof It follows from Theorems 13.7 and 13.16 immediately. □

§2. The Weak Property of Predictable Representation
373
13.18 Theorem. Assume X G S and (a,/?,v) is its predictable
triplet. Put
r _ J p/ P' is a probability measure on T, P' = P|.f0, X G 1
1 ' S(P'), (a,/3, v) is the predictable triplet of X under P1 J '
Then the following statements are equivalent:
1) X has the weak property of predictable representation,
loc
2) p' € r, p' < p => p' = p,
3) p7 € r, p' < p => P' = p,
4) p' € r, p' ~ p => p' = p,
dP'
5) P' £T,P' ~P,— €L°°=>P' = P.
loc
Proof. 1)=>2). Let P' G 7\P' <P, and Z = (Zt) be the density
process of P' w.r.t P. Since P' = P|^0, Z0 = 1 and (Z - 1) G Xic-cO^)-
By Theorem 12.31 MM[AZ|P] = 0 and (Z,XC) = 0, i.e., MM[A(Z -
\)\V] = 0 and (Z - 1,XC) = 0. By Theorem 13.17 Z = 1. This implies
P=P.
2)=>3)=>4)=>5) is trivial.
5)=^1). According to Theorem 13.17, it suffices to show
TV G A4g°, (7VC, Xc) = 0, M^[AN\V) = 0 => N = 0.
/ iV \
We may suppose \N\ < 1. Set dP' = (l + -^-jdP. It is well-known that
1 dP' 3
P* is a probability measure on T, P1 ~ P, - < -j=j- < -, P' = P|.f0, and
the density process is Z = 1 + -TV. By Theorem 12.31 we have P* e T.
Then P' = P|^, TVoo = 0, and therefore TV = 0. D
From Theorems 13.18 and 11.54 we obtain immediately the next two
results about the predictable representation property of step processes.
As to another important class of processes—Levy processes, we will
investigate their predictable representation property in §4.
13.19 Theorem. Assume that X is a step process and F = (Ft)
is the complete natural filtration of X: F = FP(X). Then X has the
weak property of predictable representation. In particular, each F-local
martingale is purely discontinuous.
13.20 Theorem. Assume that X is a point process and F = (Tt) is
the complete natural filtration of X. Then M = X — X has the strong
property of predictable representation.

374 Chapter XIII Predictable Representation Property
13.21 Theorem. Assume X G S and (a,/3, v) is its predictable
triplet. Put
p __ \ , P1 is a probability on T', X G S(P') and (a, /3, u) 1
1 is the predictable triplet of X under P' J '
Then the following statements are equivalent:
1) X has the weak property of predictable representation and Tq is the
trivial a-field.
2) P is an extreme point of T.
Proof. It is completely similar to the proof of Theorem 13.11 and is
left to readers. □
13.22 Theorem. Assume that X G S has the weak property of predic-
table representation. If P <P, then under P , X has the weak property
of predictable representation as well.
Proof. Let (a, /3, v) be the predictable triplet of X under P, and Z1
be the density process of P' w.r.t. P. Then under P', (a',/3f,i/), the
predictable triplet of X, is as follows:
a' = a + ±.(Z',XC) + [*J[W<i](y - 1)] * i/, & = P, v' = Y.v,
and Af M[AZ'|P] = Z'_(Y - 1).
Let P" be another probability measure such that P" ~ P', P" = P,\^Q
and under P", (a', /3', i/) is still the predictable triplet of X. By Theorem
13.18 it is only required to show P" = P'.
Obviously, P" <C P. Let Z" be the density process of P" w.r.t. P.
Then under P" (or P') we have
and M„[AZ"|P] = Z'L{Y - 1). Set Tn = inf {* : Z[ < - or Zf < -}.
Then J"^ T oo) = P"(Tn | oo) = 1. Set
M(n) = F(z')T"-F(z")r"' n-L
Since P" = P'\jr0, we have Z'Q = Z'J, and hence M^n) = 0. Thus JlfW €
A1ioc,o(-P)- At the same time, we have
MM[AJIfW|^| = M^jrAZ' - J^AZ")/^!^]
= {jrM^AZ'lV} - -|,M^[AZ"|^]}/io,r„] = 0,

§2. The Weak Property of Predictable Representation 375
(M(nUc) = {jr-(Z',xc) - jjr.(Z",Xc)}Tn = o.
Since X has the weak property of predictable representation, by
Theorem 13.17 we have M^ = 0, i.e.,
Because Z'Q = Z£, we conclude (Z')Tn = (Z")Tn.
Set R' = inf{* : Z[ = 0} and R" = inf{t : Z't' = 0}. Since P'(R' =
oo) = 1 and P" ~ P', we have P"(R! = oo) = 1. Then by Theorem
12.6.2) P{R' > R") = 1. By the same argument P(R" > R') = 1. Thus
P{R' = R") = 1, and P(Tn ] R' = R") = 1. Then under P,Z' and Z"
are indistinguishable, and therefore P" = P'. □
The weak property of predictable representation is closely relative to
quasi-left-continuity and complete continuity of the filtration to some
extent.
13.23 Theorem. Assume X G S. Let /jl and v be the jump measure
and Levy system of X respectively. If Mfoc = £(/z), then for F = {!Ft)
to be quasi-left-continuous it is necessary and sufficient that the following
conditions be fulfilled:
i) J = K {recall J = [a > 0], K = [a = 1], a = (u({t} x E))),
ii) There exists a predictable process H such that \H\ > 0 and
v({t},dx) = 6Ht(dx)Ij(t). (23.1)
Proof Necessity. Let D = [AX ^ 0] = Ul^n], where (Tn) is a
n
sequence of stopping times with disjoint graphs. In this case, the
accessible part T* of Tn is predictable. As the predictable support of £>,
J = U[T£] C D. But K is the largest predictable set contained in D
(Theorem 11.14). Hence J = K.
Because for each n, AXt%I[t°<oo] € Ft* = ^t°-> so
H = E^XTaI[Ts]+(l-Ij)
n
is a predictable process and \H\ > 0. Obviously, AXIj = HIj and
MU{J[H ± x]) = M„{J[H # x\) = M^(J[AX # H\) = 0,
E{Z [ u({t},dx)}=0, (23.2)
where H and x are considered as predictable functions on £2. Since J =
K,t€J=> u({t} xE) = l. Then (23.1) is deduced from (23.2).

376 Chapter XIII Predictable Representation Property
Sufficiency. Noting J = K C D, we find D = (DJ) U (D \ J) =
K U (D\ J). Since D \ J is a totally inaccessible set, for any predictable
time T we have [T] H (Z> \ J) = 0, and therefore
AXr/[r<oo] = ^^t(^a:)t^[t<oo]. (23.3)
On the other hand, from (23.1) we have
M^K[AX ^ if]) = M^K[x ^ H]) = Mu{K[x ^ H]) = 0,
AXIK = HIK.
Combining with (23.3), we obtain
AXtI[t«x>] = Ht{Ik)tI\t<<x>] € Tt-.
By Theorem 13.16.5) we know TT = TT-- This means F = (Ft) is
quasi-left-continuous. □
13.24 Lemma. Assume F = (Tt) is completely continuous. Let S
and T be stopping times. Then there exists a predictable set L such that
[%<T]JCL, [T]cLc. (24.1)
Proof. Write R = S A T. Since [R < T] G Tr = Tr-, there exists
LeV such that
I[R<T] = (Il)rI[r<oo]. (24.2)
One may suppose L C [0,72]. Otherwise, L may be replaced by L[0, R],
and (24.2) remains true. From (24.2) we know immediately
[Sf^r]] = IfyiKT]] C L* lT[R=T]i = [R[R=T\] C Lc.
On the other hand, [T[H<T]] c]i?,oo[C Lc. Thus
PI = IT[r=t\] U {T[R<T]] C Lc.
(24.1) is established. □
13.25 Lemma. Assume that F = (Ft) is completely continuous and
there is a thin set D such that for any totally inaccessible time S [S] C D.
Then for any stopping time T there exists a predictable set H such that
i) PI C H,
ii) if S is a totally inaccessible time and [5] C H, then S >T.
Proof. Let D = \J[Sn], where (Sn) is a sequence of stopping times.
n
According to Lemma 13.24, for each n there exists Ln G V such that
[(5»)[sw<T]J C Ln, IT] C L%.
Put if = D Lcn. Then H € V and
n
[T] C H, l(Sn)[Sn<T]] C Hc.

§2. The Weak Property of Predictable Representation 377
If 5 is a totally inaccessible time and [S] C H, then
IS[S<T}] C U[(Sn)[S„<T]] C Hc,
n
because [5] C D = \J[Sn]. Hence [S[S<T)] = 0, i.e., S > T. □
n
13.26 Theorem. Assume X G S. Let /z and v be the jump measure
and Levy system of X respectively. If Mfoc = /C(/x), then for F = (Tt) to
be completely continuous it is necessary and sufficient that the following
conditions be fulfilled:
i)J = K,
ii) there exists a predictable process H such that \H\ > 0 and
v(dt,dx) = 6Ht{dx)A{dt), (26.1)
where A(dt) is a random measure defined on CI x B(R+).
Proof Sufficiency. On J = K we have A({£}) = v({t) x E) = 1 and
v({t},dx) = 6Ht{dx) by (26.1). Then by Theorem 13.23 F is quasi-left-
continuous. On the other hand,
M^([AX ? H]) = M^([x * H}) = M„([x ± H])
= E[J™A(dt)JEI[xm6Ht(dx)]=0,. (26.2)
AX = HID.
If T is a totally inaccessible time, then [T] C D (Theorem 13.16.5)) and
AXtI[t<oo] = HtI[t«x>] € Ft-.
Again by Theorem 13.16.5) we find Tt = Ft-- Now it is not difficult
to show that for any stopping time T Tt = Tt-, i.e., F is completely
continuous.
Necessity. We want to construct a predictable process H such that
(26.2) holds. Let D = (JPn], where (Tn) is a sequence of stopping times
n
with disjoint graphs. By Theorem 13.23, J = K = UKl and D \ J =
(JPJI. Set #' = 2AIt°V„'|' Then #' is predictable and AXIj =
n n
H'lj.
By Lemma 13.25 for each n there exists GneV such that [7£] C Gn
and if S is a totally inaccessible time satisfying [5] C Gn, then 5 > 7£.
Put Ln = Gn\( U Gm[0,7^]). Then Ln£V and satisfies the following
two conditions:
IK} C L„, (26.3)
[7^] n Ln = 0, when n # m. (26.4)

378 Chapter XIII Predictable Representation Property
(26.4) is clear, since [I*J C Gm[0, T^] and for n ^ m [Zjj D Ln = 0. In
order to establish (26.3) it is only required to show Gm[0,T^] [7£] = 0
when n # m. Put A = [lGm(li)I[n<oo] = 1]. Then
[Tn\Gm = [(I-JxJ,
>1 € TTi, (7^)a is a totally inaccessible time and [(7^)^] C Gm. Thus
Tm < (Ji)A. However, [7$] and [T^] are disjoint. So if {Tn)A < oo, it
must be 7^ < 7*, and hence [0, 7*J[(7;)a] = 0,
K]Gm[0, IjJ = [(2J)a][0, ni] = «»-
Now since AX7*/p£<00] G J7^ = TTi_, there exists a predictable
process i/(n) such that AXTiIipx^<OQ\ = ff^ /pr*<oo]- Set
N
N
B=\hm £//W/Ln<ool, H" = IB( lim £ #(n)/Ln).
Then if" is predictable and by (26.3), (26.4) we know
AXID\j = H"ID\Jt
Set
H = H' + H"{l-Ij), H = HI{^0]+I[5=0]
Then H is predictable, |JJ| > 0 and AX = #/£>. Hence
0 = MM([AX / H]) = MM([x # #]) = Mv([x ± H]) (26.5)
and for each n, (/i([0, tAn] x {x : |x| > -})) is an adapted locally integrable
n
increasing process, its dual predictable projection is v(([0,t A n][|i/| >
-]) x E). Write A(cft) = v(dt,E). Then for each n
n
A({t:0<t<n, |Jf*| > -}) <oo,
n
and therefore A(d£) is a-finite. (26.1) follows from (26.5). □
§3. The Relation between Two Kinds of Predictable
Representation Properties
13.27 Theorem. Let M G A^ioc,o and M be the jump measure of M.
Then the following statements are equivalent:
1) Mt = C{Md),
2) i) M^ = K(n),

§3. The Relation between Two Kinds of Representation Properties 379
ii) there exist two predictable processes a^ and a^ such that
(AM - a(1>)(AM - a(2>) = 0. (27.1)
Proof. 1)=>2). Take W™ = z2/[|x|<n] € V. Then
Wt{n) = AMt2/[|AMt|<n| - fx2I[M<n]v({t},dx),
\wln)\ Kn^M^ + nyJ! x^I{^n]u{{t},dxl
< n>/p3T7 + n^x2/^,^)*^,
where i/ is the compensator of /i. Since (z2/[|x|<n])*I/ € V+, y E(Wr(71))2 E
.AjJ^ and W(n) E £?(a0- By the assumption, there exists a predictable
process H^ such that
H(n).Md = W(n)*(/x - i/), //(n)AM = W(n) = AM2/^^!^] - iyW.
_ (27.2)
Clearly, W^ ]W eV. Prom (27.2) we have
[AM = 0] C [W = 0]. (27.3)
Define A = [W = oo] and
/
I - lim * ,, if the limit exists and is finite,
[ 0, otherwise,
{lim if (n), if the limit exists and is finite,
n-KX)
0, otherwise.
Then A,X and Y are all predictable. On A we have AM ^ 0 by (27.3),
and lim |if(n)| = oo,
n—>oo
AM=(*M>W„|_gW |W
if(n) #(") n-oo if(") v '
by (27.2). On Ac we have
(AM)2 - YAM - W = 0. (27.5)
In fact, if AM = 0, (27.5) follows from (27.3). If AM ^ 0,
Letting n —> oo yields (27.5). Let 5^ and 5^2) be two predictable
processes such that they are the two roots of z2 — Yz — W = 0. Then

380 Chapter XIII Predictable Representation Property
«(!) = S^1)/^ + XIa, ofi2) = 5^Ia<= satisfy the requirement ii). i) follows
from Theorem 13.14.
2)=*1). First of all, we may suppose |a<2)| > |a(1)| and |a(2>| > 0.
Otherwise, a^ and a*-2' may be replaced by a^ and a>2', defined as
follows:
5(1) =a(1)/[|ad)|<|Q(2)|] +a(2)/[|a(i)|>|Q(2)|],
5(2) = a(1)/[|a(l)|>a(2)|] + «(2^[|a(i)|<|a(2)|>Q(2)?to] + I[aW=a(2)=Q].
Let L € M(oc0. Then there exists W € G([i) such that L = W*(n~v).
Set
^ = V=o(1)l' Y = ^[AM#a(')] = ! - -X"-
Then
AM = a^X + a<2>y,
ALt = W(t,at(1))J[Q0)#o]^ + ^CaS2))r* - ^- (2?'6)
Write Wt(1) = W(«,oS1))/[a(„/0]-Wt,Wi(2) = W(*,aj2))-^t,* > 0. Then
AL = W(1>X + W&Y. (27.7)
Taking predictable projections in (27.6) and (27.7), we obtain
a(i) px + a<2) py = o,
(27.8)
w<i) px- + w<2) py = o.
Since PX + PY = 1, PX and PY cannot vanish at the same time. Thus by
(27.8) we find
a(DW(2)_a(2)w(i)=0 (279)
W(2)
Set if = -7TT- € V. From (27.6), (27.7) and (27.9) we have
a'
(2)
HAM = Ha^X + Ha^Y = W^X + W&Y = AL.
Hence H € Lm(Md) and L = H. Md. D
Remark. In the theorem, if M is supposed to be quasi-left-continuous,
we may take a^ = 0. Indeed, we have W = 0, so that 0 and Y are two
roots of the equation z2 — Yz — W = 0.
13.28 Lemma. Assume M € A^ioc,o- Then the following statements
are equivalent:
1) £(M) = C(MC) + C(Md),
2) C(MC) C C(M),

§3. The Relation between Two Kinds of Representation Properties 381
3) Mc € C(M),
4) C(Md) C £(M),
5) Md € C{M).
Proof. 2)=>3) is trivial.
3)=*2). Assume Mc = H.M,H € Lm(M). Let L € C(MC). Then
L = K.MC,K € Lm(Mc). Thus L = #.(#. M) = (ifff). M € £(M).
Similarly, we have 4) •*=» 5).
Since M € C{M), M = Mc + Md, we have 3) «=* 5).
Finally, 1) => 2) and 2) + 4) => 1) are apparent. □
13.29 Theorem. Assume M € M\ocq. Then M has the strong
property of predictable representation if and only if M^c = £(Mc),.Mf(,c =
C(Md) and C(M) = C(MC) + C(Md).
Proof. Necessity. Let L € .Mfoc0. Then L = H.M, H € Lm(M).
Indeed, L = H. Mc. Thus M^ 0 = £(MC). Similarly, we have M(oc =
C{Md). Furthermore,
C(M) = MloC)0 = Mfoc,0 + Mt = C(MC) + C(Md).
Sufficiency. Reversing the above reasoning, we find
•M,oc,o = A<ioc,o + Mfoc = C(MC) + C(Md) = £(M). □
13.30 Definition. Let v be a predictable random measure with
„({0} x E) = 0. If
v(u,dt,dx) = G{u>,t,dx)dBt{uj), (30.1)
where i) B is a predictable increasing process with Bo = 0, ii) for fixed
(u>,t), G(u>,t,-) is a measure on (E,B{E)), iii) for fixed K G B{E),
G(v,AT) is a predictable process, then (30.1) is called a predictable
decomposition of v. Moreover, if
IA. B = 0, A = {(a;, t) : G{u, t, E) = 0}, (30.2)
the predictable decomposition (30.1) is said to be canonical.
13.31 Lemma Suppose (30.1) is the canonical predictable
decomposition of a predictable random measure v. If v has another predictable
decomposition:
v(lj, dt, dx) = G'(cj> t, dx)dB'(u). (31.1)
Then P({u; : dBt(u>) < dB[(u))}) = 1. Moreover, if the decomposition
(31.1) is canonical also, then P{{uj : dBt(u) ~ dB't(u))}) = 1.

382 Chapter XIII Predictable Representation Property
Proof. It is only required to show the first assertion. To this end, let
H G V+ and H. B' = 0. Then
E[ HHtG{t,E)dBt] = M„(H) = E[HHtG'{t,E)dB't] =0.
From (30.2) we have H. B = 0. The assertion follows from Theorem 5.14.
D
13.32 Lemma. Let fi be the jump measure of an adapted cadlag
process X. Then its compensator v has the canonical predictable
decomposition (30.1). Parthermore, if W G V* is strictly positive and
C = W*veV+, then P{{lj : dBt{uj) ~ dCt(u>)}) = 1.
Proof. For each n > 1 define
Then A\ = vn([0, t] x E) G A^ since it is the compensator of the
point process /zn([0, £] x E). There exists a predictable integrable
increasing process A such that P({uj : dAt(u) ~ dA\ (lj)}) = 1.
(In fact, if (Sk) is a localizing sequence for A™, we may take A =
Z(2kE[AisJ])-1{AM)s".) Set
Jfc=i
B= g (2nE[A{Z)])-1A{n).
n=l
Then B G •4+ is predictable, and for each n > 1 P({u : dA\n (u) <
di?t(u;)}) = 1. &Vi can be decomposed into
vn{u,dt,dx) = G(n)(<j,*,cfa)dBt(ti;) (32.1)
such that (32.1) is a predictable decomposition of the stochastic
measure vn and measure G(n\uj,t,dx) does not charge outside \x : — <
l. n
\x\ < —!—}. Set G = § G<n> and £ = /^.B, where A = {(<j,t) :
n - 1J n=i
G{u),t,E) = 0}. Then */ has the canonical predictable decomposition
(30.1).
Now we show the second assertion. For any H G V* we have
H. C = 0 ^=> H. A(n) = 0 for all n > 1. (32.2)
Analogous to (32.1), vn has the following predictable decomposition:
vn(uj,dt,dx) = G (uj,t,dx)dCt((jj),
where G (uj,t,dx) does not charge outside {x : ~ < \x\ < jjzj}, either.
Set G = Sn?=i ^ • Then v has the following predictable decomposition:
v(lj, dt, dx) = G(cj, t, dx)dCt(uj). (32.3)

§3. The Relation between Two Kinds of Representation Properties 383
Put D = {(a/,t) : G(u,tyE) = 0}, Dn = {(<j,t) : G{n){uj,t,E) = 0},
n > 1. Then D = f]nDn and for all n > lJDn.A^ = 0, ID.A^ = 0.
By (32.2) we have Id>C = 0. This implies the decomposition (32.3) is
canonical. By Lemma 13.31 we get P({(jj,dBt(uj) ~ dCt(uj)}) = 1. D
13.33 Corollary. Let fi be the jump measure of an adapted cadlag pro-
cess X, and (30.1) be the canonical predictable decomposition of its
compensator v. If(ii([0,t]xE)) e Afoc, thenP({u : dBt(u) ~ v(uj,dt,E)}) =
1.
Proof In Lemma 13.32 take W = 1, then Ct = i/([0, t] x E). D
13.34 Corollary. Assume M 6 -Mj*oc0. Let ii be the jump measure
of M, and v be the compensator of /z. If is has the canonical predictable
decomposition (30.1), then P(dBt ~ d(Xd)t) = 1.
Proof. In Lemma 13.32 take W = x2 + /[x=0] > 0, then C = (Xd). D
13.35 Theorem. Le£ M E -Mioc.o ari^ (a>/?>i/) 6c zte predictable
triplet. Assume that v has the canonical predictable decomposition (30.1).
Then M has the strong property of predictable representation if and only
«/^loc,o = £(Mc),Koc = £(Md) and P{dhLdBt) = 1.
Proof. By Theorem 13.29 and Lemma 13.28, it suffices to show
P(d/3t±dBt) = l*=>Mce C{M).
Let Mc € C(M). Then Mc = H.M,H € Lm{M), and (Mc) =
H2. (M>, #2. [Mrf] = 0. Write A = [H2 = 1]. Hence A € V and
7^.0 = /^c. (Mc) = 0. On the other hand, since H2.[Md] = 0, we
have HIq = 0, and
0 = M^H2) = M„(H2) = E[j°° H2G(t, E)dBt].
Thus IA.B = 0. Therefore P(dpt±dBt) = 1.
Conversely, assume P(d/3t^-dBt) = 1. Then there exists .A € 7s such
that IAc0 = 0 and IA.B = 0 (Theorem 5.15). Since /^-(AT) = 0, we
have IAcMc = 0, Mc = i^. Mc. On the other hand,
E[j™{IA)td[Md}t] = M„(:c2/a) = M„(x2/A)
= E[J°°(IA)t(J x2G(t,dx))dBt] =0.
Thus 7,4. Md = 0, and Mc = IA.M € £(M). □
13.36 Corollary. Assume M € A^2oc0. T/ien M has the strong

384
Chapter XIII Predictable Representation Property
property of predictable representation if and only if
Mfoc,0 = C(M%Mfoc = C(Md) and P(d(Mc)t±d(Md)t) = 1.
Proof. It follows from Theorem 13.35 and Corollary 13.34. □
13.37 Theorem. Let X € S and n be its jump measure. If Mf^. =
/C(/i), then the following statements are equivalent:
1) There exists M € Mfoc such that Mf^ = C{M),
2) There exist two predictable processes a^1' and offi such that
(AX - a^)(AX - aW) = 0.
Proof. First, we assume X € Sp. In this case, there is a predictable
process A with finite variation such that N = X — Xq — Xc — A € M^
and AN = AX - AA.
2)=>1). By Theorem 13.16 we have O = V V a{AX}. But AX =
AN+AA,AAeV. KenceO = VV<r{AN}. Set7« = a^-AA,i = 1,2.
Then 7W and 7W are predictable, and (AN - -yW)(AN - 7W) = 0. So
that by Theorems 13.6 and 13.27 we know Mfoc = C(N).
1)=»2). Let N = H.M,H € Lm{M). Then HAM = AN = AX-AA.
By Theorem 13.27, there exist two predictable processes 7^ and 7^ such
that (AM - 7<1>)(AM - 7(2)) = 0. Set a<*> = #7W + AA, i = 1,2. Then
(AX - aW)(AI - a<2>) = 0.
Now we deal with the general case. X has the integral representation:
Xt = Xq + at + Xf + / xd(n — v) + I xd^i.
J[o,t)x[\x\<i) y[o,t]x[|x|>i]
Let (p be an one-to-one mapping from (1,00) onto (1,2) and from (—00, —1)
onto (—2, —1). Let <p_1 be the inverse of <p. Define
X[ = Xq + at -r Xf + / xd(fj, — u)+ <p(x)dn.
J[0,t]x[\x\>l] /[0,t]x[|*|>l]
It is easy to see X' £ S and
\AX\ < 1 <=» \AX'\ < 1 =► AX = AX',
(37.1)
\AX\ > 1 <=* \AX'\ > 1 => AX' = <p(AX), AX = ^(AX').
Since |AX'| < 2,X' e Sp. Let // be the jump measure of X'. Then
we have A<foc = K(n'). In fact, a{AX) = a{AX'} by (37.1), and O =
V V a{AX} = V V a{AX'}. According to the result shown above, 1) is
equivalent to the following:
2') There exist two predictable processes d^1) and a^ such that
(AX'-a<V)(AX'--a<V) = 0.
It is not difficult to see 2)«=*2') by (37.1). D

§3. The Relation between Two Kinds of Representation Properties 385
13.38 Theorem. Let X be an adapted cadlag process, \i be the jump
measure of X and v be the compensator of ii. Then the following
statements are equivalent:
1) There exist two predictable processes a^ and a^ such that
(AX - aW)(AX - a<2>) = 0. (38.1)
2) v has the following canonical predictable decomposition:
v(dt,dx) = [c[l)6 (i)(cte) + C[2)6 i2){dx)}dBu (38.2)
i at at *
where CW,CW,aW and a^2) are all predictable prcoesseSy and
[aW ? 0] C [a = 1], [a(1) = 0] C [C™ = 0]. (38.3)
Proof. 2)^1). From (38.2) we have
M^([AX ? a^][AX # a<2)]) = AfM([x ^ a^][x ? a™})
= M„([xjLaW][x^aW)) = 0.
This means (38.1) holds on D = [AX ^ 0]. By (38.3)
[a^ ^ 0] C [a = 1] C D, Dc C [a™ = o].
Hence, on Dc we have AX = 0 and a*1) = 0, i.e., (38.1) holds on Dc.
1)=»2). As in the proof of Theorem 13.27, we may assume \a^\ <
\<*W\ and |a<2>| > 0. From (38.1) we have [a*1* ^0]cD. But K=[a = \\
is the largest predictable set contained in D. Hence [a^ ^ 0] C K.
Suppose v(dt,dx) = Gt(dx)dBt is the canonical predictable
decomposition of v. Then
0 = MV{[AX ± aW][AX ? a<2>]) = M^([x ? a^][x ? a™])
Set Q(1) = Gtaa^}),^ = Gt({a<2)}). Then
Gt((te) = C[l)8 w(dx) + Ct(2)£ m(dx).
Since Gt({0}) = 0, [a^ = 0] C [C^ = 0]. Note that
/ CJpdB. = [ [ I[x=aii)]v(ds,dx), i = 1,2,
«/0 «/0 «/.&
is predictable. Hence C^ may be taken to be predictable, too. □
Theorem 13.38 provides us with a predictable form of the condition
(38.1).
13.39 Theorem. Let X G S with predictable triplet (a,/3, v). If
X has the weak property of predictable representation, then the following
statements are equivalent:

386
Chapter XIII Predictable Representation Property
1) There exists M G M\oc,o such that M\OCio = C(M),
2) i) v has the canonical predictable decomposition (38.2),
ii) P{d(3t±dBt) = 1.
Proof. First, we assume that AX is bounded. In this case, X G Sp.
There exists a predictable process A with finite variation such that N =
X-X0-Xc-Ae M*£, and
(N)t = I x2dv - £ f / xv{{s},dx)]2. (39.1)
J[0,t]xE s<tlJE J
2)=>1). Set M = Xc + N. Then M G M?oc0,Mc = Xc,Md = N.
Since M$oc0 = C{XC) = £(MC), from the proof of Theorem 13.37 we
know Mfoc = C(N) = C(Md). Observe that
(N)t = f x2duc+Z { I x2v{{s},dx) - \ f xu({s}ydx)}2}
J[Oyt]xE s<t lJE lJE >
and /3 = (Xc) = (Mc) is continuous. Hence
d/3t±dBt <£=> d/3t± I x2v(dt,dx) «<=► d(3t± f x2vc(dt,dx)
Je Je
^ dfit±d(N)t.
Thus P(d(Mc)t±d(Md)t) = 1. By Collorary 13.36 we have M\oc0 =
£(M).
1)=>2). i) follows from Theorems 13.37 and 13.38. Let // be the jump
measure of M and v' be the compensator of //. Let Xc = H.M,H €
Lm{M). Then
/? = (Xc) = H2.{MC), dfit « d{Mc)u a.s..
On the other hand, let N = H'.Md = (H'x) * (// - i/), #' G Lm(M), and
v'(dt,dx) = G't(dx)dB't be the canonical predictable decomposition of i/.
Then
(AT), = / {H'x)2du' = f {{H'a)2 I x2G's(dx)}dB's,
J[0,t]xE JO L JE }
d(N)t < dB[, a.s..
By Theorem 13.35 we have P(d(Mc)t±dB't) = 1. Hence P(d(3t±d(N)t) =
1, and therefore P(d0t±dBt) = 1.
Now we deal with the general case. As in the proof of Theorem 13.37,
we introduce X' G Sp. Clearly, X' has the weak property of predictable
representation, too. As we have shown in the proof of Theorem 13.37, the
conditions that (38.1) holds for X or X' are equivalent. Thus condition
2) i) is equivalent to that the compensator of the jump measure of X' has
a canonical predictable decomposition of from (38.2). However, [AX ^

§3. The Relation between Two Kinds of Representation Properties 387
0] = [AXf ^ 0]. Hence the process B is still available for X'. This means
condition 2) ii) remains unchanged for X' (noting (X')c = Xc). In a
word, X may be replaced by X', condition 2) being unchanged. But AX'
is bounded by 2, thus the proof is completed. □
Remark. We say that the filtration F = (Ft) has the strong (resp.
weak) property of predictable representation if there exists M G M\OCio
(resp X e S) such that M (resp. X) has the strong (resp. weak) property
of predictable representation. Then Theorem 13.39 characterizes the
relation between the strong and weak property of predictable representation
of a filtration.
To conclude this paragraph, we discuss the relation between the
predictable representation property and complete continuity of a filtration.
13.40 Lemma. Assume that F = (Ft) is quasi-left-continuous and
there exists M G Affoc such that Affoc = C(M). Then F = (Ft) is
completely continuous.
Proof It is only required to show for each totally inaccessible time
T, TT = TT— Let £ G bFT- Set A = £/pr,oc[ and N = A - A. Then
N = H. M,H G Ijn(Af), and AN = HAM1. On the other hand, AN =
AA = £I[T\ since A is continuous and T > 0. Thus
£ = HTAMT, a.s. on [T < oo]. (40.1)
Taking £ = 1, we have
1 = H'TAMT, a.s. on [T < oo], (40.2)
where H' is another predictable process. By (40.2) we know AMt ¥" 0
and H'T ^ 0 a.s. on [T < oo]. Hence
f = HT/H'T a.s.. on [T < oo].
HT
But -ttj-I[t<oo] € Ft— Therefore £ G Ft— This implies Tt = Ft--
D l
13.41 Theorem. Assume that F = (Ft) is quasi-left-continuous. Let
X € S and // be the jump measure of X. If Mfoc = K>(fi), then the
following statements are equivalent:
1) There exists M G Mfoc such that Mfoc = C(M),
2) F = (Tt) is completely continuous.
Proof. 1)=>2) follows from Lemma 13.40 (even we do not need the
assumption M(oc = )C(n)).

388
Chapter XIII Predictable Representation Property
2)=>1). By Theorem 13.26 the compensator v of fi can be represented
as
v(dt,dx) = 6fft(dx)A(dt),
where H is a predictable process. Hence
M„([AX * H}) = A#M([z ± H]) = Mv([x * H}) = 0,
and therefore AX = HID, D = [AX ^ 0]. Thus
AX(AX - H) = 0,
and 1) follows from Theorem 13.37. □
The next theorem is an immediate application of Theorem 13.41.
13.42 Theorem. Let X be a step process and F = (Ft) be the
complete natural filtration of X. Assume X is quasi-left-continuous and
X G -Aioc- Then the following statements are equivalent:
1) M = X — X has the strong property of predictable representation,
2) F = (Tt) is completely continuous,
3) the Levy system v of X has the form: v(dt,dx) = 8}jt{dx)\(dt),
where H is a predictable process and A(dt) = u(dt,E).
Proof Since X is quasi-left-continuous, so is F = (Tt) (Theorem 5.64).
By Theorem 13.19 we have -Mioc,o = Mfoc = £(//), where fi is the jump
measure of X. The results follow from Theorems 13.27 and 13.41. D
§4. The Predictable Representation Property
of Levy Processes
13.43 Lemma. For any process X = {Xt)t>o,
<u J c\ v , v^ /v v \i\ n ^ Ij^OjUi,-.. u £ R 1
H = |exp(z[u0Xt0 + E MX* -*«,_,)]): 0 = t0 <h < ■ ■ ■ < tn )
is a total family in L2(Jr^)(X)), i.e., the linear subspace spanned byHis
dense in L2(Jr^>(X)), where L2(Jr^)(X)) is the Hilbert space of all square-
integrable complex T^X)-measurable r.v..
Proof. First we discuss the case of a finite number of r.v., i.e., assume
X = (Xtl,Xt2,-" ,Xtn). In this case, for any £ e L2(Jr^)(X)) there
exists a Borel function of n variables / such that £ = f(Xtl, • • •, Xtn) a.s..
Denote by F(x\, • • •, xn) the distribution function of (Xtl, • • •, Xtn). Put
dG(xi,---,xn) = /(xi,---,xn)dF(xi,---,xn).

§4. The Predictable Representation Property of Levy Processes 389
If f-LW, then for any 1x1, • • •, un G R
0 = E\£exp(-i £ UjXtj)\ = / exp(-i £ UjXj)dG(xu • • • ,xn).
Using the inverse formula of Fourier-Stieltjes transformation, we obtain
dG = 0. Hence, £ = 0 a.s.. Therefore, H is total in L2(F^(X)).
Now we discuss the general case: X = (Xt)t>o> Let £ G L2(Jr£)(X))
and f-LW. For any given e > 0, there exists a finite set {£i, • • • ,£n} and
& G L2(a(Xtl, • • •, Xtn)) such that
E[\Z - t£\2} < e.
Making use of the assertion established above, we know fJ_£e. Then
E[e] = E[tj£=Z)] < {E[?]E[\£ - 6|2]}1/2 < {eE[e]}l'\
E[i2] < e.
Letting e —► 0 yields £ = 0 a.s.. □
13.44 Theorem. Let X be a Levy process. Then for each t>0
Ttp(X) = Tf+(X) = *£(*).
Hence, FP(X) = (^(X)) is the usual augmentation of the natural
filtration of X.
Proof It is easy to deduce ^(X) = ^^(X) from stochastic
continuity of X.
For all u G U, 0 < r < s, it is not hard to calculate directly:
Mt(u,r,s) = E[e^x'-x^tp(X)} = Wvt,.vt(u)c^x«^-X^)- (44.1)
From (44.1) we know that (Mt(u, r, s))*>o is an Fp(X)-martingale, cadlag
and bounded. Let
j] = exp{iu0X0 + iui(Xtl - Xto) + • • • + iun(Xtn - Xtn_x)},
(44.2)
n> l,uo,ui,- - ,un e R,0 = to <h < - - < tn.
Using the same calculation, we have
E[f,\^(X)] = e^*°M,(tii,«o,ti) • • • Mt(un,tn-Utn). (44.3)
Denote by Yt the right side of (44.3), then (Yt) is right-continuous.
Therefore
E[V\Ttp+(X)]=Yt+ = Yt = E[V\Ttp(X)} a.s.. (44.4)
In view of Lemma 2.69, from (44.4) we obtain ^(X) = ff(X). D
13.45 Theorem. Let X be a Levy process, and T be an F (X)-
stopping time. Then
tf(X) = *{T}\/Fr(XT). (45.1)

390
Chapter XIII Predictable Representation Property
Proof. Put G = a{T) V F^X7). Q C f£(X) is obvious. Let 7? be
defined in (44.2). By right-continuity of FP(X) and (Mt{u, r, s)) we have
E[n\tf(X)] = eiuoXoMT{ui, to, h) ■ ■ ■ MT(un, tn-i, tn), a.s..
It is easy to see from (44.4) that Mt(u, r, s) is ^-measurable. Hence
E[n\T%{X)] is ^-measurable. By Lemma 13.43, for any n € L2(^(A")),
E{i)\?P{X)} is ^-measurable. This means T%{X) C Q. Hence, Q%(X) =
g. □
13.46 Theorem. Let X be a cadlag process, and T be an F°(X)-
stopping time. Then
J*_(X) = *{T}VT000(XT-). (46.1)
Proof. Put G = a{T}yJ^{XT-). Since (Xt-) is F°(X)-predictable,
we have XT- £ ^_(X). Then for each t > 0
xt~ = xti[t<Ti + xT-i[T<t) € Tt_{x).
Hence Q C J=^_{X). On the other hand, J${X) C G is obvious, and for
any 0 < s < t and Borel function /
f(Xs)I[t<T] = f(X3 ~)I[t<T\'
Hence Tr_(X) C G- Consequently, Tj>_{X) = £. °
13.47 Corollary. Let X be a Levy process, and T be an FP(X)-
stopping time. Then
?!-{X) = FP_(X) V o-{&XTI[T<oo]}. (47.1)
Proof. Since XT = XT~ + (AXr/pr<oo])Jp\oo|, (47.1) follows directly
from (45.1) and (46.1). □
13.48 Theorem. Let X be a Levy process. Then FP(X) is quasi-
left-continuous.
Proof. By Theorem 11.36 we have already known that a Levy process
is quasi-left-continuous. If T is a predictable time, then &XtI[t<<x>) = 0
a.s.. By (47.1) we have T£(X) = f£_(X). □
13.49 Theorem. Let X be a Levy process and X £ So. Let F =
FP(X). Then X has the weak property of predictable representation.
Proof. Since the predictable triplet of X and the law of Xo determine
completely the probability measure on ^"oo, according to Theorem 13.18,
X has the weak property of predictable representation. □
Remark. For any Levy process X we conclude that FP(X) has the
weak property of predictable representation.

§4. The Predictable Representation Property of Levy Processes 391
13.50 Theorem. Let X be a Levy process with X$ = 0 and F =
FP(X). Let T be a stopping time. Then
1) T is totally inaccessible <==> [T] C [AX ^ 0],
2) T is predictable <=> [T] C [AX = 0].
Proof. 1) If T is totally inaccessible, then by Theorems 13.49 and
13.16.5) [71 C [AX £ 0]. If [71 C [AX £ 0], then T is totally
inaccessible, since X is quasi-left-continuous.
2) follows from 1), since F is quasi-left-continuous, predictable times
are just accessible times. D
13.51 Theorem. Let X be a Levy process with Xo = 0, and F =
FP(X). Then the following statements are equivalent:
1) F = FP(X) is completely continuous,
2) there exists a Borel function g ^ 0 such that
v(dt,dx) = 6gt(dx)A(dt), (51.1)
where v is the Levy system of X and A(dt) is a a-finite measure on ii+,
3) there exists a Borel function g ^ 0 such that AX = gh&x^o] •
Proof. 1)=>2). Without loss of generality, we may assume X is a semi-
martingale. Then X has the weak property of predictable representation.
Since v is non-random, 2) follows from Theorem 13.26.
2)=»3). From (51.1) we have
M»([AX ± g\) = MM([x ± g]) = Mv([x ± g\) = 0.
Thus AX = gI[Ax?o].
3)=»1). We may also assume X G S. Since AX(AX - g) = 0. By
Theorems 13.37 and 13.41 we know that F is completely continuous. □
13.52 Theorem. Let X be a Levy process with Xo = 0 and F =
FP(X). Let (a,^, u) be the predictable triplet of X. Then F has the
strong property of predictable representation if and only if the following
conditions are fulfilled: i) There exists a Borel function g ^ 0 such that
v{dt,dx) = 6gt{dx)A{dt), ii) d/3t±A{dt).
Proof. Without loss of generality, we may assume X G S. If F has the
strong property of predictable representation, then by Lemma 13.40 and
Theorem 13.48 F is completely continuous and i) follows from Theorem
13.51. ii) follows from Theorem 13.39. Conversely, if i) and ii) hold, then
AX = gI[AX^o]^ and again by Theorem 13.39 we know that F has the
strong property of predictable representation. □
13.53 Theorem. Let X be a homogeneous Levy process with Xo = 0
and F = FP(X). Let v be the Levy system of X. Then

392
Chapter XIII Predictable Representation Property
1) F is completely continuous if and only if
v(dt,dx) = \6a(dx)dt, A > 0, a E R\ {0}, (53.1)
or equivalently, AX = a/^x/o]?a € R\ {0}.
2) F has the strong property of predictable representation if and only
if X = bY + at, where Y is a standard Wiener process or a homogeneous
Poisson process and a,b G R.
Proof In this case, X G S and v(dt,dx) = XdtF(dx), where A > 0
and F is a a-finite measure. Then 1) follows from Theorem 13.51. At
the same time, 2) follows from Theorem 13.52, since the fact that d/3t =
a2dt,A{dt) = Xdt and d/3t±A(dt) implies a2 = 0 or A = 0. D
13.54 Corollary. Let X be a homogeneous Levy process with Xo = 0
and F = FP(X). Assume that X is a martingale. Then X has the
strong property of predictable representation if and only if X is a standard
Wiener process or a compensated Poisson process, up to a constant factor.
Problems and Complements
13.1 Let Cq(R+) be the collection of all continuous functions on R+
with compact support. If M G M\oc 0 and
{exp{(/. M)oo - \U2- (M)U} : / € C0(R+)}
is a total family in L2(Too), then M has the strong property of predictable
representation and To is the trivial <r-field.
13.2 Let M,N G M\oc,o such that (M, N) exists. Write X = M -
(M, N). Then M has the strong property of predictable representation if
and only if for any L G M\oc with Lq = 1, LX G M\oc =>► L = £(N).
13.3 Assume that X G M\QC 0 has the strong property of predictable
. loc
representation. Let P -C P, and X = M + A be the canonical
decomposition of X under P'. Then there exists a unique L G ^foco(^/) such
that under P/, A = (L, M)(P/).
13.4 Let A* = Mt + /0 Hsds, t > 0, where M is a Brownian motion, i/
is a predictable process such that for all t > 0, /q H2ds < oo. Let F = (Ft)
be the usual natural filtration of X. Suppose Po is a probability measure
on T^ such that under Po,X is a Brownian motion w.r.t. F. Then for
allt >0,P|?f «Po|yf.
And hence, if X = M + A is the canonical decomposition of X w.r.t.
P, then M (it is also a Brownian motion w.r.t. F) has the strong property
of predictable representation w.r.t. F.

Problems and Complements
393
13.5 Let M G Mioc0 with (M)^ = oo a.s., and the reference filtration
F = (Tt) is the usual natural filtration of M. Set rt = inf {5 : (M)3 > t}
and Bt = MTt, t > 0. Then M has the strong property of predictable
representation if and only if B has the strong property of predictable
representation w.r.t. (TTt)'
13.6 Let M G A4foc 0 with (M)^ = 00 a.s., and the reference filtration
F = (Ft) is the usual natural filtration of M. Set n = inf {s : (M)s > t}
and Bt = Mrt, t > 0. Then the following conditions are equivalent:
1) vt > o, n e ^oo(B),
2) V* > 0, (M)t e foo(B),
3) Vt > 0, (M)t is an (Jrt(B))-stopping time,
4) ^00 = Foo(B).
If these conditions are in force (then M is said to be pure), M has the
strong property of predictable representation.
13.7 Let W be a standard Brownian motion, F = (Ft) be its usual
natural filtration, let H G V* such that for almost all lj the Lebesgue
measure of {t : Ht(uj) = 0} is zero. Set M = H.W. Then M has the strong
property of predictable representation w.r.t. (^(M)). In particular, so
does M = Wn. W for each n G N.
13.8 Assume that M G A^ioc,o is quasi-left-continiuous. Denote
C'(M) = {H. M : H is an optional process such that H. M exists}.
Then the following statements are equivalent:
1) -Mioc.0 = £'(M),
2) for any L G Moc,o, [L, M] = 0 => L = 0,
3) Mg° C £;(Af),
4) for any L G A*§°, [L, M] = 0 =► L = 0.
13.9 Assume that M G -Mioc,o is quasi-left-continuous. If M has the
weak property of predictable representation, then A^ioc,o = C'(M), where
C'(M) is defined in the previous problem.
13.10 Let X be a step process and F = (Tt) be its complete natural
filtration. Then M\oc = Wioc-
13.11 Let VT be a Brownian motion. We have \W\ = M + A, M =
sgn{W)W, A = L°(W). Then 1) the usual natural filtration of \W\,
denoted by G, coincides with that of M; 2) M is a Brownian motion, w.r.t.
both F and G, and has the strong property of predictable representation
w.r.t. both F and G as well.
00
13.12 Let X = Xo + £ £n-fpr„,oo[ be a step process, where Tn ] 00,
71=1
for each n > 0, Tn < 00 => Tn < Tn+i(r0 = 0) and for each n > 1,
[Tn < 00] = [£n ¥" 0]. Let F = (^i) be the complete natural filtration of

394
Chapter XIII Predictable Representation Property
X. Then F = (Tt) has the strong property of predictable representation
if and only if there exist Borel functions fn (Eo,ti,xi, • • - ,tn,xn,tn+i),
i = 1,2, n = 0,1, • • •, such that for n > 0
i) £n+i = fn {Xo,Ti,£i,- • • ,Tn,£n,Tn+i) a.s. on [arn+1 < l,Tn+i <
oo],
ii) on [aTn+1 = l,Tn+i < oo]
[£n+l - fn (-Xo,Ti,fi,« • • ,Tn,£n,Tn+i)]
Kn+i " /i2)(Xo,Ti,6, •'' ,Tn,fn,Tn+i)] = 0 a.s.,
where a = (v({t} x £")), v is the Levy system of X.
13.13 Let X = I[t,oo[i T > 0, be a single point process and F = (Ji)
be its complete natural filtration. Let G be the distribution function of T
and c = inf {t: P(T > t) = 0}. Then
1) M G Mo if and only if there exists a Borel function h on ]0, oo]
such that \h\. G^ < oo, h. G^ = 0 and
M, = I^HT) - Vxl ofl^j) /„,„ MB.. * * «■
2) If M G A4ioc,o, then (M*)o<*<c is a martingale.
3) If c < oo and P(T = c) > 0, then Moc,o = Mo-
13.14 Let Y = lo + M + A, where M is a martingale with Mo = 0,
A G Vo satisfies the condition that for all t > 0, E[f0 \dA3\] < oo. Let
X be an adapted step process, and G = (Gt) be the complete natural
filtration of X. Then Z = (Zt = E[Yt\Gt\) (it is called the filtering process
of Y w.r.t. G) has a cadlag modification, and
Zt = Z0 + At + / (U(s,x) + -—— I\a9<\\(^(ds,dx) - v(ds,dx)),
J[o,t]xE 1 - as l J
where i) A = (j4$) is the G-compensator of A,
ii) // is the jump measure of X and */ is the G-compensator of /z,
iii) U = M^[Z\V(G)\ -Z--AA (indeed, we have Mfl[Z\V(G)]
= M»[Y\V(G)}).
13.15 Assume that Xa and Xb are homogeneous Poisson processes
with rate a > 0 and b > 0 respectively, £ be a r.v. with P(£ = b) = p,
P(f = a) = 1 - p, 0 < p < 1, and £,Xa,Xb are independent. Let
7% = a{£,X2,X*,s < t}, t > 0, and F = (Tt) be the completion of
F° = (J?). Set X = XaI^=a] + XbIfe=b]. Let G = (ft) be the complete
natural filtration of X. Then the filtering process Z = (Zt = P[£ = b\Gt\)
satisfies the following stochastic differential equation
Zt=p+ f {\Za)^'~H~r'~){dX. ~ [bZs- + a(l - Z8-)]ett), i > 0.
Jo bZs- + a(l — Zs-)

Problems and Complements
395
13.16 Let W be a Brownian motion and F = F (W). Then for any
stopping time T and any r.v. f G Tt, there exists H G Lm(W) such that
£ = tf.Wra.s..
13.17 Let TV be a compensated Poisson process and F = FP(N).
Then for any finite predictable time T and any r.v. £ G Tt (resp. £ G .^oo),
there exists i/ G Lm(N) such that £ = H.Nt (resp. £ = H. Noo).
13.18 Suppose X1 and X2 are two semimartingales (resp. local
martingales), having the weak (resp. strong) property of predictable
representation w.r.t. filtrations F1 = (T^) and F2 = (J7?) respectively.
Let (a1,/?1,!/1) and (a2,/?2,*/2) be the predictable triplet of X1 and X2,
a{ = (*/({*} x £)), J* = [a* > 0], and u{{dt,dx) = Gi{t,dx)dB\ be the
canonical predictable decompositions of */, i = 1,2, respectively. Assume
that ^q and T%> are independent and J1 f) J2 = 0. Set
X = XX + X2,
F = (*i), Tt = T\\lT2, t>0.
Then X has the weak (resp. strong) property of predictable representation
w.r.t. F = (Jrt) if and only if
d/31 _L d/32, di/1 _L d/32 a.s. (resp.d(i91 + B1) _L d{02 + B2) a.s.).

Chapter XIV
Absolute Continuity and Contiguity of
Measures
Absolute continuity and singularity of measures induced by stochastic
processes are a classical problem of stochastic process theory. Semimartin-
gale theory and stochastic calculus provide a completely new approach to
it. In §1 we introduce the basic tools—Hellinger processes. Then in §2 we
discuss absolute continuity and singularity of measures. The
generalizations of absolute continuity and singularity—contiguity and entire
separation of measures, and the associated problem of convergence in variation
of measures are discussed in §3. Finally, the application to Levy processes
is given in §4.
§1. Hellinger Processes
In this and next paragraphs, we always suppose (fi, J7) is a measurable
space, P, P' and P are probability measures on T such that
P <^P, P1< P.
14.1 Definition For any a e]0,1[ define
w.*>-*[(SnSn-
It is not difficult to see that /ia(P,P') does not depend on the choice of
P, but on P, P' and a only. In fact, if P = -(P + P'), then
*[(5)(Sn-*K£nsr
14.2 Theorem 1) 0 < ha(P,P') < 1,

§1. Hellinger Processes 397
2)MP,P') = 1^P = P',
3) ha(P,P*) = 0 «<=► PIP',
4) lim MP, P') = 1 <«=► P7 < P.
aj.0
Proof. For any a €]0,1[ we have
uat>1-a < au + (1 - a)v, u>0,v>0, (2.1)
and the equality holds if and only if u = v. Hence
(^ (-^) <a^ + 1-a^, P-a.s.. (2.2
vdP' VdP' dP dP V y
Then 1) follows immediately by taking expectations in (2.2). And
Mp, po = i <=> p(^ = 4£) = i <=> p = p',
2) is established. Obviously,
/iQ(P, P') = 0 <^ pt^-^C = o) = 1 «=» PIP',
vdpdp '
/dP\a/dP'\l-a dP'
we have 3). Finally, since lim I —=-1 I —=-1 = —^-I.ap ,, by (2.2)
aioVdp; VdP^ dP lg>o]' J v ;
and the dominated convergence theorem we find
limha(P,P') = £[^-/dp .1 = W^ > 0).
Hence
p' ^p ^ P'(^r >o) = 1 «=» lim MP, P*) = 1,
W / alO QV '
i.e., 4) is established. □
14.3 Theorem. For any a E]0,1[ there exists a constant Ca > 0
such that
2[1 - ha{P,P')} < ||P - P'|| < [Ca(l - /iQ(P,P'))]1/2, (3.1)
where \\P - P'\\ = 2sup \P(A) - P'(A)\ is the total variation of P - P'.
A
Proof. Take P = \(P + P'). Then ^E + ^E- = 2,
2V dP dP
rdP\a(0 dP\i-Qi ,~
^dP^ ~dP>
2d-MW)../[l-(;)'P-5)>.
||P-P'||=2/|l-g|dP.
It is easy to verify 1 - za(2 - z)1_a < |l - z\ for z € [0,2]. Whence the
left inequality of (3.1) follows. On the other hand, we have already known

398 Chapter XIV Absolute Continuity and Contiguity of Measures
that olz + (1 — a)(2 — z) — za(2 — z)l~a has only one zero point z = 1 on
[0, 2]. And since
lim(2 - l)~2[az + (1 - a)(2 - z) - za(2 - z)l~a} = 2a(l - a) > 0,
there exists a constnat Ca > 0 such that
a* + (1 - a)(2 - z) - za{2 - z)1'01 > ±C~l{z - l)2, z G [0,2]. (3.2)
Thus the right inequality of (3.1) follows by substituting z with —— in
(3.2) and integrating against P. □
Remark. Usually, /ii/2(P, P') is called the Hellinger integral, and
denoted by f VdPdP'. It is easy to check that 2(1 -hx^iP, P')) is a metric
on the space formed by all probability measures on (fi,^7). It is called
Hellinger-Kakutani metric, and also denoted by j{y/dP-\fdP,f.
Theorem 14.3 illustrates that the convergence in Hellinger-Kakutani metric is
just the convergence in variation.
Prom now on a right-continuous filtration F = (Ft) is given such that
T -=\l Tu and Fp is taken as the reference filtration. Let Z and Z' be
t
the density processes of P and P' w.r.t. P respectively. Denote
Rk = mf{t : Zt < ^}, R!k = mf{t : Z[ < ±}, Sk = Rk A R*k,
R = inf{t :Zt = 0}, Rf = inf{t :Z't = 0}, S = R A Rf,
r = u[o,sj = [o]u[Z->o,#.>o],
A:
y(a) = Z"{Z')l-«, q6]0,1[.
14.4 Lemma. Under P,Y(a) is a non-negative supermartingale of
class (D).
Proof. Since 0 < Y(a) < aZ + (1 - a)Z', under P, Y(a) is non-
negative and of class (D). It is not hard to justify that under P, W =
Z1
—I[z>o] is a supermartingale. By Jensen's inequality, for 0 < s < t
E[Wl~a\Ts\ < (E[Wt\Fa])l-a < W}-°, P-a.s.,
i.e., Wl~a is a P-supermartingale. Then Y(a) = W1~aZ is a P-super-
martingale. □
14.5 Theorem. There exists a unique (up to P-indistinguishability)
predictable increasing process H(a) on T with Ho(a) = 0 such that Y(a) +
Y-{a)-H{a)eM(P).

§1. Hellinger Processes 399
Proof. According to Doob-Meyer decomposition we have
Y{a) = Y0{a) + M-A, (5.1)
where M G Mo{P) and A e Aq(P) is predictable. Observe that
Jr. • Y(a) = Km I]SnM ■ Y(a) = lim [Y(a) - (Y(a))s«] = 0.
By the uniqueness of Doob-Meyer decomposition we have
ITc • M = 0, 7r< * A = 0.
Set
Then f?(a) satisfies the requirements in the theorem. Indeed, for each
n, i5[i/sn(a)] < nl£[,Asn] < oo, and hence H(a) is a predictable increasing
process on I\ On the other hand, by the uniqueness of the canonical
decomposition Y-(a) • H(a)Sn is uniquely determined by y(a)5n, so does
H(a)Sn. Thus the uniqueness of H(a) on T is established. D
Remark. In the above proof H(a) has been defined on the whole
R+ and is uniquely determined by the requirement: H(a) = Ir • H(a).
But it may happen that H(a) takes the value +00 on Tc. In other words,
if (a) is the unique predictable increasing il+-valued process such that
H(a) = Ir • H(a) and Y(a) + Y-(a) • H{a) e M(P).
14.6 Theorem. Let P be another probability measure on (fi,^7) and
P <& P. Let H(a) be the predictable increasing process on F uniquely
determined as in Theorem 14.5 under P. Then H(a) is
P-indistinguishable from H(a).
Proof Let W be the density process of P w.r.t. P. Then the
corresponding Y(a) = Y(a)W, and by (5.1)
7(a) = {Y0{a) + M- A)W = Y0{a)W + WM -AW-W-A. (6.1)
The first three terms on the right-hand side of (6.1) are all P-local
martingales. Hence under P
Y-(a) • 17(a) = W- • A = Y-(a) • H{a).
Under P we have W > 0, and hence T = [0] U [F_(a) > 0]. Thus H(a)
is P-indistinguishable from H(a). □
14.7 Definition. H(a) is called the Hellinger process of order a
between P and P'. Observe that H(a) is not symmetric w.r.t. (P,P'),
unless a = -. In the sense of Theorem 14.6 H(a) is independent of

400 Chapter XIV Absolute Continuity and Contiguity of Measures
the choice of P (so are «S, S' and T). Hence we may consider H(a) as
P + P'-a.e. uniquely determined.
14.8 Theorem. Under P we have AH (a) < 1 and on ]0,S[, AH (a)
<1.
Proof. From (5.1) we have AY(a) = AM - Y-(a)AH(a), and
0 < Y(a) = AM + y_(a)[l - Aff(a)]. (8.1)
Taking predictable projections in (8.1) and noting P(AM) = 0, we find
?(Y{a)) = y_(a)[l - AH{a)} > 0.
Since Y-(a) > 0 an m]0,oo[, we have AH(a) < 1. On the other hand,
T = inf {t± AHt(a) = 1} is a predictable time, and T < oo => p(lr(a))T =
0. Thus E[YT{a)I[T<oo]] = 0. This implies T > S, and AH (a) < 1 on
JO, S[, in consequence. □
In the sequel we turn to the calculation of Hellinger processes.
14.9 Theorem. On T, H(a) is the P'-compensator of K(a) defined
as follows:
*(.) = ^{^ ■ <*•> - jrzrV.*-) + ^ • <«*>}
where
ipa(u, v) = au + (l- a)v - uavl~a. (9.2)
Proof. On [0, Sn\ we may apply Ito formula to Za, and obtain
Z" = Zg + {aZa_-lI^M) ■ Z + a(a2"^r2 • (Z<)
+ E(=[(1 + £)«-,-„£]).
If 0 < Sn < oo, one can verify directly that the jumps of the two sides of
(9.3) at Sn are the same. Hence (9.3) holds on [0, «Sn]. Analogously, on
{Z'?-a = {Z'Qf-° + (1 - a)((Z'_)-°I]0yOo[) ■ Z' - ^^L{Z')-l-Q ■ (Z'c)
Using the formula of integration by parts,
Y(a) = {Z1I]QM).{Z'f-a + {{Z'_?-aI\o,oo{).Za + [Za, {Z'f-%

§1. Hellinger Processes 401
on [0, Sn] we have
Y(a) = r„(a) + (1 - Q)(^/|0,oo|) • Z' + a(^/,0iOo() . Z
-4if2y-<«). [£. (zo) - _i_. (z«, n + £. (n]
+ £(y_(a){(l + — ) -i-(i-a)—+ (! + —)
AZ r/ AZ\« ir/ AZ'\i-q n\
_!_a_+[(1 + _) -l][(l + —) -l]})
Because [Y(a)+Y-(a).K(a)]Sn G Moc(-P), it is easy to see that {H(a))Sn
is the P-compensator of (K(a))Sn. Consequently the proof is complete.
D
14.10 Corollary. Assume P = -{P+P'). Let \i be the jump measure
of Z, and v be the P-compensator of p. Then
H(a) = ^^(^ + jrY ■ (ZC) + ^(A, V) * v, (10.1)
where
A = l + ^, A' = l--^. (10.2)
Proof. In this case, Z + Z' = 2, Zc + Z'c = 0, AZ + AZ' = 0. Thus
(Z'c) = (Zc), (ZcyZ'c) = -(Zc). Since Zt = 2 for t > R', we have
Ire • Z = 0, Irc • {Zc) = 0 and ITc v = 0. (9.1) reduces to
^) = ^^(^ + ^r)2-(^c> + ^(A,V)*/i,
and (10.1) follows by Theorem 14.9. □
14.11 Lemma. Let A and B be two predictable processes with finite
variation and Aq = Bo = 0. // A and B are P-indistinguishable, then
I[Z_>o] ' A and I[z_>o] * B are P-indistinguishable.
Proof. For any H G V+ we have
E[(HI[Z_>0]) ■ Aoo] = E{Zoo[(HI[z_>0]) ■ Aoo]} = E[{HZ.I[Z_>0]) ■ A^}.
By the same argument,
E[{HI[Z_>0] ■ BU] = E[(HZ_I[Z_>0]) ■ Boo}.

402 Chapter XIV Absolute Continuity and Contiguity of Measures
Hence Z_/^_>0] • A and Z_/^_>0] • B are P-indistinguishable, and so are
I[Z->o] ' A and /[z_>o] * B. D
14.12 Theorem. Assume that X G S(P) and under P, X has the
weak property of predictable representation. Let the predictable
characteristics of X under P,P' and P be (a,/?,v), (a',/3f,i/) and (a,0,V)
respectively such that
v = Y.v, Y€V+, [a=l]c[a = l], (12.1)
( v' = Y'.v, Y' G V+, [a = 1] C [a' = 1].
Then on T we have
H(<*) = a{1~a)K2 ■ 0 + <pa(Y, Y')*v + Z<pa(l-a,l- a'), (12.2)
K = i{/r ■[a'-a + (xl^y) * (i/ - i/)]}. (12.3)
dp
In particular, on T
H{\) = \K2 ■ 0 + \{ W - VY>)2 * v + \ £(v^^ - n/T^)2.
Proof. Since X has the weak property of predictable representation
under P, we have
Z = Z0 + H-XC + W* (p-v),
where /z is the jump measure of X,
H = d^L, W = U + j^Ifcq, U = M^Z\f\.
Under P we have
a - a - (x/[|x|<i)) *(u-u) = — -(Z, Xc).
Set
K = dd^I[z~>0]' ta ~ 5 ~ (x/[i^i<i)) * (^ - ?)]}-
Since (Z,XC) = I[Z_>0].(Z,XC), by Lemma 14.11 under P we have
{Z,Xc) = Z--(K-/3), H = Z_K.
On the other hand, U = MM[AZ|P] = Z-(Y - 1),U = Z-{Y - a) =
Z-(a — a). Then we obtain
z = z0 + (z.k) ■ x< + [z_ (y -1 + YrfW] * (" " *)•

§1. Hellinger Processes 403
Similary, we have
z' = z'0 + (z'_k>) ■ x< + [z'_ (r -1 + ^rf/P<1])] * (m - *),
where
K' = TgfexflV ~ 5 ~ (x/[kl<i]) * ("' - £)]}•
Thus (Zc) = {Z-K)2-]$, (Z'c) = {Z'_K')2-p, (ZC,Z'C) = (Z-Z'_KK')0,
and on T
k{ZC)-z^-{z^z'C)+W
^■{Zc)--—.(Zc,Z":) + -^(Z'c) = (K-K')2.0 = (KY.p, (12-4)
where K = K - K' on T.
Denote £> = [AX ^ 0], J = [5 > 0]. Then
l + ^ = l + r/D-/D + f^/D-(a-5)-^|a
Z_ 1 - a 1 - a
= lr/i> + :rI4/D«, (12.5)
1 — a
1 + ^ = Y'ID + \^4lDc (12.6)
Z_ 1 — a
In (12.5) and (12.6), y and y' are the abridgements for (Y(t,AXt)) and
(Y'(t,AXt)) respectively, and - = 0 by convention. Thus
AZ _ AZ'^
The JP-compensator of the right-hand side of (12.7) is
•.ttV). » + £[*.(£§, £f)(i-i)]
= v>a(y, r') * v + £ Va(l - a, 1 - a'). (12.8)
Then (12.2) follows from Theorem 14.9, (12.4) and (12.8). □
loc
14.13 Corollary. Assume P' <%:P,X e S(P) and under P,X has
the weak property of predictable representation. Let the predictable char-
aderistics of X under P1 and P be (a',(5',v') and (a,/?, v) respectively
such that
(3' = I3, v' = Yv, Ye f>+, [a = 1] C [a' = 1].

404 Chapter XIV Absolute Continuity and Contiguity of Measures
Then on T we have
H(a) = a^~a)K2 ■ (3 + Va(l, Y)*u + Z <pa(\ - a, 1 - a'), (13.1)
where
K=|?{/r'[a'~a _ (x/[isi<i]) * v - »)))•
Proof If P' <C P, we may take P = P. Then the assertion follows
immediately from Theorem 14.12. In general, being restricted on [0, Sn A
n], (13.1) holds. Hence (13.1) holds on T = (J[0, Sn A n]. D
§2. Absolute Continuity and Singularity
Now we start to discuss the absolute continuity and singularity of
measures under the same assumptions as in the previous paragraph. We
also use the same notations.
14.14 Theorem. Under P we have
[R > 0,ZR. > 0] = [R > 0, Jj- • (ZC)R. + (l - I\ + -|-)2 * vR_ < oo],
(14.1)
[Zoo > 0] = [R = oo, -^ • (Zc)oo + (l - Jl + -^-f * i/oo < oo]. (14.2)
{Recall that n is the jump measure of Z and v is the P-compensator of
Proof. Let £ = [Z- > 0] U [0]. Then L = ^-.Z € (Moc(P))B.
Z — Z
On [0, R\ we have AL = —-—— > — 1. By the exponential formula, on
[0,R[
Z = Z0exp{X},
X = L-L0- \{LC) - £[AL - log(l + AL)\.
Let u(y) = (y A 1) V (-1). Define
X* = L-L0- \{LC) - £[AL - «(log(l + AL))]
4-(»-«-l^-(n-s:f-.^(.^))]

§2. Absolute Continuity and Singularity 405
= j--(Z-Z0)-A.
We are going to show
A- r k •«+n[£ -(* (i+f))] e mj.<*»«.
{ If -=— = —1, uflog (l + -y—)) = —1-) To this end, it suffices to show
K = £ [y - "(log (l + £-))] € G4£c(P))B. It is easy to check that
for y > —1
U,|2
0 < y - u(log(l + y)) < c-
i + lvl'
where c > 0 is a constant. Thus on [0, Rn]
Since Z € -Mioc(P), # € (^(P))*.
Observe that on [0, R{
AX = log(l + AL), AXU = u(log(l + AL)),
(14.3)
X - Xu = £[log(l + AL) - u(log(l + AL))].
If Xr- (resp. X%_) exists and is finite, then {t : 0 < t < R, |log(l +
AL)| > 1} is at most a finite set. Thus by (14.3) we have
[R > 0, Xr- exists and is finite] = [R > 0, Xr_ exists and is finite].
(14.4)
Because \AXU\ < 1, by Theorem 8.33 we know
[R > 0, X%_ exists and is finite] = [R > 0, Ar- + (XU)R- < oo], (14.5)
where A is the P-compensator of A. Note that
A + [X«] = \ ■ j^.{Zc) + E[«2(log(l + AL)) - «(log(l + AX)) + AL]
and for y > — 1
ci(l - VT+y~)2 < u2(log(l + y)) - «(log(l + y)) + y < c2(l - s/TTy)2,
where ci > 0 and c2 > 0 are constants. Hence
[R > 0, AR- + (XU)R- < oo]
= [R > 0, ^.{ZC)R- + (l - ^/l + |-) * i/r- < oo]. (14.6)

406 Chapter XIV Absolute Continuity and Contiguity of Measures
Therefore (14.1) follows from (14.4), (14.5) and (14.6). (14.2) follows from
(14.1), since Z^ > 0 => R = oo. D
14.15 Theorem. Put N = [S < oo or #oc(~) = oo]. Then
i) on TV, P'±P,
ii) on Nc, P' ~ P.
Proof. Take P = ~(P' + P). By Corollary 14.10 we know
B(l)-5(i + i),-<^ + 3<^-^""
where A = 1 + ——, A7 = 1 — ——. By Theorem 14.14 we have
Zt— Zt _
[Z^ > 0] = [R = oo, Zr2.(Zc)oo + (1 - y/X)2 * i/oo < oo],
[Z^ > 0] = [#' = oo, Zr2.(Zc)oo + (1 - v'A7)2 * Voo < oo].
Since 1 is between y/\ and v/A7, (1 - \/A)2 < (y/\ - y/X1)2, (1 - n/A7)2 <
(v/A - -y/A7)2. Hence
[5 = oo, #00(1/2) < 00] = [Zoo >0,Z'oo> 0], (15.1)
[5 < 00 or Hoo (l/2) = 00] = [Zoo = 0] U [Z^ = 0]. (15.2)
Because P(Zoo = 0) = 0 and P'iZ'^ = 0) = 0, the assertions follow from
(15.1) and (15.2) immediately. □
14.16 Theorem. P' <C P if and only if the following conditions are
fulfilled:
i) Pj) < Po (P'o o,nd Po are the restrictions of P and P on T$
respectively),
ii) Pf(H00(^) < OO) = 1,
iii) P(/[I=_Z_] * Voo > 0) = 0.
Proof. Prom Theorem 14.15 we know
p'<p<-»P'(5 = oo,ffoo(^) <oo) = l.
Necessity, i) and ii) are trivial.
E' [/[x=-z_] * "00] = E\Z'oa{l{x=_zA * uoo]
= E[(Z'_I[X=-Z_)) * Voo\ = HZLl^-z.]) * Moo]
= E[ £ Z't_I[0jiAZt=-Zt-]\ = E[Z'r_I[0<r<OOiZr_>0]], (16.1)

§2. Absolute Continuity and Singularity 407
since 0 ^ AZt = — Zt- is possible only when t = R. Because P(R <
oo) = 0 and P' < P, so P'{R < oo) = 0. Hence iii) follows from (16.1).
Sufficiency. In order to obtain P'(S = oo,iJoo(-) < oo) = 1, by ii)
and P'(R' < oo) = 0 it suffices to show Pf(R < oo) = 0. From iii) and
(16.1) we have
J5[Zfi-/[0<JR<oo,Z/2_>0]] = 0- (16.2)
But P(R < oo, R' = oo) = 0. Thus R < oo => Rf = oo and Z'R_ > 0 P-
a.s.. It is deduced from (16.2) that P(0 < R < oo, Zfi_ > 0) = 0. Hence
P/(0 < i? < oo, ZH_ > 0) = 0. By i) we know P\R > 0) = 1. By
Theorem 14.14
[R > 0, ZR_ > 0] = [R > 0, #«-(|) < oo],
and therefore by ii) P'{R > 0,ZR- > 0) = 1. At last, P'(i? < oo) = 0.
D
14.17 Remark. If P = -(P + P'), condition iii) in Theorem 14.16
is equivalent to the following
iii)' For all A € V, (IAX) * ^oo = 0 P'-a.s. =► (IA\') * ^oo = 0 P'-a.s..
(Recall that in this case A * v and A7 * v are the compensators of \i under
P and P' respectively.)
Proof. iii)=*iii)'. Let A G V and (IA\) * i/^ = 0 P'-a.s.. Then
^[A>o] * ^oo = 0 P'-a.s.. By iii)
(IA\f) *!/«,= (/^[A>o]A/) * ^oo = [V * (/^[a>o] * ^)]oo = 0, P'-a.s..
iii)/=>iii). Obviously, we have (AJ[a=o]) * ^oo = 0. By iii)7
[7[A=0](A + A')] * *„ = (/[a=o]A;) * ^oo = 0, P'-a.s.. (17.1)
But J[A+A'=o] * ^oo = 0, P-a.s. and P-a.s.. Prom (17.1) [/[a=o,A+A'>o](* +
X)] * ^oo = 0, P'-a.s., and hence /[a=o] * ^oo = 0, P'-a.s.. □
14.18 Theorem. P'_LP if and only if
P'(Z0 = 0 or Hcoi^) = oo or /[x=_z_] * Moo > 0) = 1. (18.1)
Proof, we have
PIP
<=» P'(S = oo or JJoo(l/2) < oo) = 0
<=> P'(i? < oo or ifoo(l/2) = oo) = 1
<=» P'fZo = 0, or 0 < i? < oo and #00(5) < 00. or ^00(5) = 00) = 1

408 Chapter XIV Absolute Continuity and Contiguity of Measures
<=> P'{Z0 = 0, or 0 < R < oo and ZR- > 0, or H^l/2) = oo) = 1
<=» P'{Z0 = 0, or I[x=-Z_] * //oo > 0, or ffoo(l/2) = oo) = 1,
noting that [Z0 = 0] = [R = 0],
[i? > 0, JJ*_(l/2) < oo] = [i? > 0, Z*_ > 0] = [/[x=_z_] * Moo > 0]. □
It should be pointed out that (18.1) is not a predictable criterion.
Seemingly, it is hard to find a predictable criterion for singularity in the
general case.
loc
14.19 Theorem. Assume P<^P. Then
1) P' < p ^> P'(Hoo(l/2) < oo) = 1,
2) P'-LP <=» P'(ffoo(l/2) = oo) = 1.
Proof. In this case, P,(R = oo) = 1 and hence P'(S = oo) =* 1. Then
in Theorem 14.15 we may take TV = [#00(1/2) = 00], and the assertions
follow. D
14.20 Theorem. Assume that X E S(P) and under P, X has the
weak property of predictable representation. Let the predictable
characteristics of X under P, P' and P be (a, /3, v), (a', /3', 1/) and (5, /?, v)
respectively such that
v = Y.v, YeV+, [a = l]c[a = l],
1/ = Y'. v, Y' e V+, [a = 1] C [af = !]•
Set
ti = M{t: / \d(as - as - {xl[\x\<i]) * {y - v)s)\ = 00},
r2 = inf{* : J \d(a's - 55 - (z/[|x|<i]) * W - ?)5)| = 00},
r = t\ A r2,
if = [a7 - a - (x/[|x|<i]) * (1/ - i/)]/[0,t[ + (+«>)^[rfoo[-
Le£ if 6e an R-valued predictable process such that if on [0, t] H is abso-
lutely continuous w.r.t. (3, then H3 = / Kud/3U,Q < s < t; and if on [0,£]
it is not absolutely continuous w.r.t. (5, then Kt = +00. Define
A = K2.0 + {VY - y/Y')2 * v + E(vf^ - vT^7)2,
TV = [Z0Z'0 = 0] U (|J[A 7^ #]) U [Aoo = 00] U [I[Yy>=o] * Moo > 0]
U[ £ J[AXi=0]J[at=l]U[a't=l] > OJ ,

§2. Absolute Continuity and Singularity 409
where /z is the jump measure of X. Then 1) on iV, P'LP, 2) on iVc, P' ~
P.
Proof. 1) Set
ff(1> = [a - 5 - (x/[|x|<l]) * [y ~ ?)]/[0,tiI + (+°°)/[ri,oo[,
and
F(2) = a' - 5 - (z/[|x|<i]) * (i/ - ?)]/[o,r2[ + (+oo)/[T2j0o[.
Let IfW, i = 1, 2, be an ii-valued predictable process such that if on [0, t],
#W is absolutely continuous w.r.t. /3, then i/i*' = / K$d/3Uy0 < s <t;
Jo _
and if on [0, t], i/W is not absolutely continuous w.r.t. /?, then /Q1' = H-oo.
Define
A& = {KW)2.J3 + (i - y/Y)2 * v + ^{y/T^L - vT^S)2,
^(2) = (KW)2.J3 + (1 - y/Y1)2 * i? + E(yr^ - n/1^5)2.
It is not difficult to check TV C N\ U JV2, where
iVi = [Z0 = 0] U (UlA ^ AD U [A& = oo] U [7y=0 * //oo > 0]U
t
[T, I[AXt=0,at=l] > °J>
AT2 = [Z'0 = 0] U (U[fl ^ A]) U [a£> = oo] U [7y/=0 * Moo > 0]U
t
[EWt=0,aJ=l] >0]'
(In fact, if/3 = p = J3 and d^W « d&, z = 1,2, then tf(2) - K™ = AT
andA<2(A(1) + A(2)).)
Since P C P, we have
P(Z0 = 0) = 0,
t
P(A<£ < oo) = 1
(by Theorems 14.12, 14.16 and comparing A^ with the Hellinger process
of order 1/2 between P and P),
E[I[Y=0] * Moo] = E[I[Y=o\ * "oo] = E[(I[y=0)Y) * "oo] = 0,
P(\J[AXt = 0, at = l}) = 0 (under P [o = 1] C [AX # 0]),
t
and hence P(JVi) = 0. Similarly, P'(JV2) = 0. Then on N,P'±P.

410 Chapter XIV Absolute Continuity and Contiguity of Measures
2) We want to show Nc C [S = oo, i/oo(l/2) < oo]. Then the assertion
follows from Theorem 14.15. As shown in the proof of Theorem 14.12,
Z = Z0€{L),L e MiocoiP) and
ALt = (Y(t,AXt) - 1)/[ax,/o] - T^We=o],
(see (12.5)). According to the definition of N, on Nc for any t, if AXt ^ 0,
Y(t,AXt) >0andALt = y(t,AXt)-l > -1; and if AXt = 0,at < land
AL* = —-—^ > -1. In a word, on Nc AL > -1, and hence Z > 0 and
1 - at
Ri = oo. Similarly, on Nc we have i?2 = oo. Therefore Nc C [5 = oo]. It
is easy to see from Theorem 14.12 that iVc C [JJoo(l/2) < oo]. D
Remark. The assertion 1) doesn't need the assumption that under
P, X has the weak property of predictable representation.
oo
14.21 Theorem. Let X = Xo+ £ £n-f[rn,oo[ &e a s^eP process, where
71=1
Tn | oo, for each n > 0,Tn < oo => Tn < Tn+i(T0 = 0), and for each
n > 1, [Tn < oo] = [fn ^ 0]. Le* F = F°(X), P7 and P 6e *wo probability
measures on T = V-^i- The Levy systems of X under P1 and P are
t
denoted by v1 and v respectively. Then P1 «C P if and only if the following
conditions are fulfilled:
i) P'0 < P0,
ii) there exists W € V+ such that v' = W • v, [a = 1] C [a7 = 1] and
P'(A,00 < oo) = 1, w/iere A; = (1 - VW)2 * i/ + E(V/TT^ - >/T^)2.
In this case, the density process of P1 w.r.t. P is
f^( n W(TniU))( n i^i)e(»-^f, on [A' < oo],
0, on [A' = oo],
(21.1)
where vc is the continuous part of v.
Proof Sufficiency. Take P = -(P + P'). Then Theorems 14.12
and 14.20 are used. It suffices to check P'(N) = 0, where N is denned
in Theorem 14.20. Indeed, P'{ZQZ'Q = 0) = P'{Z'0 = 0) = 0, because
dP'
Z'o = Zo-TFT -P'-a-s.; P'(Aoo = oo) = 0, because (1 - \/W)2 * v =
dPo
(y/7 - y/Y7)2 * v;
E'[I[YY'=0] * Moo] = E'[I[Y>=0] * Moo] = E[(Ipl=0]Y') * ?oo] = 0,

§2. Absolute Continuity and Singularity 411
because Y' = YW(under P');
E' [ E^[ax1=o]^(=i]uK=i]]
= e'[ Eo Wt=o]K=i]] = B'lgW1" 4)1 = °-
The necessity is clear by Theorem 14.16 ii), because H(l/2) is P'-
indistingnishable from £(1 - y/W)2 *v + £(\/l -a - y/l - a1)2. Prom the
proof of Theorem 14.12 we know that the density process Z' of P' w.r.t.
P (take P = P) satisfies
Z> = Z>0+[zL(W-1 + ^)]*(vl-v).
L = (W - 1 + ) * (// - v) is a P-local martingale on U[0> -R^l C
>> 1 — a J n
[A! < oo], and
a'rp — aT„ a!a — a,
Lt = {W-l)*{n-v)t+ £ f w ~ S i-Z1^
r„<t 1 — ar„ .s<* 1 — a>s
Tn<t 1 - aTn s<t 1 -«5
= -(W - 1) * ut + E« - as) + (W - 1) * & - E y-f
* s<t l ~ a.
+ E -r—-
Tn<t 1 ~ aTn
a's -a*
s
as - a's
= -(w-i)*v< + £ (w(rni«n)-1) + £ t^
Tn<l S<t,S?Tn 1 - a-
= -(W-l)*^c+E ALa.
Denote by Z the process defined in (21.1). Clearly, for each n,ZRn =
Z^(L^) = (Z')^. If for some n,^ = R' < oo, then Z*/ = ZH/n =
Z^ = Z'R = 0. If for all n,R'n < R' < oo, then Z/*_ = UmZ^ =
liniZfl, - = Z'r' = 0- Hence from the definition of Z we know Z = 0 on
[Ji'jOof. Therefore Z = Z', and the proof is complete. D
14.22 Corollary. In addition to the assumptions of Theorem 14.21,
suppose Pf0 < Pq and there exists W efi+ such that v' = W.v, [a = 1] C
[a' = 1], and for each t > 0
p'((i - Vw)2 * i/t + E(x/T^I - \Ji-<? < oo) = i.
Then P'_LP if and only if
P'((l - y/W)2 * i/oo + E (v^^ - Jl - a{)2 = oo) = 1.

412 Chapter XIV Absolute Continuity and Continuity of Measures
Proof. By Theorem 14.21 we know P\ < Pt for all t > 0(P{ and Pt
are the restrictions of P' and P on Tt respectively). Then the assertion
follows from Theorem 14.19.2). □
§3. Contiguity, Entire Separation
and Convergence in Variation
In this paragraph we always suppose (ft71, ^rn)1Tt > 1, is a sequence of
measurable spaces, Pn and P/n are two probability measures on (ft71,^771)
for each n > 1.
14.23 Definition. We say that (P'n) is contiguous to (P71), and
denote it by (P/ri) < (P71), if for all An G F1
Pn(An) - 0 =► P'n(An) - 0.
Obviously, \\P'n - Pn\\ -* 0 =► (P,n) < (P71). We say that (P71) and
(Pfn) are entirely separated, and denote it by (P/7l)A(Pn), if there exist
a subsequence (n^) and Ank G TUk such that
J"n*(>lnJ -»0, Pn'(^)-+0, asfc^oo.
Let £„ € .F\n > 1. If for all e > 0
Pn(|^| > e) - 0,
we say that (£n) converges to zero in (P71), and denote it by £n —► 0. (£n,
Pn) is said to be tight if
Urn EK Pn(|£J > TV) = 0. (23.1)
N-*oo n—>oo
If£n,n > 1, are all finite-valued, (23.1) is equivalent to
lim supP7l(|^|>AT) = 0,
N-*oo n
i.e., the family of the distributions of £n is tight.
The concepts of contiguity and entire separation are the generalizations
of the concepts of absolute continuity and singularity respectively. In fact,
if (fi71,^71) = {Sl,f),Pn = P and Pln = P', then (P,n) < (Pn) is just
P' < P, and (P/n)A(Pn) is just P'_LP. We suggest the reader to justify
it.
14.24 Theorem. The following statements are equivalent:
1) (P'n) < (P71),
2)/ora//^nG^Tl,^^0=>^Cl0,
3) for all £n G F1, (fn> P71) is %A* =► (£n, P'n) w tight

§3. Contiguity, Entire Separation and Convergence in Variation 413
■-{."!'
Proof. 2)=»1). Let An € F1 and Pn(An) -» 0. Write f„ = IAri. Then
pn p/n
£n ^ 0, and hence £n *-* 0, i.e., P'n(An) -+ 0.
1)=> 3). Suppose there exists £n G ^rn,n > 1, such that (£n,Pn) is
tight, but (£n,Pfn) is not tight. Then there exists e > 0, a subsequence
(nfc) and &nfc -► +oo such that for all k > 1, P/rifc(l£nJ > &nfc) > £• Define
[IfnJ > hk], n = njt, fc> 1,
n £ (njt).
Then Pn(An) -► 0, but P'n{An) -f* 0. This is a contradiction.
3)=>2). Suppose there exists 7]n € .P\n > 1, such that (r]n) converges
to zero in (Pn), but does not converge to zero in (P/n). Then there exists
e > 0, a subsequence (njt) and anje [ 0 such that for all k, Pfnk(\r]nk\ >
e) > e, but Pnk(\r]nk\ > ank) —► 0 as k -♦ oo. Let
{r}nk/a>nk, n = nk, k>\,
0, n £ K).
Then te^P71) is tight. Indeed, we have Urn Pn(\£n\ > 1) = 0. But
n—*oo
(f^P771) is not tight. In fact, for each TV when njt is large enough, we
have e/onk > N and
BS P'n(\£n\ >N)>h^ P'nk(\tnk\ >—)>€.
n-+oo fc—oo V * dnk'
It is a contradiction. □
Denote P* = Up11 + P'71), and
Clearly, Pn(/n < oo) = 1, and the Lebesgue decomposition of P'n w.r.t.
Pn is as follows:
p*(B) = I lndPn + P,n{B[ln = oo]), BeF1.
JB
ln is determined to be Pn + P/n-a.s., just like the uniqueness in Lebesgue
decomposition. In fact, in the defintion of Zn, P may be replaced by any
<r-finite measure /z71 whenever Pn < nn and Pln < /j,n.
14.25 Lemma. (/n,Pn) a^d (—,Pn) are ft^W.
Proof. We have

414 Chapter XIV Absolute Continuity and Continuity of Measures
14.26 Theorem. (P/n) < (Pn) if and only if (ln,Pn) is uniformly
integrable and Pfn(ln = oo) -» 0.
Proof. Prom Lebesgue decomposition we know for any Bn G T*
J lndPn < Pfn{Bn) < I lndPn + P,n(/n = oo). (26.1)
Sufficiency. Assume Pn(Bn) -» 0. Since (ln,Pn) is uniformly
integrable, / lndPn -♦ 0 (Theorem 1.9). Then by (26.1) we obtain
JBn
P'n{Bn) - 0.
; Necessity. Since Pn{ln = oo) = 0, by the contiguity P'n(ln = oo) -►
0. From (26.1) we know f lndPn < 1 and for any Bn G Tn,n > 1,
Pn{Bn) - 0 => P'n(Bn) - 0 => / lndPn - 0. Hence (/n,Pn) is
JBn
uniformly integrable. □
14.27 Theorem. The following statements are equivalent:
i) (p*1) < (p»),
2) (Zn, P'n) is tight,
3) (—,P/n) is%H
4) lim Urn ha(Pn,P1n) = l.
a—O n-»oo
Proof. 1)=^2) and l)=>-3) follow from Lemma 14.25 and Theorem
14.24.
2)=3-l). We are to justify the two conditions in Theorem 14.26. By
(26.1) we have
/ lndPn < Pln{ln > N).
Hence, noting P71^ < oo) = l,(ln,Pn) is uniformly integrable. On the
other hand, P'n(ln = oo) < P'n(ln > N). Thus
B15 p"l(/n = oo) < lim Em" P'n(ln > N) = 0.
n—►oo N—►oo n—*oo
3)=>1). Let Pn(£n) -♦ 0, 5n € ^rn. Then, noting zn + z'n = 2, we
have
= P'"(2„<l)+(2iV-l)P'l(I?n).
Letting n —► oo and N —> oo consecutively yields P'n(Bn) —► 0.

§3. Contiguity, Entire Separation and Convergence in Variation 415
3)-<=^4). It is not hard to see that there exist three functions ^*7i
and 72 on ]0,1[ such that if) > 0,72 > 71 > 0, Um tp(a) = Um 71(a) =
a—►() a—►()
Um 72(a) = 0 and for x G [0,2]
-tl>(a) - 2/[l<72(a)] < xa(2 - x)1-" - (2 - x) < ^(a) - 2/(l<7l(a)]. (27.1)
Substituting x by zn in (27.1) and integrating against P , we find
-V(a) - Pn(zn < 72(a)) - P'n(zn < 72(a)) < ha(Pn,P'n) - 1
< V(a) - Pn{zn < 71(a)) - P^izn < 71(a)).
Since lim lim Pn{zn < 71(a)) = 0 (Lemma 14.25), we have
a—>0 n—»oo
- lim fim" P/n(2n < 72(a)) < lim lim ha(Pn, P'n) - 1
< - lim liin P'n(zn < 71(a)).
a-*oon—>oo
Now 3)-£=>4) is apparent. □
14.28 Theorem. The following statements are equivalent:
1) (P'n)A(Pn),
2) /or aff JV > 0, Em" P/n(/n > N) = 1,
n—►oo
3) /or a// e > 0, Urn P/n(z„ > e) = 0,
n—►oo
4) /or some a e]0, l[, Hm ha{Pn,P'n) = 0,
n—*oo
5) /or any a €]0, l[, lim /ia(Pn, P'n) = 0.
n—►(»
Proof. Let 1) hold. There exists a subsequence (n^) and 2?nfc G ^7Tlfc
such that Pnfc(BnJ -♦ 0,P'nk{Bnk) -> 1. For any TV > 0
P'nk(Bnk) = I lnkdPnk + P,n"(Bnk[lnk > N})
JBnk[lnk<N]
<NPnk{Bnk) + P'n*(lnk >N).
Hence lim P'nk{lnk > N) = 1, and Um Pn{ln > N) = 1. 2) is estab-
k—►oo n—^oo
Ushed.
For any e > 0
Pmfc (*nfc > e) < P'n" {Bcnk) + I ^dPn"
^Bnk[znk>e\ Znk
<P>^(B<nk) + -ePn*{Bnk).
Thus 3) follows.

416 Chapter XIV Absolute Continuity and Continuity of Measures
For any a e]0,1[, by Holder's inequality
WB^ ' KJB<nk '
< (Pn"{Bnk))a + (P**(BSk))l"a - 0.
Thus 5) follows. 5)=S>4) is trivial.
2)=>1). There is a subsequence (nk) such that P'nk(lnk > k) > 1 --£■
Let Bnk = [lnk > k}. Then P'nk(Bnk) -> 1. By Lemma 14.25
lim Pnk(lnk >k)< lim Urn Pnk{lnk >N) = 0.
k—►oo N—►oo fc—>oo
Hence Pnfc(5nJ - 0,P'n*(£nJ -► 1, i.e., (Pn)A(Pm).
3)=>-l). There is a subsequence (n^) such that P'nk\znk > —J < r-
On the other hand, Pnk(znk < ^) < i. Let £n/t = [znjfc > ^]. Then
*"n*(£nJ^0,P"HSnJ-l.
4)=>1). There is a subsequence (nk) such that lim hQ{Pnk, P'n*) = 0.
Let Bnk = [2nfc < 4J. Then
P"H5„J = / ^JP* < f (znknz'nky-«dPnk
<ha{Pnk,P'nk),
p,nk(Bcnk) = ( z>nkdp"k < f {z^n^y-od^
JBK JBnk
<hQ{Pnk,P'nk).
Hence Pn*(B„fc) - 0, P"1*{Bnk)^\. U
14.29 Theorem. The following statements are equivalent:
1) \\Pn - P'n\\ -» 0,
2) J\ln-l\dPn^0,
3)l„-l£o,
4)*n-l£o,
5) /or some a e]0,1[, Urn ha(Pn, P/n) = 1,
6) for any a e]0, l[, lim°/ia(Pn,P/n) = 1.
71—►OO
Proof. We have
||P" - P'l = J \Zn - 3>n\dP" = j\ln~ l\dPn + P'n(ln = OO).

§3. Contiguity, Entire Separation and Convergence in Variation 417
1) => 2) follows immediately. 2) =>> 3) is trivial by Chebyshev's
inequality. 3) => 4) is easy. Indeed, Zn — 1 = 2( 1).
4) => 1). For any given e > 0, if 0 < 6 < e < 1, then
(\zn-l\<e)= f -dPn> I -dPn
J[\zn-l\<e] Zn J[\*n-1\<6] *n
^ Yhpn{lzn "1{ -6)'
Letting n —> oo and <5 j 0 consecutively yields 2n - 1 -» 0. Since |zn - 1| <
l,\\Pn-P'n\\=2J\zn-l\dPn^0.
1) => 6) => 5) => 1) follows from Theorem 14.3. D
Prom now on we suppose for each n a right-continuous filtration Fn =
(JJ1) is given such that T11 = \j T? and take (F71)^ as the reference
t
filtration. Let Zn and Z,n be the density processes of Pn and P'n w.r.t.
P™ respectively. Denote
RjJ = inf {* : Ztn < i}, Iff = inf {t: Zt/fl < ±}, S? = i?£ A Iff,
JT» = inf{t : Ztn = 0}, #n = inf{* : 2*» = 0}, S" = i?n A i?,n,
rn = ij[o, S£] = [o] u [zu > o, z1? > o],
/in—the jump measure of Zn,
vn—the compensator of /j,n under Tf,
if"—the Hellinger process of (Pn, P'n) of order 1/2,
and for N > 2
in(Ar) = (Am/[NAn<Vn])*^,
X . X
where \n = 1 + -=—, Am = 1 —. It is not hard to check directly
pn 7 7'n °
that \n.vn and Am.i/n are the compensators of \in under Pn and P'n
respectively.
In the sequel, for the sake of convenience, we omit the index n
constantly. It appears only in the case, where it is indispensable.
14.30 Lemma. If (P'n) < (Pn), then
lim Gm P'n{S? < oo) = 0.
Jfc—^00 71—>°0
Proof. We have
P'(R'k<oo)<P'(Z'Rlk<l)<l,
P'(Sk < oo) < P'(Rk < oo) + P'(R'k < oo).

418 Chapter XIV Absolute Continuity and Continuity of Measures
Hence it suffices to show lim Um Pfn(Ri < oo) = 0. If it is not true,
there exist 6 > 0 and a subsequence (n^) such that for all k
P*nk{Rlk <oo)>6. (30.1)
But similarly we have P{Rk < oo) < -. Then Pnk{R^k < oo) -♦ 0, and
by the contiguity, Pf7lk(R^k < oo) -♦ 0, which contradicts (30.1). □
14.31 Theorem. (Pfn)< {Pn) if and only if the following conditions
are fulfilled:
i) (P'0n) < (P£), where P'0n = P'n_\^ and P% = P»|^,
ii) (H^,P'n) is tight, i.e., Iw^Kn^P"1^ >N) = 0,
iii) for any e > 0 , Um Em P'n(i" (N) > e) = 0.
n—*oo n—►oo
Proof. Necessity, i) is obvious. Noting that
HSk <kY..HSk
(recall V = \fZ~Z1), Y < 1 and P' < 2P, we have
P'(#oo > AT) < P'(Sk < oo) + P'(F_. ^ > j)
2k
<p'(5ifc<oo) + -£;[y_.H00]
= p'(5fc<co) + ^£;[yo-yoo]
2k
<P'(5fe<cx>) + ^. (31.1)
Letting n —+ oo, TV —+ oo and A; —► oo successively by Lemma 14.30 yields
ii). Set
jn(N) = (I[NXn<x,n])*nn.
If u < -v,ipi/2{u,v) = -(u + v) - Y/m; = -{y/v - y/u)2 > cv, where
Ht1-;??)2-™"5
2 '^1/zv ' ' T ' v 2
i(AT) < c-1(<Pl/2(A, A/)/[AfA<v]) * v < c-'H, N>2.
Hence i(N)Sk is P-integrable, and hence P'-integrable. Thus i(N)Sk is
the P'-compensator oij(N)Sk. By Lenglart's inequality for e > 0
P'(isk(N) >e)< -£E'[(Aj(N))*Sk] + P'Us^N) > 0). (31.2)
Z 1 Z'
Since Aj(iV) < 1, and on [0, Sk[, -=->—, -=- < 2k, when N > 4k2 on
Zi_ 2k ju_

§3. Contiguity, Entire Separation and Convergence in Variation 419
Z1 N Z
[0,Sfc[ we have — < 2k < — < N—. Thus
Z_ 2k Z—
3skJN) = 0, (Aj(N))*Sk < I[Sk<oo].
Hence, when N > 4k2, by (31.1) we have
P'(ioo(N) >e)< P'(iSk(N) > e) + P,(S% < oo)
<(±+2)P'(Sk<oo).
Letting n —+ oo, N —► oo and k —► oo successively by Lemma 14.30 yields
ui).
SuflBciency. Prom the exponential formula, on [0, Sk\ we have
-^■s+i^-sM'+if)-!!]}
+(A' -\)*(fi-u) + [log A' - (A' - 1)] * n - [logA - (A - 1)] * n)
+(A' - A) * n + [log y - (A' - A)] * m}
= ^exp{A + B}, (31.3)
where Zc>p' = Zc - -=j-.(Zc, Z'c) = Zc + -^-.(Zc) is the continuous mar-
z _ z_
tingale part of Z under P\ and
B = (A' - A) * {n - u) + [log j - (A' - A)] * n
= (Vi|>&) lo6 ~) * /* + [7[|p-i|>6](p ~ 1)] * "'
+ (/[Ip-i|<'»] lo6 ") * (/* _ "') + [7[|p-i|<t] (lo6 " _ * + *>)] * "'
= B1 + B2 + B3 + B4,

420 Chapter XIV Absolute Continuity and Continuity of Measures
where p = —, v' is the P'-compensator of /z, and b E]0,1[ is a constant.
A
Below we will discuss under P', and estimate (31.3) term by term.
Firstly, since (Ptf) < (Pg), we have
lim Em" P'n(Z'0n/Z2 >N) = 0. (31.4)
n—>oo n—>oo u
Secondly, since (ZC-P')(P') = (Zc), by Lenglart's inequality
P'(({j-_ + jr)-Zc'P'ySk >")< L/N2 + ^'(SHoo > L),
P'(A*Sk > 2N) < L/N2 + P'(8iJoo > L) + P'^H^ > N).
Letting k —+ oo, n —+ oo, iV —► oo and L —► oo successively yields
lim Em P'n( lim (/T)£n > 2N) = 0. (31.5)
Thirdly, using | log(l + x)| < |x|/(l — \x\) for \x\ < 1, we have
((j[|P-i|<6] log -)*(/*- "')) < (7[|p-i|<6] lo62 P) * "'
£ < V*h{&£) • ' * (^T^)'^ - »* * -' * «*,
where c& is a constant, dependent of b only. By Lenglart's inequality
P'((B3)*Sk >N)< L/N2 + P^cfcffoo > L).
Letting A;—+00, n—* 00, TV —+00 and L —► 00 successively yields
lim Bm P'n( Urn (£n'3)cn > n) = 0. (31.6)
Fourthly, using | log(l + x) - x\ < x2/2(l - |x|) for |x| < 1, we have
\B4\<(Ii\P-i\<b]\-logp + p-l\)*v'
* (V**^) • ■> * ^^^F^ - "2 * ' * <*•
P'((£% > N) < P'(c6ffoo > N).
Hence
lim fim P/n( lim (BnA)*Sn > n) = 0. (31.7)
N—00*1—00 \Jt_ooV '*k - J v '
Fifthly, we have
-(1+7m3T)(^-1)2*I//-C6^
^'((52)sfc > *) < P'icbHn > N).

§3. Contiguity, Entire Separation and Convergence in Variation 421
Hence
lim Em" P'n( lim (Bn'2)*Sn > N) = 0. (31.8)
Sixthly, take 0 < 6 < 1 - 6, then
51 < (l[s<P<i-b) log+ -) * M + (^[P<«] log+ -) * H
< (/[p<i-6] !og ^) * M + (J[P<«] log+ -) * M,
^((exp^1})^ > AT)
< f((VH * M)5fc > j^) + P'((/[p<6] * /i)5fc > 0). (31.9)
Clearly,
I[p<l-b) *"' < *[|p-l|>6] * "' < ^(/[|p-l|>6]|P " 1|) *»' < c&#-
By Lenglart's inequality
i"(< W* • ri* ^)<«?t ^H. > I), (mo)
On the other hand,
I[p<6\ *V* < I[K\<\'} * v' + J[0<A'</a,p<6] * */
< i(K) + KI[x,>0^p<s] * u'
-W + (T^(^-1)2^'
. .,Tjr^ 2Kb __
< %(K) + t=^H.
~ (1-v^)2
Note that fi is integer-valued. Again by Lenglart's inequality
P'dllpKS] * »)sk > 0) = P'ttlfrKS] * IM)sk > 1)
<V + P'((I[P<6\*i/)sk>v), V>0. (31.11)
Prom (31.9)-(31.11) we obtain
P'dexpiB1}^ >N)< ^i^ + P'{cbH00 >L) + V
+P'(ioo(K) > ±v) + P'(^oc > »?(1 - V6)2/4K6).
Letting k —♦ oo, n—+00, TV—+00, L —♦ 00, 6 —» 0, if—+00 and 77 —► 0
successively yields
lim Hm P'n( lim (exp{Bn>1})i» > n) = 0. (31.12)

422 Chapter XIV Absolute Continuity and Continuity of Measures
Prom (31.4)-(31.8) and (31.12) we get
(/ 7iln \ * \
lim ( — ) > TV) = 0.
{Zln\* /Z'n\*
In fact, it is not difficult to see lim I -=— J = ( -=— J . Hence
k->oo \Zn'Sk \Zn'oo
(Zln \
-22. > TV) = 0.
- " %So ~ '
This imphes (P'n) < (Pn) (Theorem 14.27). D
14.32 Remark. In Theorem 14.31 condition iii) can be substituted
by
iii)' For all An € T*
(lAnXn) * C - 0 => (IAn A'n") * C - 0.
Proof. iii)^iii)'.
(IAnXn) * i& < (VA«<A'«]A'n) * C + (/A„[NA">A-]A'n) * C
<iSo(iV) + 7V(/AnA")*C.
iii)'+ii)=>iii).
(VA"<A'«]An) * C < (v^T - l)-2#£>-
Take JVn -» oo, An = [NnXn < X'n], then tJ^JV) ^ 0. □
14.33 Theorem. If for all N > 0, Em P'n(H" > N) = 1, then
n—kx)
(Pm)A(Pn).
Proo/. By (31.1)
2k
P'iHoo >N)< P'(Sk <<*>) + -
2k
< P'{Rk < oo) + P'(i4 < oo) + —
< P'(Rk < oo) + i + ^.
Whence lim Um Pfn(Ri < oo) = 1. Comparing with Lemma 14.25, we
k—^oon—kx)
know (P,n)A(Pn). D
Obviously, if (P,0n)A(Pg), then (P,n)A(Pn). But either (P,0n)A(P5)
or the condition in Theorem 14.33 is far from necessary for entire
separation.

§3. Contiguity, Entire Separation and Convergence in Variation 423
Remark. Naturally, Theorem 14.16 can be deduced from Theorem
14.31, and also one can deduce from Theorem 14.31 that if P^H^ =
oo) = 1, then P'±P. The details axe left to the reader.
14.34 Lemma. The following statements are equivalent:
1) \\Pn - P'n\\ -+ 0,
2) (Z"-1)^0,
3) (VZ^Z^- 1)^0.
Proof. 1)=>2). By the maximal inequality of martingales, for e > 0
P((Z - 1&, > e) < jElZ^ - i| = 1||P - p>\\.
The assertion follows immediately.
2)=>1). Obviously, Z^-1^0. For any given £>0and0<<5<£<l
P(\Zoo - 1| < e) > f ^-dP > T^PdZoo - 1| < 6).
J[\Zoo-l\<S) ^oo 1 +0
P™
Letting n —» oo and 6 —+ 0 successively yields Z^ — 1 —♦ 0, and hence
||P'" - Pn\\ = 2En[\Z20 - 1|] -» 0, since |Z£ - 1| < 1.
Noting that 1 - (VZZ1)2 = (1 - Z)2 and 0 < \fZZ~' < 1, we have
(i - \fzz>y < [(i - z)*}2 < 2(1 - y/zz1)*.
Hence 2) <*=» 3) follows. □
14.35 Theorem. The following statements are equivalent:
1) ||Pn - P,n|| -» 0,
2)i)||Pg-Pgl||-.0;ii)if^^0,
3)i)||Pg-Pgl||-.0;ii)ffS)^0.
Proo/. 1)^2). i) is trivial. Recall that Y = Y0 + M - A, where
M is a martingale with Mq = 0 and A = Y-.H. By Lemma 14.34,
(Yn - r0n)* ^ 0. A is dominated by (Y - Y0)*, A(Y - Y0)* < \AY\ < 1.
By Lenglart's inequahty we know A1^ —► 0.
Observe that on [inft>oyi > 1/2],
Ho°= (yZ ~*)' Ao°+ Ao° ~2(r " 1)°°Ao°+ Ax"
and on [inft>0 Yt < 1/2], (Y - 1)^ > 1/2. Therefore for any e > 0
P(Hoo >e)< P((Y - 1)^ > \) + P([2(r - 1)^ + lMoo > e).
Hence #" ^0.

424 Chapter XIV Absolute Continuity and Continuity of Measures
2)=>3) is trivial.
3)=>1). At first, observe that
2#oo > (a/A - n/A7)2 * i/oo > \Uj - lfl[NXf<x] * i/oo
- \V iv ~ / ^'^i * I/°0,
Hence
Applying Theorem 14.31 and Lemma 14.30 we know (Pn) < (P'n) and
lim Urn Pn(S% < oo) = 0. (35.1)
Now define L = '°° .y = ^r>M — H. Using Ito formula, on T we
have
l-.M=1-(I^.Z+I^.z)-1-(^-^)^^-u)
= \{j:-jr)zc+(^-*)*(»-")
= \{j:-jr)-zc>p + (^-v*(»-*■»)
+\[(iz ~ ¥) h\ -{zc) + [(A"1){V™ "1)] * "• (35-2)
where ZCjP is the continuous martingale part of Z under P. It is easy to
see that on [0, Sk\ we have
l(r" idh-^ * (h+^)2-<*C) *8H- <35-3>
In/AA7- 1| < |\/A - n/A7!,
|(A- IXn/AA7- 1)1 < (V\ + 1)1 v^A - 1\\V\- v/A7!,
|(A - 1)(>/AA' - 1| * i/ < [(>/A + 1)(V\ - v/A7)2] * v
< 2(\/l + 2Jfc + 1)H, (35.4)
since 1 is between \/A and n/A7, and \y/\ — 1| < |\/A — v/A7!- Under P we
have
<\{^~ + ^rf-(ZC) + (2k + !)(^- S*)2 * " < (2 + 4k)H. (35.5)

§3. Contiguity, Entire Separation and Convergence in Variation 425
From (35.2)-(35.5) and Lenglart's inequality we obtain
(Ln)Sn^0. (35.6)
On the other hand, since (Lc)sk < 2HsK,
(L^snCo. (35.7)
By the exponential formula, on [0, S*] we have
Y = Y0S{L) = Y0exp{L - \{LC) + S(log(l + AL) - AL)}.
Noting that 0 < x - log(l + x) < x2 for |x| < - and
we have
0 < E(AL, - log(l + A^))/[|ALt|< i, < £(^)2/[|AL,l<±]
< {y/\~X - l)2 * Moo < (VX - V\')2 * Moo-
Since [(v^ - y/X'fX] * n < 2(1 + 2k)H, using Lenglart's inequality again,
we obtain
(£(AL» - log(l + AL"))/^^)^ - 0. (35.8)
For any e > 0,
[(£|log(l + AL)-AL|)Sfc>£]
C [(ALySk > \] U [(£(AL - log(l + AL))I{^L^Sk > e}. (35.9)
According to (35.6), (ALn)*Sn ^0, and from (35.7), (35.8) and (35.9) we
get
(Ln)*Sn - \(Ln>c)sz + (E| log(l + AL) - AL\)Sn ^ 0. (35.10)
Since r0n - 1 ^ 0 by i) and
Yn -1 = r0n - 1 + r0"{exp(Ln - ^(Ln'c) + E(log(l + ALn) - ALn)) - 1},
from (35.10) we have
(Yn-l)*Sn^0. (35.11)
For any given e > 0
Pn((Yn - 1)^ > e) < Pn(S% < oo) + Pn((Yn - iySn > e).

426 Chapter XIV Absolute Continuity and Continuity of Measures
Letting n —► oc and k —► oo successively, from (35.11) and (35.1) we know
At last, \\Pn - P'n\\ -»• 0 follows from Lemma 14.34. □
14.36 Theorem. Le2 P be a probability measure on f71 such that
Pn «C p" and P'n < P". Assume JTn € S(Pn) and under ?", Xn
has the weak property of predictable representation. Let the predictable
characteristics of Xn under Pn, P"1 and p" be (an, /?", un), (a'n, 0'n, i/n)
and {an,pn,vn) respectively such that
l/n = Yn. pnj yn € Vn+, [an = 1] C [an = 1],
„/n = y/n . ?n) y/ta € £n+) [gn = j] c [fl/n _ j]
Se*
An = (iT1)2 • j9" + (>/F* - v/Y^')2 * £n + SCv/T^a^ - Vl - a'")2,
u>/iere iiT71 = -i{/r".[a'n - ctn - (x/[|x|<i])] * ("'" - "")}, and
d(in
P{N) = I[NYn<Y>»] * »'n + E[(l - a'B)/[iv(i-«A')<(i-a'»)]]> * > 2.
Then (Pfn) <J (Pn) i/ and only if the following conditions are satisfied:
i) (P?) «J_P£),
ii) lim Urn P,ri(A" > N) = 0,
iii) /or any e > 0, Urn lim P'n(FL(N) >e) = 0.
N—>oon—><x>
Proof First of all, we want to calculate in(N). In fact, i(N) is the
P'-compensator of
Ir- £/. * z*Jl&z*o] = Ir-{I[NY<Y'] *M + E^i=ft i^/d^o]}'
where /z is the jump measure of X and D = [AX ^ 0]. Hence i(N) =
Ir.I(N), and actually it is also independent of the choice of P. In the
sequel, we will use Theorems 14.12 and 14.20 fully.
Necessity. By Theorem 14.20, on [Sn = oo] we have An < 8Hn. Hence
Pfn{A^ >N)< P'n(Sn < oo) + P,n(8ff£ > TV). (36.1)
Since Pn(Rn < oo) = 0, Urn P'n{Rn < oo) = 0, hence lim P,n(Sn <
71—>00 71—>00
oo) = 0. Thus by Theorem 14.31 and (36.1) we obtain ii). By the same
reason, for any e > 0, Pfn{I^(N) > e) < P'n(Sn < oo) + P'n{i^(N) > e),
and we obtain iii).
Sufficiency. On n[#n = Pt] we have Hn < An. But P/n(U[#n ?
t t
(P}) = 0. FVom ii) we obtain lim Urn P/n(#£> > N) = 0. Similarly,
N—>oo 7i—>oo

§4. Measures Induced by Levy Processes 427
since in(N) < In{N), from iii) we obtain lim fiin P"1^" (N) > e) = 0
N—*oo n—k»
for any e > 0. Then by Theorem 14.31 (P"1) < (Pn). □
14.37 Remark. Just like Remark 14.32, condition iii) in Theorem
14.36 can be substituted by the following
iii)' For alli„GF
pfn r>tn
(IAn * E[(l - a»)/pB>0]])oo p-+ 0 =* (7^ * S[(l - <)/pn>0]])oo - 0.
In fact, we have iii)=>iii)' and iii);+ii)=^iii) as well. The proof is left to
the reader.
14.38 Theorem. Under the assumptions of Theorem 14.36, \\Pn -
Pfn\\ —♦ 0 if and only if the following conditions are satisfied:
i)l|P?--Poll|->0,
ii) AS, - 0.
Proof It is similar to the proof of Theorem 14.36, but we should use
Theorem 14.35 instead of Theorem 14.31. D
§4. Measures Induced by Levy Processes
In this paragraph we apply the general results to the measures induced
by Levy processes. Assume that X = (Xt) is a cadlag process. Let
F = F°+(X) and T = \J Ft = V'^t(X). Let P and P' be two measures
t t
on T. Suppose under P or P', X is a Levy process. Then
£[eiu(Xt-X0)] = eXp{iU/t - ±u2/3t + (eiux - 1 - iuxl^^) * ut),
and
Eyu(Xt-X0)] = exp{iu/t' _ Ijft + (C<«x _ 1 _ iu^^) * 4},
where (/, /?, */) and (f,f3f,vf) are all non-random, continuous in £. P
(resp. P') is completely determined by Pq and (/,/?,*/) (resp. Pq and
(//,/3/,^/)). They are the measures induced by Levy processes. Let fi be
the jump measure of X and Xc (resp. X'c) be the continuous martingale
part of X, if X is a semimartingale under P (resp. P').
14.39 Lemma. Let g be a {non-random) Borel function on i£+ x E.

428 Chapter XIV Absolute Continuity and Contiguity of Measures
1) If / g^dv < oo, then
JR+xE
Elexpl f gdM = exp( / (e9 - l)dv\. (39.1)
L W/UxE jj lJR+xE j
2) // / 9 , xdv < oo, then
J Jr+xe1+ 9
E[exp{g * (ii - !/)«,} = exp{ / (e9 - 1 - g)dv\. (39.2)
KJR+xE }
n
Proof. 1) If g is a simple function, g = £ 0*/^, #i € #(#+) x #(£)>
i/(Bi) < oo, z = 1, • • •, n and B\Bj = 0 for i ^ j, then
tffexpj / sdM}l = tffexpj £ a^Bi)}] = ft ^exp^^)}]
L lJR+xE )j L Li=l JJ i=l
= ft exp{(e^ - l)u(Bi)} = exp { £ (e*< - l)i/(Bi)}
i=i L i=i J
= exp< / (e9 — l)dv\,
1JR+xE >
i.e., (39.1) holds. If g > 0, g can be approximated by an increasing
sequence of non-negative simple functions. Then (39.1) remains true by
the monotone convergence theorem. If g < 0, g can be approximated by a
decreasing sequence of non-positive simple fimctions. In this case (39.1) is
still true by the dominated convergence theorem (exp< / gdfi\ < 1)
J R+ x E
and the monotone convergence theorem. For a general #, since
e\ / S+^l = / ff+A/<cx>,
1JR+xE J JR+xE
I gdjji makes sense. Because / gI\a>o]d^ and / gI\a<o]du
JR+xE JR+xE w J JR+xE w J
are independent, (39.1) remains true:
= expf / [(e^>°l - 1) + (e9li9<o) _ i^dl/\
= exp{ / (e9 - l)dv\.
2) We have already known that for any b > 0

§4. Measures Induced by Levy Processes 429
For gl[\g\>b] (39.2) can be deduced from (39.1). Now we may assume
\g\ < b. There exists a constant c& > 0 such that
\ex - l-x\ <cbx2, \x\ <2b.
Choose a sequence (gn) of simple functions such that gn —► g and
\9n\<\g\, n>i, / \gn-g\2dv-*0.
JR+xE
Write <q = g*(v- u)^ r]n = gn*{n~ !/)«>. Then
E[\Vn-v\2} = / \9n-9?dv-+^
JR+xE
E[exp{Vn}] = exp{ / (e9n - 1 - gn)du\
J R+ xE
< exp|c6 / g2duj < oo, (39.3)
E[exp{2fin}] = exp{ / (e2^ - 1 - 2gn)dv\
lJR+xE }
< exp{4c6 / g2dv}
1 Jr+xE }
< oo.
lR+>
This implies (exp{7?n}) is uniformly integrable. Letting n -» oo in (39.3),
we obtain (39.2). D
14.40 Lemma. Assume X G S(P) and g is a Borel function on R+.
TOO
U I 92d(3s < oo, then g.X^ is a normal r.v. and
Jo
E[exp{g.Xcco}}=exp{-jo g2sd/33}. (40.1)
n
Proof. If g = J2 a>ihti_l tx\ is a simple function, 0 < t\ < • • • < tn <
oo, then obviously g.X^ ~ N[0, - I g2d(33J and (40.1) holds. For a
general g, the assertions follow by similar procedure of approximation as
in the proof of Lemma 14.39.2). D
14.41 Theorem. P' <£ P if and only if the following conditions are
satisfied:
i) P'0 < P0,
ii) i/ = y.i/, y g (B(«+) x B{E))+, f (i - \/F)2di/ < oo,
JR+xE
i") /?' = /3,
roo
iv) /' - / - xl[]xm * (y' -u) = K.0, K € B(R+), Jq K2sd0s < oo.

430 Chapter XIV Absolute Continuity and Contiguity of Measures
In this case, the density process of P' w.r.t. P is
+ [(logF)/[|y_1|<6]] *(ii-V) + [(l-Y + log Y)/[|y _!,<,,]] * V
+ [(l0gy)/ny_1|>j]] * H + [(1 " nV-l|>6]] * "}> (41.1)
where b G]0,1[ is a constant
Proof. Without loss of generality, we may suppose X G S(P). Since
there exists a continuous function g such that X — g G S(P), X may be
replaced by X — g, the filtration F being unchanged.
The necessity follows from Theorems 14.12 and 14.16 immediately.
Sufficiency. For all t > 0
/
Jo
1 \d[(xi[M^) * (i/ - i/)a]i < (|x(y - i)|/IW<i]) * vt
< {[(x2/[|x|<i]) * "tW - l)2/[|y-i|<6] * "t}}1/2
+(\y - i\i[\Y-n>b}) *Ut <°°
This implies /' is a function with finite variation as well and X € S(P').
Let
HP'
L = K.XC + (Y - 1) * (n - i/), Z = Z0£(L), Z0 = -^.
By the exponential formula and ALt = (Y(*> AXt) — 1) ^[AX't^o] we naye
Z = Z0exp{K.Xc + {Y-l)*{ii-v)-lK2.[3+(logY-Y+ !)*»}. (41.2)
Observing that log+y < Cft|y — l| when \y — l| > b, log2y < ct|y — 1|2
when \y — 1| < b, and |logy — y + 1| < c\,\y — 1|2 when |y — 1| < b, where
Cft > 0 is a constant, dependent on 6 only, we find
(Y-l)*(n-v) + (logy - y + 1) * m
= [(^ - 1K[|Y-1|<6]] *(/*-") + [(^ - 1)/[|V-1|>6]] * (M - ")
+[(iogy)/[|y_1|>6]] * n + [(-y + i)/[|y_i|>6]] * n
+[(iogy - y + i)/[|y-i|<6]] *(/*-") + [(logy - y + i)/[|y-i|<t]] *"
= [(iogy)/[|y-i|>6]] * m + [(iogy)/[|y_i|<6]] * (m - ")
-[(y - i)/[|y_i|>6]] * i/ + [(iogy - y + i)V-n<6]] * *■ (41.3)
Since K.Xc, [QogY)I[\Y-i\>q] * M and [(logy)/[|y_i|<6]] * (n - u) are

§4. Measures Induced by Levy Processes 431
independent, from Lemmas 14.39 and 14.40 we obtain
£?[S(Z)oo] = £[exp{AT.X£, - |A'2.)8oe}]«[exp{[(logy)JI|y_1|>j)] *Moo}]
■E[exp{[(logY)I[]Y-i\<b]] * (/* - ^)oo}
•exP{[(-r + i)i[iY-i\>b}) * "°o + [(logr - y + i)V-n<6]] * M
= exp{[e(,ogy)V-n>6] - 1 + e(logy)/iiy-n<6] _ j _ (]ogY)I[]Y-i\<q
+(-y + i)/[|y_i|>6] + (iogy - y + lj/py-in^] * «/«>} = i,
ElZoEiLU] = E[E[6 (LU^o] = 1.
Set P" = [Zof(L)oo].P. Then P" is a probability measure such that
P'q = P'0 and P" «C P. From the assumptions it is easy to check that
under P" the predictable triplet of X is just (/',/?', v'), and hence X is
a Levy process under P". Then P" = P', (41.1) follows from (41.2) and
(41.3), and we are done. □
14.42 Theorem. P' and P are not singular if and only if the
following conditions are satisfied:
i) P'q and Po are not singular,
ii) u' = Y'.V, (1 - W7)2 * Fqo < oo, where V = -{v + u'),
iii) 0' = 0,
iv) /' - / - (xIN<i]) * («/ - «/) = #./?, # € BiR+lK2.^ < oo.
Proof. Let P be a probability such that under P, X is a Levy process
andP0 = i(P0 + Po),
£[eiu(*'-*0)] = exp{iujt - ±u20t + (eiux - 1 - iuxl^]) *Vt,
where7=|(/ + A?=^ + )9')^=i(i/ + ^).
Necessity. From Theorem 14.20.1) we have conditions i), iii), iv), and
(y/Y - W7)2 *»70O < oo, where y € (B(R+) x B(£))+ such that v = Y.V.
Since y + Y' = 2, 1 is between Y and y'. Hence ii) holds:
(1 - y/Y')2 *V00 < {y/Y - y/Y1)2 *VX < oo.
Sufficiency. Observe that
(l-V/y7)2*Foo<CO
-«=► (\Y' - l|2/[|y_i|<6]) *^oo + fly' - l|/[|V-i|>t]) *F°° < °°'
where b €]0,1[. Since Y' - 1 = 1 - Y, we have (1 - v/F)2 * F,*, < co as
well. Because
/3 = /7 = £,

432 Chapter XIV Absolute Continuity and Contiguity of Measures
/-7-H|I|<1))*(^-f) = -^a:./3,
f'-7-(xI[lxl<1])*(i/-V) = ±K.0.
By Theorem 14.41 we know P < P and P' < P.
Now by Theorem 14.12 we have
H(a) = /r.p^^tf2./? + <pa(Y, Y') * v\.
Hence the Doob-Meyer decomposition of Y(a) (see Theorem 14.5) is
Y(a) = Y0(a) + M(a) - Y.(a). [a{1~a)K2.0 + <pa(Y, Y') * u].
Denote ht = E[Yt(a)] = ha(Pt, P't). Since Y(a) is of class (D),
h = h0 - h-\a{l~a)K2.p + <p<*{YX)*v\
and therefore
hQ(P, P') = ha(P0, P'0)exp{ ~ [a(1~a)IT2.flx, + <Pa(Y, Y') *«/«,]}.
(42.1)
In particular,
hi,2(P, P') = h1/2(P0t P,0)exp{-[^2./3oo + \{s/Y - W7)2 * V^}} > 0.
This implies that P and P' are not singular. □
14.43 Theorem. Assume that P and P' are not singular. Put
Nl = \-aW = °1 U [/[y=01 * M°° > °]'
N2 = [-^ = 0l U [/[y-o] * Moo > 0],
L«-r 0
N = NiU N2.
Then P(Ni) = 0, P'(N2) = 0, and on JVC iue /mt/e P' ~ P,
^ = ^eXP(KX~ " \K2fi°° + [(lo6 f)%-H>^] * ^
+ [(log jr)/[|y-l|<6]] * (M - ")oo + /[|V_i|>6] *{v- l/)oo
+ [(l - y + loS -pr)/Bv-i|<*]] * "oo}.
Proof. This is a consequence of Theorems 14.20 and 14.42. It is only
required to calculate the derivative on Nc. To this end, by making use of

§4. Measures Induced by Levy Processes 433
(41.1), on Nc we have
dP' dP' IdP dP'0 r^c r/. Y\f i
~Ip=1p/ip = dp-0exp\KX->+ K^fJ^-h^J *^
+ [( lQg jr) V-l|<»l] * (/* - F)°° + [(F - r')^[|y-l|>6]] * ^oo
Y'
+ [{Y-Y' + log y)/[|y_l|<4]] **oo}-
y
i^- 2
By Girsanov's theorem Xc = ~X° - (Xc, - — .X°) = Xc + -K.0. Besides,
[(^yHM^]*^-")
y
= [(lo6y)V-i|<t]] *(A*-")+ [(^-!)(log-pr)/[|y-i|<a]] *?•
Whence (43.1) is deduced. □
14.44 Definition. Set
if PIP',
{+00,
d(P,_ , . ,_ ,_
""-Poo + (VY - VY7)2 * Foo, otherwise.
In the sequel, we discuss the contiguity, entire separation and
convergence in variation of the measures induced by Levy processes. We only
need to add the index n to all notations.
14.45 Theorem. (P/n) < (Pn) if and only if the following conditions
are satisfied:
i) (P'0n) < (Pg),
ii) (i/») < {vn),
iii) Um d(Pn, P'n) < 00.
n—►oo
Proof. If there exists an infinite number of n such that Pn J_P/n, then
(Pn)A(P/n) and lim d(Pn,P/ri) = +00. Thus, without loss of generality,
n—*oo
we may suppose for all n, Pn and P'n are not singular. Then as in the
proof of Theorem 14.42, we have Pn < P* and P/n < P*. Hence
Theorem 14.36 and Remark 14.37 apply. It is only required to note that
A£o = d{Pn, P'n) is non-random. D
14.46 Theorem. (Pn)A(P/n) if and only if (PJ)A(P/0n) or
US d(Pn,P/n) = +oo.
n—►oo
Proo/ We may also suppose for all n, Pn and Pm are not singular.

434 Chapter XIV Absolute Continuity and Contiguity of Measures
Necessity comes from (42.1), in this case we have
0= Urn h1/2(Pn,P'n)
n—*oo
= Urn h1/2(PlP'0n)exp{-[l(Kn)2.^ + hy/Y^-VY^f *!%]}.
n—*oo o Z
Then either lim h1/2(P%,Ptf) = 0 i.e., (P%)A(P'<?), or Em" d^P"1)
n-»oo n-»oo
= +00.
The sufficiency follows directly from Theorem 14.33. □
14.47 Theorem. \\Pn - P'n\\ -+ 0 if and only if ||P£ - Pg»|| - 0
and lim d(Pn,P'n) = 0.
n—kx)
Proof. We may also suppose for all n, Pn and Pln are not singular.
Then Theorem 14.38 applies. □
9 Problems and Complements
14.1 Let P and P' be two probability measures on (fi,^7). Then
MiM*>-tof{$iWiW-: ^^(n^r6 }'
14.2 Define (by using the notations in §1)
*t(a) = e"^(Q) n [(1 " &Hs(a))eAH°M], t > 0.
s<t
Then ^(a) = 7V(a)$(a), where 7V(a) > 0 satisfies the following
conditions:
i) If T is a stopping time and 3>r_(a) > 0, then N(a)T G M\oc(P),
ii) N(a) is a P-supermartingale.
14.3 Assume Pf <&P,X e S(P). Let (a,/3,i>) and (a7,/?7,*/) be the
predictable characteristics of X under P and P7 such that vf = Y.v,Y €
V+, and [a = 1] C [a7 = 1]. Then
p\k2.^ + (i - v/F)2 * i/oc + E (vT^ - \A - <*'t)2 < oo) = i,
*>o v
where AT = Tg[JrV ~a~ (xI[\*\<i}) * ("' " ")]•
14.4 Let X be a step process with X0 = 0 and F = F°(X). Let P
and P7 be two probability measures on T = \Jt Tt. Assume that v and i/
are the Levy systems of X under P and P' respectively, and
P{y{R+ x £) < oo) = 1, PV(#+ x E) < oo) = 1.

Problems and Complements
435
Then P/ <^ P if and only if the following conditions are fulfilled:
i) 1/ = Y.v,Ye V+, and [a = 1] C [a' = 1],
ii) PXA+ x E) < oo) = 1.
14.5 Let X be a point process and F = F°(X). Let P and P' be two
probability measures on T — V Ft- Assume that under P, X is a Poisson
process with parameter A > 0 and A is the P'-compensator of X. Then
loc r*
P^<P if and only if dht < dt, At = / Asds. In this case, the density
. Jo
process of ™ w.r.t. P is
n (^)e*-\
where Tn is the n-th jump time of X.
14.6 Give an example that for each n, Pn ~ P'n, but (Pn)A(P'n).
14.7 Let fin = £1, (f71) be an increasing sequence of a-fields, T —
V J71, P and P' be two probability measures on T, Pn = P\jm, P'n =
PV„. Then 1) (Pm) < (Pn) <!=► P' < P, 2) (Pm)A(Pn) <^ P'IP.
14.8 Let Pn and P'n be probability measures on (ftn,.Fl), P" =
dP'n
l^pn + p'n), ^ = ^=fi-, Fn be the distribution law of z'n on [0,1] under
P". Then (P'n) < (Pn) if and only if for any limit point F of (F„) we
have F({1}) = 0.
14.9 Let Pn and P'n be probability measures on (^.F1),?71 =
1 dPn ldP'n
/pn + p'n)in = ^_ / __.j ^n an<j pv be the distribution law of /„
* dP I dP
on it+ under Pn and P'n respectively.
1) The following two statements are equivalent:
a) (P/n) < (Pn) and (Fn) weakly converges to a distribution law on
«+,
b) (F'n) weakly converges to a distribution law on it+.
2) If (Fn) weakly converges to a distribution law F on ii+, then
(P,n) < (Pn) <=> fxF{dx) = 1 <^ F({0}) = 0.
14.10 Let X be a continuous process with Xo = 0 and F = F^_(X).
Let P and P' be two probability measures on T = V «^i such that under
P or P' X is a Levy process. Then either P ~ P' or P±P'.
14.11 Let X be a cadlag process with Xo = 0 and F = F^(X). Let
P and P' be two probability measures on T = \/t Tt such that under P
and P' X is a homogeneous Levy process. Then either P ~ P' or PJ_PX.
Find the necessary and sufficient conditions for P ~ P'.

436 Chapter XIV Absolute Continuity and Contiguity of Measures
14.12 Let Xn be a step process with X% = 0 and F71 = F°{Xn).
Let Pn and P'n be probability measures onF,i/n and v,n be the Levy
systems of Xn under Pn and P'n respectively. Then
||^_^||C0=>||Pn-P,n||-*0.

Chapter XV
Weak Convergence for Cadlag Processes
In the last two chapters of our book, we shall discuss the weak
convergence of distributions of cadlag processes, especially, the weak convergence
of distributions for semimartingales. In this chapter, we will introduce
some fundamental facts about the weak convergence of distributions for
stochastic processes. In §1 we establish that the collection Dd of all cadlag
functions from R+ to Rd, equipped with the Skorokhod topology, is a
Polish space and the Borel a-field of Dd coincides with the a-field generated
by the canonical process on Dd. Some deeper properties of convergent
sequences under the Skorokhod topology will be discussed in §2. The
general results of weak convergence of measures on Polish space and the
conditions of tightness for stochastic processes will be given in §3. In §4
we characterize the weak convergence of step processes in terms of the
weak convergence of jump times and jump sizes. This approach is simple
and elementary.
§1. D[0, oo[ and Skorokhod Topology
15.1 Definition. For a e]0,oo], denote by Dd = D{Rd, [0,a]) the
totality of iid-valued cadlag functions on [0, a] and by Dd = D(Rd,R+)
the totality of Rd-valued cadlag functions on it+. For d = 1, denote
simply Da = D^, D = D1.
Similarly, denote by Cd = C(Rd, [0,a]) the totality of iJd-valued
continuous functions on [0, a] and by Cd = C(Rd,R+) the totality of Rd-
valued continuous functions on ii+.
15.2 Definition. For each Rd-valued function x on ii+ and A C JZ+,
define
uj(A, x) = sup{|x(s) — x(t)\ : s,t € ^4}

438 Chapter XV Weak Convergence for Cadlag Processes
(jj(6,x,a)=s\ip{bJ([t,t + S],x) • 0 <t < t + 6 < a},
=sup{|x(s) - x(t)\ : 0 < s,t < a, \t - s\ < <5},
u/(M,a)=inf|m^([^^^
(2.1)
where | • | is the Euclidian norm in Rd.
Obviously, uj(6,x,a), u/(<5, x,a) are nondecreasing in 6.
Note that (2.1) is different from the following definition of Zf(6,x,a)
in Da(cf. Billingsley [1]):
—i(c v -rj —n. . r x 0 = t0 <ti <'-<tr = a,\
uj (o,x,a) = inf < max u( [ti-i,U ,x) : . - f . - >.
\l<i<r U L mfi<i<r(ti - U-i) > 6 J
the difference comes from the following fact: for fixed a, point a plays a
particular role in Da, while in D, point a dose not play any essential role.
15.3 Lemma. 1) uf(6,x,a) < cj(2<5, x,a).
2)Ifxe Ci, thenu)(6,x,a) < 2uj'(6,x,a).
Proof, 1) For each partition {tj}o<j<r of [0,a] satisfying tj - tj-\ > 6}
j = 1, • • • ,r - 1, by adding some points (if necessary), we may assume
that tj - tj-i < 2<5, thus uj([tj-\,tj[,x) < u;(2<5,x,a), hence u/(<5,x,a) <
cj(2£, x, a).
2) Owing to (2.1), for each rj > 0, there is a partition of [0, a] satisfying
mini<j<r_i(tj - tj-i) > 6 and uj([tj-i,tj[,a) < u'(6,x,a) + 77, 1 < j < r.
Now for 0 < t — s < <5, either s, t belong to the same interval [tj-i, tj[ and
\x(t) -x(s)\ <uj([tj-utjlx) < J(8,x,a) +r],
or s,t belong to two adjacent intervals [<j-i,£j[, [<j,<j+i[ respectively and
\x(t)-x(s)\ < \x(t)-x(tj)\ + \x(tj-)-x(s)\ < 2u\6,x,a) + 2r).
In sum, cj(<5,x,a) < 2uj'(6,x,a) + 2r\. Since r\ is arbitrary, 2) holds. D
15.4 Theorem. 1) x e DjJ if and only if lim^o^A^^a) = 0.
2) x € Dd if and only if for all N G N, lims->0vf{6,x, N) = 0.
Proof 1) Due to (2.1), lim6_>oLj'(6,x,a) = 0 is equivalent to the
following fact: For each e > 0, there exists a partition {tj}o<j<r of [0,a]
satisfying maxi<j<r_i \tj — tj-\\ > 6 and
Lj{[tj-1,tj[,x)<e l<j<r. (4.1)
Necessity. For e > 0, let
r = r{e)
= sup a : u ' r
[ maxi<i<ra;( [tj-i,t3; ,

§1. Z?[0,oo[ and Skorokhod Topology
439
Since x(0) = x(0+), we have r > 0. Since x(r—) exists, [0, r[ may be
decomposed into a finite number of intervals on each of which the
oscillations of x are less than e. If r < a, while 7] is small enough so that
u7([t, r + 77[, x) < £, then [0, r + r\[ also has the above property. It
contradicts the definition of r. Therefore r = a and (4.1) holds.
Sufficiency. Owing to (4.1), x is right-continuous. If x fi D^, then
there is to G]0,7V] such that x(to—) does not exist or is infinite. So
lim*fto x(') ~~ lim^o x(*) > £o > 0. Then for this £0 (4.1) does not hold.
Hence x is cadlag on [0, N].
2)xeDd<=>xeD%, ViV G N *=> lim^o^'to*, W) = 0, VN e
N. D
15.5 Theorem, x e Dd if and only if x is the uniformly convergent
limit on each compact interval of a sequence of cadlag step functions with
a finite number of jumps.
Proof Sufficiency. Cadlag step functions with a finite number of jumps
belong to D^ for all a > 0. Therefore their uniformly convergent limits on
each compact interval belong to Dd for all a > 0, and hence x G Dd.
Necessity. By Theorem 15.4, there is a 6n such that (*/(<$#, x,N) < jj
for N G N. Let {tf} be the corresponding partition of [0,7V] satisfying
msx^iKMlt^tflx) < £. Set ^
xN(t) = t xitfLjHt?^ < t < t?) + x(N)I(t > N).
Then xjy is a cadlag step function with a finite number of jumps and
sup \xN(t)-x(t)\ < —.
t<N-l ™
Therefore x is the uniformly convergent limit of (xn) on each compact
interval. □
15.6 Definition. Put
_ J % A is a strictly increasing continuous function 1
0 "" J from ft+ to ii+, A(0) = 0, lim^oo X(t) = +oc J '
A = sup
AM-AM
t-S
A = {A:A€A0 ||A||A < oo}.
A€ A0,
1' For convenience of typesetting, here and in the sequel, we also write I(t" < t <
t,n+i) and 1(A) instead of I[t",tn [ and I a respectively.

440 Chapter XV Weak Convergence for Cadlag Processes
Denote by e the identity mapping from R+ to R+ and by A-1 the inverse
mapping of A.
Prom the above definition it is easy to deduce the following facts:
Ia = ||A-1||a, ||AoM||a<||A||a + |WIa,
sup|A(*)-*|<a(el|A|lA-l), (6.1)
t<a
sup \fi(\X(t) - t\)\ < sup *-*-*• sup \\(t) -t\< sup \\(t) - tic1
t<a s s t<a t<a
15.7 Definition. For x,y € Dd, let
||z||a = SUp \x(t)\, \\x\\ = SUp |x(t)|,
t<a t
p(s,y)=inf (||A||A+ £ 2~N(1 A \\(xkN) o \ - ykN\\)\ , (7.1)
where
fl, t<N,
kN(t) = <N + l-t, N<t<N + l, (7.2)
(o, t>N + l.
From (7.1) it is easy to verify that p(x, y) satisfies
P(x, y) > 0, p(x, y) = p(y, x), p(x, z) < p(x, y) + p(y, z). (7.3)
15.8 Lemma. Suppose p(xn,x) —► 0 as n —► oo, £/ien £/iere is a
sequence (\n) C A suc/i £/ia£
||An-e||—►O, as n —► oo, (8.1)
VNeN, \\xno Xn - x\\N -►(), asn -► oo. (8.2)
Proof. By Definition 15.7, if p(xn,x) —► 0, there is (/zn) C A such that
||Mn||A->0, asn->oo, (8.3)
VAT g iV, \\(xnkN) o p,n - xkjy\\ —► 0, as n —► oo. (8.4)
Write mn = (e^A - 1 + n"1)"1/2, then mn -> oo by (8.3). Set
^ (/\ = \V"n(t), t < mn,
\t - mn + nn(mn), t>mn.
Then An G Ao, and from (6.1) we know
||An - e|| = ||/xn - e||m„ < mn(e^«^ - 1) < — - 0. (8.5)
mn

§1. £>[0, oo[ and Skorokhod Topology 441
Thus (8.1) holds. For each fixed iV, if n is large enough, from (8.5) and
(8.4) we have
\\xn ° An - x\\N<\\(kN+iXn) O \n - kN+ix\\N
<\\{kN+ixn) ofin- fc^+ix|| -► 0.
Thus (8.2) holds. D
Remark. From the above proof it is easy to know that if p(xn, x) —► 0,
then there is a sequence (An) C Ao such that
||An - c|| —> 0, as n—► oo,
VN eN II xn — x o An||7v —► 0, as n —► oo. (8.6)
15.9 Theorem, p is a distance on Dd.
Proof. Due to (7.3), it suffices to deduce x = y from p(x,y) = 0.
Suppose p{x,y) = 0. By Lemma 15.8, there is (An) C A such that (8.1)
holds and ||x oAn- y\\x —► 0, VAT G N. If x is continuous at £, i.e.,
Ax(t) = 0, then
\x(t) - y(t)\ < \x(t) - x(Xn(t))\ + \x(Xn{t)) - y(t)\ -> 0,
thus x(t) = y(t) at every continuous point t of x. By Theorem 15.5 the
set of discontinuous points of x is at most coimtable and the continuous
points of x are dense everywhere. Hence x = y. □
15.10 Theorem. Suppose {x,xn,n > 1} C Dd, then the following
statements are equivalent:
1) p{x,xn) -► 0,
2) There is (An) C A and (8.1), (8.2) (or (8.1), (8.6)) hold,
3) For each N G N, there exists (A^) C Ao such that as n —► oo
HA? - e||*-> 0, (iai)
||*noA*r-a:||iv->0 (or \\xn-xo\%\\N^0).
Proof. 1) =^2) is the conclusion of Lemma 15.8. 2)=> 3) is obvious.
3)=> 1): At first, we will prove that for each fixed TV there exists a
sequence (//?) C A such that (for simplicity we suppress the superscript
N)
||/*n|U - 0, (10.2)
lim ||x„ - x o Hn\\N < -—. (10.3)
n—*oo JiIS
Let (^) be the sequence defined as follows:
*o = 0,

442 Chapter XV Weak Convergence for Cadlag Processes
" \+oo,
Then tk —► oo, as x G Dd. Set
_ ,^L„ >tfc: \x(t)-x(tk)\ > ^}, iftfc<oo,
Vn(0 " \£(N)
Xnit)~*\(N) + t-N, t>N.
Unk = A~ (tk) fin(Unk) = h)-
Let
{tk + (t - unk)u**k^*nk, unk<t< unyk+i A TV, unM\ < oo,
tk + t - unk, unk<t< uUyk+i = oo, t < TV,
Hn(N)+t-N, t>N,
then /in is piecewise linear, \in G A and by (10.1)
Unfc -► tk, ||/in||A -► 0, ||^n - e|| = ||/in - e\\N -► 0 as n -► oo,
i.e., (10.2) is true. On the other hand, if t G [unk,unyk+i[r\[0,TV], then
An(t)> M*) € [*fc,*fc+i[and
|x(An(t))-x(Ain(0)l<l/(2iV),
M*) - x(AXn(*))l < MO " *(An(*))l + MAn(*)) " x(/in(«))|
<||xn-xoAri||7v + l/(2TV),
fim \\xn -xo Hu\\n < 1/(2TV).
n—>oo
i.e., (10.3) holds.
Secondly, for all TV G TV, if (//^) satisfies (10.2), (10.3), then there is
an increasing sequence [tin) such that for n > tin
ll/^IU < 1/iV, \\Xn-XO^\\N<l/N,
Now take ~pn = /z^, n;v < n < nw+i, then
^11^^ = 0, (10.4)
Urn \\xn - x o /in||jv = 0, VTV G TV.
Hence for fixed TV G TV, if n is large enough, we have
\\kNxn- (kNx)ofln\\
< \\kNXn- kN(xoJLn)\\ + \\kN(xojIn)- (kNx)oJ[n\\
< \\xn-xo]in\\N+1 + \\kN - kNojin\\\\x\\N+2
< \\xn -XO /Zjltf+i + ||Pn - c|U+l|k||iV+2-
Using the above inequality and (10.4), it is easy to get p(xn,x) —* 0. □

§1. D[0, oo[ and Skorokhod Topology 443
Remarks. 1) For x,y eDd , let
oo
p(x,y)= inf {||A-e||+ £ 2~N(1 A \\(xkN)o \-ykN\\)}. (10.5)
A€Ao N=l
Then it is also a distance on Dd. According to Theorem 15.10, p and p
define the same topology in Dd. This topology is called the Skorokhod
topology in Dd.
2) Dd is a linear space, but it is not a topological linear space under p
(orp).
oo
15.11 Example. Suppose xn{t) = £ a?I(t? < t < t^+1), where
i=0
oo
to = 0 and t% | oo as k —► oo, x(t) = J2 aj(ti < t < ^+i), where to = 0
and tk T °° QS fc —► oo, i.e., xn, x are step functions. If
lim t? = *i, i > 1, (11.1)
n—*oo
lim af = aj, if t» < oo,
n—»oo
then it is easy to verify that limn-_oo p(#n> #) = 0.
In fact, for TV e N,\ltk < N < tk+i, take
xN(t) = itj + tt^{t~t^ tl^t<t^^k-1^
\tk + t-tnk, t>tnk,
then A^ G A. Using (11.1), if n is large enough, we have
Un ~ 4n < max |tj - t]\, \\xn -xo \%\\N < max^ \c$ - aj\.
Hence (10.1) and (8.6) hold, and by Theorem 15.10 we have p(xn,x) —> 0.
15.12 Theorem. 1) The Skorokhod topology is weaker than the
topology induced by uniform convergence on each compact interval.
2) If p(xn,x) —> 0 and x is continuous at t^, then xn(to) —► x(to).
3) If x € Cd, then p(xn,x) -*0i/ and only if
\\x - xn\\a-* 0, Va>0. (12.1)
Proof 1) If ||x - xu\\n -* 0, V7V G iV, then taking An = e, from
Theorem 15.10 we get p(xn,x) —♦ 0.
2) Suppose (An) C A and (8.1), (8.6) are satisfied. Then
\x(t0) - xn(t0)\ < \x(t0) - x(\n(t0))\ + \x(\n{t0)) - xn(t0)\. (12.2)
Due to the continuity of x at to and limn_>oo An(£o) = £o> the first term on
the right-hand side of (12.2) converges to 0. Owing to (8.6), the second
term tends to 0 also.

444 Chapter XV Weak Convergence for Cadlag Processes
3) By 1) it suffices to prove that (12.1) is necessary. Suppose (An) C A
and (8.1), (8.6) are satisfied. Since
\\x ~ xn\\a < \\x ~ X O \n\\a + \\X o\n- Xn\\a
< ^(11 An - e||a> x, a + ||An - e||a) + \\x oAn- xn||a> (12.3)
by the uniform continuity on compact and (8.6), we know that the right-
hand side of (12.3) tends to zero, and hence (12.1) is true. □
15.13 Remark. For x,y G Cd set
00 AT
Pu(x,y)= E 2-N(lA\\x-y\\N).
N=l
Then the ^-convergence is equvalent to uniform convergence on compact
and it is easy to verify directly that Cd is a Polish space under pu.
15.14 Lemma. For x G Dd set
i„(t) = x(^An), (14.1)
n
where [a] is the integer part of a. Then \imnp(xn,x) = 0.
Proof. It is obvious that xn G Dd. For given e > 0, take N and 6 > 1/2
such that
2~N < e/4, and u/(<5, x, TV + 1) < e/4.
If {tj, 1 < j < r + 1} is a partition of [0, N + 1] satisfying tj - tj-\ > 6,
1 < j < r, U > N + 1/2 and
LJ([tj-1,tjlx)<e/4. (14.2)
Take n>n0 = aV(8/e:<5) V4/6 and set s? = -[-ntj]/n, then 0 < s^-tj <
1/n, s£ > TV. Let An be the following piecewise linear function:
wt) = Uj ~{t- tj)'i££, tj<t< tj+1, j < r - 1,
\*-*r + s?, t>tj.
Then
||A„||a < sup
j<r
log^
s< sj-l
s i"*1 - 3» * s < £•
tj - tj-i
While t G [tj-i,tjl Xn{t) e [s?-V8?[ and xn(Xn{t)) G {x(s) : s €
[tj-i,tj[}. Hence |xn(An(t)) - x(t)\ < e/4 by (14.2). Therefore
Ikn ° An - x||jv < e/4,
p(^n,x)<||An||A+ E2"Ac(lA||XnOAn-x||Jt)<- + -+ £ 2"* < C,
fc=l * 4 Jfc=W+l
so the claim is true. D

§1. D[0, oo[ and Skorokhod Topology 445
15.15 Lemma. Dd is separable under the Skorokhod topology.
Proof. Set
j x is a step function with 1
eDd * l
-(■
a finite number of jumps J '
-{■
. the jump times of x and the 1
' values taken by x are all rationals J
It is easy to know that C is countable. If C denote the closure of C in Dd,
then by Example 15.11 we obtain A CC C Dd. Meanwhile, Theorem 15.5
means A = Dd, hence C = A = Dd and Dd is separable. D
15.16 Lemma. Dd is complete under p.
Proof. Suppose that (xn) is a p-fundamental sequence. It must include
a subsequence (yi = xnnl > 1) such that
p(y*,y/+i)<2-2/, />i.
Hence there exists a sequence (A/) C A such that
||Ar1||A = ||AH|A<2-2',
IIVI o A* - M+1||, < \\(ki+lyi) o\i - kl+1yl+1\\ < 2/+12"2' = 2"'+1.
Set
/W"\t-/ + A,(Z), t>J,
Then for (A/) we still have
||Ar1||A = ||Ai||A<2-2',
\\yio\i-yi+1\\l<2-l+1, (16.1)
thus (6.1) and (16.1) yield
IIAf1 " e\\ = \\\i - e\\ = ||A, - e||, < l(e"x'^ - 1) < 2"',
IIAii+i o AfA o • • • o Af1 - Afi o • • • o Ara|| = 1^+1 ~ e|| < 2~^+1K
Hence for each / there is a nondecreasing continuous in such that
Urn ||A/-1fco...oAj-1-W||=0.
We have
K+k ° • • • ° Arx(<) - K+k ° • • • ° KH*)
log
t-s
<iiAr+1fciu + --- + iiAr1iiA<2-2('-1)
<HAfio...oAriiiA

446 Chapter XV Weak Convergence for Cadlag Processes
Let k —► oo in the above inequality, we get ||/i/||A < 2~2^l~l\ So fn G A.
By the definition of p,\ and (16.1) we have
W = Mz+i ° Aj"1, //j"1 = A/ o fif+1,
\\yi o /ij"1 - M+1 o Aif+ill,-! < llM ° A« - M+iH, < 2-'+1.
Hence (j// o /xj"1) is a fundamental sequence in the topology induced by
uniform convergence on compact, and there exists x G Dd such that
||W o /.f1 - x||i < 2"'+2.
Now applying Theorem 15.10, we have Um/_00p(y/,x) = 0, and
furthermore, limn^00p(xnix) = 0. Therefore Dd is complete. D
15.17 Theorem. Dd equipped with the metric p is a Polish space.
Proof. This is a direct consequence of Theorem 15.9, Lemmas 15.15
and 15.16. D
15.18 Theorem. // we denote by V the Borel a-field of Dd equipped
with the Skorokhod topology and
Poo = a{Xt : Xt(x) = x(t),x £Dd,te fl+},
i.e., Vqq is the a-field generated by the canonical process on Dd, then
Poo = V. (18.1)
Proof. Let g be a bounded continuous real function on R and for
fixed t write hk(x) = k ft* ' g(x(s))ds. While p(xn,x) —► 0, we have
g(xn(s)) —♦ g(x(s)), except for at most a countable number of s, and
g o xn is uniformly bounded. Hence hk{xn) -» /ifc(x), i.e., hk is a
continuous function on Dd. Thus hk G V. Since x G Dd is right-continuous,
limjt—oo hk(x) = g(x(t)). Then for fixed £, g(x(t)) is a D-measurable
function on Dd. Therefore by the monotone class theorem we have V^ C
V.
For x, y G Dd set
/ x /Ml x / x /Ml
*n(0 = *(— A n), yn(«) = y{l—± A n).
n n
Then xn is determined by {x(^),k < n2}. For fixed z G Dd, define
ff(x) = p(xn,z) = h (x(-J,0 < fc < n2) ,
where /i is a function on Rn +1. Obviously,
'lfc\ /fc>
lff(z) " ff(v)| = \p(xn, z) ~ PiVn, z)\ < max
0<k<n2
'(;)-'(=)

§1. P[0,oo[ and Skorokhod Topology 447
Thus for x G Dd, g(x) is a continuous function of {x(£),0 < k < n2}.
Then for fixed 2, as a function of x, p(xn, z) G Poo- Now by Lemma 15.14
we have
p(x,z) = lim p{xn,z) G Poo-
n—>oo
Furthermore,
0(z,e) = {x : p{z,x) < e} G Poo-
Since Dd is separable, every open set in Dd is Poo-measurable. Thus
P C Poo- O
15.19 Definition. On Dd set
P° = a(x(u) : u < t), P° = V A°, £>° = (A%>0-
t>o
(Dd, P°) is called the canonical measurable space on Dd. If there is a
probability P on (Dd,P°), let
3>t t
then the $£> = (Dd, P, D, P) is called the canonical probability space with
filtration or canonical filtered probability space. Recall that the stochastic
process {Xt)t>o defined by
Xt(x) = x(t), xeDd, te ii+,
is called the canonical process.
15.20 Lemma. 1) For x G Dd, set
<j>\{t,x) = X*(t) = SUp|x(s)|,
s<t
fafax) = sup|Aa;(s)|.
s<t
Then for fixed t, (f>\ and $2 are upper semi-continuous functions of x in
Dd, that is
(t>i(t,x)> Urn <t>i{t,y), i = l,2.
p(x,y)->0
Furthermore, if Ax(t) = 0, then <f>{ is continuous at x, i = 1,2.
2) uf(6,x,N) is an upper semi-continuous function of x.
Proof. 1) Since x G Dd is right-continuous in t, so are (j>\ and fa. If x
is continuous in t, so are (j>\ and fa.
At first, we suppose p(xn, x) —♦ 0 and x is continuous at t. Then there
is a sequence (An) C A such that
IIAnlU - 0, ||An - e\\N -* 0, ||xn -10 Xn\\N -* 0, V7V G AT.

448 Chapter XV Weak Convergence for Cadlag Processes
Meanwhile,
K(t) - X*(t)\<\x'n(t) - X*(K(t))\ + \x*(\n(t)) - X*(t)\
<\\xn -xo \n\\N + \x*(\n(t)) - x*{t)\, t < N.
Thus <f>i is continuous at x. In the general case, take e > 0 such that x is
continuous at t + e, then limn-^ x* (t + e) = x*(t + e). Since x* is right
continuous in £, letting t + e tend to t along the continuous points of x,
we have
x*(t) = lim x*(t + e) = Urn lim x*(t + e) > Kin x*(t).
Thus 0i is upper semi-continuous. Similarly, 02 has the same property.
2) For e > 0, there is a partition {tj}i<j<r satisfying
tj - tj-i ><5, 1 < j < r - 1, (20.1)
LJfttj-utjlx) <uj'(6,x,N) + £/2. (20.2)
Now take rj > 0 satisfying <5 + 27? < tj — tj-i, 1 < j < r — 1, rj <
| A (AT — £r_i). If p(x, y) < r]2~N, then there exists AG Ao such that (see
(10.5))
||A_e|| = ||A-i_e||<7?,
\\xo\-y\\N<ri. (20.3)
Set Sj = \~l(tj), then {$i,•••, sr-iyN} is a partition of [0, TV]. By (18.1)
|5j-5j_l| > -|5j-A(5j)| + |ti-^_i|-|A(5i_i)-5i_i| > (5, 1 < j < r-1.
Owing to (20.3) and (20.2), we have
u([sj-i,Sjly) < S7([A(«j_i), A(sj)[,x) + 2t?
< u/(<5, x, TV) + e/2 + 2t? < c/(<5, x, TV) + e, l<j< r.
This means u/(<5, y, TV) < u/(<5,x,TV) + e, as p(x,y) < r]2~N. Therefore
u/(<5, x, TV) is an upper semi-continuous function of x. □
15.21 Lemma. Suppose F is a relatively compact set in Rd and
{{x{t) : t > 0} C T, and x is a step \
x G D : function for which the lengths of intervals > ,
between adjacent jump times are > 6 I
then H(T,6) is relatively compact in Dd.
Proof It suffices to prove that each sequence (xn) C i/(T,<5) has
a convergent subsequence. Denote by tk(xn) the fc-th jump time of xn.
Using the diagonal method we may choose a subsequence (yn) of (xn) such
that for each £, {tk{yn)) satisfies one of the following conditions:
(a) limn_00^(yri) = sk < oo and lim^oo yn(tk(yn)) = ak,
(b) limn^oo tk{yn) = sk = oo.

§1. £>[0, oo[ and Skorokhod Topology 449
If tk-i{yn) < oo, then tk(yn) - tk-i(yn) > 6. Thus sk - sk-X > 6 while
sk-i < oo. Set
A:
It is easy to verify directly that p(yn,y) -♦ 0 and H(T,6) is relatively
compact. D
15.22 Theorem. Under the Skorokhod topology a set K C Dd is
relatively compact if and only if the following conditions hold:
sup ||x||at <oo, VTVgTV, (22.1)
xeK
lim sup u/(<5, x, TV) = 0, VTV G TV. (22.2)
Proof Necessity. For fixed TV G TV, by Lemma 15.20, 0i(x) = \\x\\n
is an upper semi-continuous functional on Dd, hence it is bounded on
compact K and (22.1) holds.
Also for fixed TV, owing to Lemma 15.20, u/(<5,x,TV) is upper semi-
continuous in x and is nondecreasing in <5, lims-^o^'iS^x^N) = 0. Hence
by Dini's theorem u/(<5, x, TV) uniformly converges to zero on compact K
as 6 -> 0, so (22.2) is true.
Sufficiency. For each TV G TV, take Tn = {#($) : x G if,s < TV},
then by (22.1) Tw is a relatively compact set in Rd. From (22.2), there is
f*N < 1 such that
sup J{6N, x, TV + 1) < -j-. (22.3)
Using the notations of Lemma 15.21, write Kn = H(TN,6]y), then Kyv
is a relatively compact set in Dd. From (22.3), for x e K, there exists a
partition {tj}i<j<r+i of [0,TV + 1] such that
tj - tj-x ><5, 1 < j < r, tr>N,
57([tj_i,*i[,x)< —, 1<J <r + l.
Set A = e G A,
2/(0 = £ *(«j-i)/(tj-i < * < ^) + x(tr)/(t > tr),
then y G lf/v and
oo 12
P(x,y)< E2-n(lA||x-y||n+1)<-+ £2"'<-.
n=l JV ,>w JV

450 Chapter XV Weak Convergence for Cadlag Processes
Thus x G K2JN = {z : p{z,KN) < £}. Therefore K C K2JN for all N
and
k c n k%
N>1
2/N
Meanwhile fliv>i ^n 1S compact, so K is relatively compact. □
15.23 Definition. For x € Dd, define
u,"(6,x,N) = sup ||x(t) - x(h)\ A |x(*a) - x(t)\ : ° " ^Vt^s' ^1
(23.1)
15.24 Theorem. Under the Skorokhod topology a set K C Dd is
relatively compact if and only if the following conditions hold:
sup \\x\\N < oo, V7V G AT, (24.1)
limsupu;([0,<5[,x) =0, (24.2)
lim sup <j"(«, x, TV) = 0, ViV G AT. (24.3)
Proof It suffices to prove that if (22.1) (i.e., (24.1)) holds, (22.2) is
equivalent to (24.2) and (24.3).
Necessity. If N > <5, u;([0,<5[, x) < lj'(6,x,N), so (24.2) is necessary.
For given e,6 > 0, there is a partition {sj}i<j<r of [0,7V] such that
Sj+i - Sj > <5, 1 < j < r - 2, 5r_i > N - 1 and Zj([sj,Sj+i[,x) <
u'(8,x,N)+e. Now for 0 <tj<t< tj+\ < 7V-1, fy+i-fy < 6, at least one
of [tj,t[ and [£,fy+if is included in some [s^, S{+i[. Thus u/'(<5, x, TV — 1) <
uj'(6,x,N) + e. Therefore (24.3) may be deduced from (22.2).
Sufficiency. For given e > 0, take 6 > 0 such that
u/'(£,x,7V) < e, u7([0,<5[,x) < e, Vx G tf. (24.4)
Now we will prove
<-/(«/2, x, iV) < 6e, Vx G X. (24.5)
At first, for t\ < s < t2 < h + 6 it must be that
u([t1,s],x)ALJ{[s,t2],x)<2e. (24.6)
In fact, for t\ < r\ < r2 < s if |x(ri) - x(t2)\ > e, then by (24.4) for
^1,^2 € [5,^2] we have \x(ai)-x(r2)\ < e, i = 1,2 and |x(ai)-x(<72)| < 2e.
Hence (24.6) holds.
Secondly, from (24.6) we may get |Ax(ti)| A|Ax(t2)| < 2e, as \t\—t2\ <
6. Now take a sequence (sj) such that
6/2 < Sj+i — Sj < 6 and |Ax(s)| < 2e, as s £ {sj}.

§2. Continuity for Skorokhod Topology 451
Finally, for each fixed j set
a\ = sup{s : lJ([sj-i,s[,x) < 2e}, o2 = inf{£ : uj([t,Sj[,x) < 2e}.
Owing to (24.6), we have a\ > a2. Now if a\ < Sj,
uj([sj-i, Sjl x) < u([sj-i, ailx) + \Am(°i)\ + v(Wu*jl *)
< 2e + 2e + 2e = 6e, (24.7)
If (T\ > Sj, then by the definition of o\ (24.7) is also true. Thus (24.5)
holds, and hence (22.2) may be derived from (24.2) and (24.3). D
15.25 Corollary. Suppose L C Dd and
sup||x||tv < oo, VAT g N.
xeL
Then L is not relatively compact if and only if there exists a sequence
(xn) C L such that at least one of the following conditions holds:
a) There are {t\), (t^) such that
lim tln = 0, lim xn(tln) = au i = 1,2,
n—►oo n—►oo
and a\ ^ a2,
b) There are t\<t^< ij* such that
Um tln = t < oo, lim xn(tln) = a*, i = 1,2,3,
n—>oo n—>oo
and a\ ^ a2, a2 ^ a$.
§2. Continuity for Skorokhod Topology
Throughout this paragraph, the convergence of (xn) to x in Dd always
means the covergence under the Skorokhod topology (unless otherwise
stated), and it is denoted simply by xn —► x.
15.26 Lemma. Suppose D . Then for each t > 0 there
exists a sequence (tn) such that tn —> t and
lim lim sup \xn(s) — x(t)\ = 0, (26.1)
*lon-ootn<a<tn+0
lim Em sup \xn(s) - x(t-)\ = 0. (26.2)
*1° n^°° tn-6<s<tn
In particular,
Xn{tn) - X(t), Xn(tn-) -► x(t-), (26.3)
Axn(tn) - As(t), (26.4)
lim lim p([tn-6,tn + 6],xn)- |Aar(t)M =0,
610 n—*oo
lim Um uJ([tn - 6,tn[yxn) Vu([tn,tn + <5],xn) = 0. (26.5)
£|0 n—►oo

452 Chapter XV Weak Convergence for Cadlag Processes
Besides, set yn(s) = xn(s) - Axn(tn)I(s > tn), y(s) = x(s) — Ax(t)I(s >
t), then yn -> y.
Proof. Suppose that (Xn) C A satisfies (8.1) and (8.2). Take tn =
\n(t), then A"1^) G [M + 6fn], while s G [tn,tn + <5], where 6 > 0 and
&n = K^ttn + 6) - A"1^). According to (8.1), if n is large enough,
tn + 6 < t + 2(5, 6'n < 26 and for s G [*n, tn + 6]
\Xn(s) ~ X(t)\ < \xn(s) - X(X-\S))\ + MX-'is)) - X(t)\
<\xn(s)-x(X-1(s))\+ZJ(%t + 6fnlx)
< \\xn -xoX-1 \\t+2S + w([t,t + 26], x),
now by (8.2) and the right continuity of x we get (26.1). Similarly, (26.2)
holds also. (26.3)-(26.5) can be deduced directly from (26.1) and (26.2).
Finally, we have
MM*)) - y(*)\ = MM*)) " X(S) + (**n(tn) ~ Ax(t))I{s > t)\
< K{K{s)) - x{s)\ + \Axn(tn) - Ax{t)\.
Hence from (26.4) we know that (Xn) satisfies (8.1) and (8.2) for (yn) as
well, and yn —» y. □
Remark. From (26.5) it is easy to know that if Ax(t) ^ 0, then
the (tn) satisfying (26.4) is essentially unique, i.e., if t'n —♦ t and limn_>oo
Axn(t'n) ^ 0, it must be t'n = tn while n is large enough. But if Ax(t) = 0,
then (26.3) and (26.4) hold for every (tn) satisfying tn —♦ t.
15.27 Theorem. Suppose that xn —► x, yn —► y and for each t > 0
there exists a sequence tn —* t such that Axn(tn) —* Ax(t) and Ayn(tn) —►
Ay{t). Then
xn + yn-+x + y, (27.1)
(xn,Vn)-+(x,y) (in D2d). (27.2)
Proof. It suffices to prove that (xn + yn) is relatively compact, because
the convergence of (xn + yn) at the common continuous points of x and y
guarantees the uniqueness of limit points of (xn + yn).
Since xn —♦ x, yn —♦ y we have supn ||xn + yn\\N < oo for all N G N.
If (xn + yn) is not relatively compact, then one of a) and b) in Corollary
15.25 holds. If a) holds, there are tln —♦ 0, i = 1,2, such that
lim (xn + yn){tn) ^ lim (xn + yn){tn).
n—*oo n—>oo
But limn_00 xn(tn) = x(0), limn_00 y(tln) = y(0), therefore a) is
impossible.
If b) holds, there are tn < t\ < tn, tn -♦ t, and (xn + yn){tn) -* (k,
but a\ ^ a,2 ^ as. Let (£n) be the sequence in the assumption. Now by

§2. Continuity for Skorokhod Topology 453
Lemma 15.26 and its remark, (xn), (yn) satisfy (26.5), and for (xn + yn)
we get
Urn lim uJ([tn - <5, tn[, xn + yn) V aJ([tn, tn + <5],xn + yn) = 0. (27.3)
Now there are an infinite number of n, then such that either t„ < t or
*n ^ *• For *^e f°rmer case we have
lim lim u^([^n — d, tn[, xn + yn) > |cii — c^21 ^ 0.
6[0 n—>oo
For the latter case we have
lim Em S7([tn, £n + <5], xn + yn) > \a2 - a3| ^ 0.
6[0 n—>oo
These contradict (27.3). Hence b) is impossible also. Therefore (xn + yn)
is relatively compact and (27.1) holds.
Write x^ = (xn,0) G D2d, jm = (0,yn) G D2dJbhen^£; -♦ (x,0) = x,
2M -♦ (0,y) = y. Now by (27.1) we get (xn,yn) =x; + y^ -*x + y = (x,y).
D
15.28 Corollary. Suppose xn -» x G Cd, yn -» y, J/ien xn + yn —►
x + y, {xn,yn) -♦ (x,y).
Proof. Since x G Cd, hence xn(£n) —♦ x(£) for every (tn) converging to t.
Thus from Lemma 15.26 we know that (xn), (yn) satisfy the assumptions
of Theorem 15.27 and the claims are true. □
15.29 Definition. For x G Dd define
J(x) = {t > 0 : Ax(t) ^ 0}, (29.1)
U(x) = {u > 0 : \Ax(t)\ = u for some t}. (29.2)
Then J(x), the collection of all discontinuous points of x, is at most
countable.
For u > 0, denote
t°(x,u) = 0,
tv{x,u) = inf{t > tp{x,u) : \Ax{t)\ > u},
xu(t) = x(t) - E Ax{tP(x,u))I(t > t*(x,u)).
tp(x, u) is the p-th jump time of x with the norm of jump size greater than
u. Because x G Dd, limp_oo tp{x, u) = oo.
15.30 Theorem. For u > 0 and p > 1, £/ie following mappings on
Dd.
/i(x) = <p(x,u), /2(x) = x(*p(x,u)),
/3(x) = x(ip(x,«)-), /4(x) = Ax(*p(x, u)),

454
Chapter XV Weak Convergence for Cadlag Processes
iftp(x,u) < oo, and
are continuous at x while u fi U(x).
Proof. Let xn -► x, u $ U(x). Write tp = tp{xn,u), tp = tp{x,u).
We will prove the continuity of /i, 1 < i < 4, by induction on p. It is
apparent for p = 0.
Assume that the continuity of /», 1 < i < 4, has been established for
p. Then lim^^ *p+1 > lim^oo tp = tp. If tp = oo, limn_oo t^1 = oo =
£p+1 immediately. Below we assume tp < oo. If Um^^ t£+x = fp, then
there is a subsequence (n') satisfying t^" —♦ £p, meanwhile Axn/(££7 ) >
u. By the remark of Lemma 15.26, tf£ = t^f while n' is large enough.
This contradicts tip1 > £p, therefore linan_KX>i£fl > *P- 0n the other
hand, for any closed J C]ip,ip+1[, supt€/ |Ax(£)| < u. In view of Lemma
15.20.1),
lim supjc/ |Axn(t)| < u.
n—*oo
Hence lim^^ tip1 > £p+1. Again by Lemma 15.26 and u £ U(x) we have
£H _ tp+1 and xniq*1) - x(^1), inltf1-) - *(*p+1-), Ax^1)
-* Ax(*p+1) if tp+x < oo. This means for all p > 1, /i(x), 1 < z < 4, are
continuous at x.
Using the above results and Lemma 15.26.3), it is easy to prove by
induction that for q > 1
*?(.) = *„(•) - £ Axn(«)/(. > «)
- x(-) - £ Ax(*p)J(- > tp) = xuq.
p=l
On the other hand , x% = x%q on [0, t%[ and xu = xn<? on [0, £*[. Meanwhile
for each N e N, while n, g are large enough we have t% > N, tq > N.
Thus x£ -♦ xw, due to Theorem 15.10. □
15.31 Corollary. Suppose that g is a continuous mapping from Rr
to Rh and for some u > 0,
g(x) = 0, |x| < u.
Write
*(t) = E j(Ax(5)),
s<t
then x i—► (x, x) is a continuous mapping from Rd to R*+h.

§2. Continuity for Skorokhod Topology 455
Proof. Take a positive u £ U(x) such that g(x) = 0 while |x| < u.
Suppose xn —♦ x. Using the notations of Theorem 15.30, write
*£ = E g(Axn(t%))I(t > %), 5* = £ g(Ax(t'))I(t > ?).
p=l p—\
Then, similarly to the proof of Theorem 15.30, we have
x£ -» x*7, xn -» x, as n -» oo.
Moreover, since the jump times of Xfi, x are that of xn, x respectively,
from Theorem 15.27 we get (xn,xn) —► (x,x) and hence the claim holds.
D
15.32. Recall that if x is a step function, then it has the following
canonical representation:
x(t) = £ x(U)I[tai+A(t), t>0,
i>0
where i) 0 = to < t\ < - - - < tn < - - -, tn ] oo,
ii) U < oo => ti < £i+i,
iii) U < oo <=> x(U) ^ x(U-i), i > 1.
If n = inf{fc : ^ = +00} < 00, then for k > n, x(£fc) = x(£n_i). fy,
j > 1, are the jump times of x.
15.33 Theorem. Suppose that x,xn, n > 1, are step functions, (tj),
(ft) are the jump times of x, xn respectively, to = t% = 0. Then the
following statements are equivalent:
1) For all j > 1
f(t^xn(t^)) - f(tj9x(tj))> as n-> 00 (33.1)
where /(£, x) is a function on ii+ x R to ii+x] — |, |[ defined by
/(*,*) = (i,5E^£). (33.2)
2) xn —+ x as n —+ 00, and /or all N > 0
inf{|Axn(t?)| : 0 < t7] < 7V,n > 1} > 0. (33.3)
Proof It is apparent that for / defined by (33.2), (33.1) is equivalent
to the following fact:
tni - U, i > 1, (33.4)
xn(t?) -> x{U), i e{j>0 : tj < 00}. (33.5)
Meanwhile (33.3) is equivalent to the following fact: For all N > 0, there
exists en > 0 such that if 0 < t" < N then
\&xn(t])\ > eN. (33.6)

456
Chapter XV Weak Convergence for Cadlag Processes
1) => 2). Define a piecewise linear function \n(t) as follows:
K(t) = t» + (t- $)2^f, t] < t < $+1.
tj+i - tj
Then by (33.4) and (33.5) it is easy to verify that for each N > 0,
\\K - e||* - 0, \\xn o \n - x\\N - 0.
Hence according to Theorem 15.10, p(xn,x) -» 0. On the other hand, x
has at most a finite number of discontinuous points in [0, N] and
min |Ax(*?)l > 0. (33.7)
o<tj<Nl w" '
(33.3) is deduced from (33.4),(33.5) and (33.7).
2) =► 1) For TV > 0, let eN satisfy (33.6). Take 6 < eN A inf{|Ax(^)| •
0 < tj < N}. Using the notations of (29.3), write tj = tj(x,6), tj =
V{xn,8). Then we have t™ < N when tj < N and n is large enough.
Hence Theorem 15.30 yields
t] = tj(xn,6)-+tj(x,6) = tj,
xn{t]) =xn(tj{xnJ)) -+x{ti{x,6)) =x{tj).
Since N may be an arbitrary positive number, (33.4) and (33.5) hold, and
thus 1) is true. □
Remark. If (33.3) does not hold, then (33.1) cannot be derived from
xn -* x. For example, x(t) = 0, xn(t) = ^/[i/n,oo[(*)> p{xU^x) -♦ 0. But
tf = ^ ti = +oo and (33.1) is not true.
§3. Weak Convergence and Tightness
15.34 In this paragraph we suppose that 5 is a Polish Space, i.e., there
is a distance p on S such that under p, S is a completely separable metric
space. Denote by B = B(S) the Borel a-field of S. Set
Cb(S) = the collection of all bounded continuous functions on 5,
CU(S) = the collection of all bounded uniformly continuous functions
onS.
V(S) = the collection of all probability measures on (5, B).
For / G Cb(S) denote
11/11= sup |/(x)|.
xes
Then || • || is a norm, under which Cb(S) is a Banach space and CU(S) is
separable. For / G Cb(S) and p, G V(S) denote
M/) = / f(x)tM{dz).
Js

§3. Weak Convergence and Tightness 457
Recall that for //, v G V(S), if p(F) = v(F) for all closed F, then
\i = v\ if /x(/) = i/(/) for all / G CU(S), then /z = 1/ as well.
15.35 Definition. Suppose that /in,/z G P(«S), if
Jtin^MnCf) = M/), V/ G C6(S), (35.1)
then we say that (y^) weakly converges to /z and denote it by fin —* /z.
15.36 Definition. Let /z G ^(S), / be a mapping from S to another
Polish space S' and Df be the set of all discontinuous points of /. If
fi(Df) = 0, then / is called /z-a.s. (or a. s.) continuous. For ACS, denote
by dA the boundary of A. If 7,4 is /z-a.s. continuous, i.e., {i(dA) = 0, A is
called a /z- continuous set.
Remark. If S" = R, then Df e B. In the general case, /<*(£>/) = 0
means Df C A € B and //(A) = 0. Thus a /z-a.s. continuous mapping
/ may not be measurable w.r..t. B, but it is measurable w.r.t. the fi-
completion #M of B.
The following theorem gives some equivalent statements of weak
convergence (cf. Billingsley [1]).
15.37 Theorem. Suppose that fin, fi G V(S)y then iin^> ft is
equivalent to each of the following statements:
1) For every bounded fi-a.s. continuous f,
Blimb/iB(/) = /i(/), (37.1)
2) V^€ Cu(5), (37.1) holds,
3) limLin(F) < v(F), V closed set F,
n
4) !im/in(G) > fi{G), V open set G,
n
5) lim/4n(j4) = fi(A), V/z-continuous set A.
n
In particular, if /in —► /z and h is a fi-a.s. continuous mapping from S
to another Polish space S', then fj,n o h~l -^//o h~l.
From the above theorem it is easy to verify that if (<Pk)k>i ls a dense
subset of unit sphere of CU(S). Let
oo
d(w)= £ 2-n\n(Vn)-v(<pk)l
71=1
then d is a distance on V(S) and the topology induced by d coincides with
the topology of weak convergence.
15.38 Definition. Let A C p{S). If for every e > 0 there is a compact

458 Chapter XV Weak Convergence for Cadlag Processes
Ke C S such that
inf n(Ke)> 1-e, (38.1)
neA
then A is said to be tight.
15.39 Theorem. Let A C V(S). Then under the topology of weak
convergence A is relatively compact if and only if A is tight
This is a fundamental result of measure theory on metric spaces, due to
Y. V. Prokhorov, its proof may be found in many textbooks(cf. Billingsley
[i])-
15.40 Theorem. A subset A of V(Dd) is tight if and only if the
following conditions hold:
lim sup//({x : \\x\\N > a}) = 0, VN e N, (40.1)
a^°°^eA
Urn sup fi{{x : u/(<5, x, N) > r)}) = 0, WV G iV, rj > 0. (40.2)
Proof Necessity. Since A is tight, for any given e > 0 there exists a
compact Ke such that ml^A^{Ke) > 1 — e. Hence Theorem 15.22 implies
sup \\x\\n < oc and
xeKc
Urn sup c/(<5, x, N) = 0. (40.3)
HO xeKe
Thus for a > sup \\x\\n we have {x : \\x\\n > a} C K%,
xeKe
sup/i({x : ||x||7v > a}) < sup//(#£) < e,
/x€i4 y.eA
i.e., (40.1) holds.
By (40.3) for any 77 > 0 there is 6V such that sup uj^S^.x^N) < r),
xeKe
hence {x : u/(^, x, AT) > 77} c ATg and
sup//({a; : (j,(5t?,x,iV) > 7?}) < sup//(#£) < £,
i.e., (40.2) holds.
Sufficiency. By (40.1), (40.2) for any given e > 0, n e N and k > 1
there are a# and <5^Jt such that
sup/x({x : ||x||tf > a^v}) < e2~N~l,
neA
supii{{x:uj'(6Nk,x,N) > l/k}) < e2~N-k-1.
neA

§3. Weak Convergence and Tightness
459
Put
CNe = {x : 11*11* < aN} H Q {* = "'(***,*,#) < !/*},
N=\
Then Theorem 15.22 implies that Ce is relatively compact. But we have
inf n(CNe) > 1 - e2~N, TV > 1, inf n(C£) > 1 - e.
Therefore A is tight. D
15.41 Definition. Suppose that for each n, Xn is an S-valued
random element on a probability space (fl71, J"71, P71), C{Xn) = Pn o (X71)"1
is the distribution of X71, X is an S-valued random element on a
probability space (fi,^7, P) and C(X) = P o X~l is the distribution of X. If
C(Xn) ^ £(^0, we sav that {Xn) converges in distribution (or in law) to
X and denote it by Xn —>X. If (£(Xn)) is tight, we say that the set of
the distributions of (Xn) is tight, or (Xn) is to/At.
It is easy to know from the definition of weak convergence that Xn-+X
is equivalent to that for every / G Cb(S)
En[f(Xn)] -, E[f(X)],
where JE?71, E denote the expectations corresponding to P71, P respectively.
By Theorem 15.39 {C(Xn)} is relatively compact if and only if {C(Xn)}
is tight.
In the definition, for different n the probability space (ttn,Jrn,Pn)
may be different. But it is not hard to see that we may find a probability
space (fi,^7, P) on which a sequence (Yn) of S-valued random elements
is defined such that C(Xn) = C(Yn), n > 1. Therefore in the sequel
for simplicity we always assume that the sequence of random elements is
defined on a common probability space.
15.42 Theorem (Skorokhod's representation theorem). Suppose that
(MOjMtu™ > 1) C V(S) and /4n"-*Mo> then there exist a probability space
(fi,^7, P) and a sequence (Xn, n > 0) of S-valued random elements defined
on it such that fin = C(Xn), n > 0, and
lim p(Xn, X0) = 0, a.s..
n—>oo
Proof Take Q = [0,1], T = B([0,1]) and denote by P the Lebesgue
measure on [0,1]. Firstly, we divide S into:
oo
i j=\

460 Chapter XV Weak Convergence for Cadlag Processes
(here £ denotes the union of disjoint sets) and
dia{Siu...ik) = sup{p(x,y) : x,y G Sir..ik} < 2"*\ fc > 1,
Mn(aSilf...zJ = 0, n>0.
Due to the separabiUty of S, such a division of S exists (for example
we may divide S by means of the /^-continuous balls with centers in a
countable dense subset of S and radii less than 2~k~x).
Secondly, we divide [0,1] into the sum of intervals as follows:
[0,l] = £Aln), A^ = g A^,tJ, *>1, n>0,
i j = l
where | A| denotes the length of interval A and A^...iifc are arranged in the
lexicographical order.
Thirdly, we define random elements Xn as follows. Set
where x is a fixed point of S, Xir..ik is a point of 5^...^, the interior of
Siy...ik. Then each Xn is an 5-valued random element. Since x^...^^ 6
SJUfc+p c S°-V we have p(Xnk)H,Xnk+p)(uj)) < 2~k, p > 1. Due to
the completeness of «S, there is an S-valued random element Xn such that
lim Xnk)((j) = Xn{uj), u e ft, n > 0.
k—»oo
We axe going to prove that Xn —► Xo P-a.s.. For any £ > 0, take fc such
that 2-* < e/2. If w € (A^J0, since
\*l?.4,\ = M*,-*,) - MSiv.4,) = IA^.,.1, j > 1,
by virture of the arrangement of A^...{ , there is an njt such that for n > rik
(n)
we have uG A) ; ,• and
XA'(w) = Xi,...ifc, X^ '(w) = xiv..ik, n > 0,
p(*»,X0) < p(Xn,X<fe)) + p(xik\X{0k)) + p(X{0k\z0)
< 2 • 2~k < e.
Therefore Xn - X on f)( U (A,^.ifc)°).

§3. Weak Convergence and Tightness 461
Finally, we prove that C(Xn) = \in. Denote C = {Sir..ik, ij > 1,
1 < J < k, k > 1}. Then C is a 7r-class and
P(X£+* G 5ir..ifc) = |Af },J = /i»($,~«»). p > 1,
Thus £(Xn) and /zn coincide on C and by the montone class theorem we
obtain C(Xn) = /zn. □
15.43 Definition In the remainders of this paragraph we consider the
cadlag processes only, their distributions are probability measures on
Polish space Dd. Below X and Xn all stand for i2d-valued cadlag processes,
unless explicity stated.
Similar to (29.1) and (29.2), define
J(X) = {t>0: P{AXt ^ 0) > 0}, (43.1)
U{X) = {u > 0 : P(\AXt\ = u for some t > 0) > 0}, (43.2)
T0(X,u) = 0,
Tp+i(X,u) = inf{* > Tp(X,u) : \AXt\ >u}, p> 0. (43.3)
15.44 Lemma. J(X) and U(X) are at most countable.
Proof. Obviously, the conclusion comes from the following identities:
J(X)= U U:P(Tp(x,-)=t)>0},
n,p>l l V V n' ' '
U(X) = ^ {u : P{\±XTAxb\ = U,TP(X, I) < oo) > o}. D
15.45. Theorem. Suppose that Xn —> X. Then
1) The finite-dimensional distributions of Xn converge to that of X
along D = R+ \ J(X), i.e.,
(X-,---,^)-^,...^) ueD,p>\.
{We also denote it by Xn —> X.)
2) Suppose that g is a continuous function on R+ x R x Rr, satisfying
5(00, x, y) = 0 and u G U(X). Then
(g(Ti(Xn, «), *£(x»,tt), AX»(JfBfU)), 1 < i < k)
£(9(T(X,u),XTi{XiU),AXTi(x,u)), l<i<k).
3) If g is a continuous function on Rr, vanishing in a neighbourhood
o/0, then
(Xn,Xg(AXn))Mx,X9(*X))-

462 Chapter XV Weak Convergence for Cadlag Processes
Proof. By virture of Theorem 15.37, if h is C(X)-a,.s. continuous then
h(Xn)±h(X).
1) Take h(x) = (x(*i),---x(tp)), U G D, then from Theorem 15.12.2)
h is C(X)-sl.s. continuous.
2) Use the notations in (29.3) and take
h(x) = (j(t*(x, u), x(t*(x, u)), Ax(f (x, u)), 1 < i < fc).
Then from the assumption concerning g and Theorem 15.30, /i is C(X)-a,.s.
continuous.
3) Take h(x) = (x, £ 9(^x(s)))- Then hY Corollary 15.31 h is C(X)-
a.s. continuous. □
15.46 Lemma. X71 —> X if and only if the following conditions hold:
i) {Xn) is tight,
ii) All possible limit points of (C(Xn)) are identical.
Proof. Due to Prokhorov theorem, it is obvious. □
By virtue of the previous theorem, in order to establish the weak
convergence of (C(Xn)) we may verify the tightness and uniqueness of limit
points for (C(Xn)) separately.
Suppose D is a dense subset of ii+, if Xn —► Xy then it is easy to
know that the possible limit point is unique.
Prom Theorem 15.40, we also have the following theorem.
15.47 Theorem. {Xn) is tight if and only if the following conditions
hold:
lim supP( sup \X?\ > a) = 0, V7V G JV, (47.1)
a-*°°n>l Kt<N J
]imsupP{uj'(6,Xn,N) > t?) = 0, V7V G iV,7? > 0, (47.2)
or equivalently,
lim IS P( sup IX711 > a) = 0, V7V G JV, (47.3)
lim Im P(<-/(«, X71, N) > tj) = 0, V7V G iV, t? > 0. (47.4)
15.48 Definition. If (/zn) C ^P(Dd) is tight and for every possible
limit point fi of (fin) /i(Cd) = 1, then (fin) is said to be C-tight. (Xn) is
said to be C-tight, if (C(Xn)) is C-tight.

§3. Weak Convergence and Tightness 463
15.49 Lemma. The following statements are equivalent:
1) (Xn) is C-tight,
2) (Xn) is tight and
Urn P(suplAXDI > e) = 0, VAT G N,e > 0, (49.1)
n—►oo \+^?v7 /
kt<JV
3)
lim limPf sup \X^\ > a) = 0, WV € JV, (49.2)
a—►oo n \ *<fj '
lim E5P(ti;(«, Xn, N) > 17) = 0, VJV € TV, 77 > 0. (49.3)
Proo/. 1) => 2). Under 1) {Xn) is tight, hence it suffices to prove
(49.1) for any convergent subsequence of (Xn). For simplicity we may
assume that Xn —+X. Then X is a continuous process and J{X) = 0.
Thus Lemma 15.20 and Theorem 15.37 entail
sup \AX?\ £ sup \AXtl V7V > 0.
t<N t<N
But X is continuous and sup \AXt\ = 0, so (49.1) holds.
t<N
2) => 3). Due to the tightness of (X71), (47.3) implies that (49.2) is
necessary. (49.3) is deduced from (47.4), (49.1) and the following easy
inequality
cj(<5, x, N) < 2u/(<5, x, N) + sup |Ax(t)\.
t<N
3) => 1). It is apparent that (49.2) and (49.3) imply the tightness
of (Xn). Assume that (Xn ) is a convergent subsequence of {Xn) and
Xn' ^>X. Then Lemma 15.20 and Theorem 15.37 entail
sup |AXsn| -£ sup | AXS|, V* £ J(X).
s<t s<t
But sup|AXsn| < uj(6,Xn,t), hence (49.3) implies sup|AXs| = 0 a.s.
3<t 3<t
while t G J{X). Therefore X is continuous and (Xn) is C-tight. □
15.50 Lemma. Suppose that for all n,q G N, the process Xn has the
following decomposition:
Xn = Unq + Vnq + Wnq
where i) for each q, (Unq)n>i is tight, ii) for each q, (Vnq)n>i is tight and
there is a sequence (aq) of real numbers such that \imaq = 0 and
lim P(sup \AVtnq\ > aq) = 0, V7V G JV, (50.1)
n_KX> t<N

464 Chapter XV Weak Convergence for Cadlag Processes
iii) _
lim limP(sup \W?q\ > rj) = 0, ViV e TV, r) > 0. (50.2)
Then (Xn) is tight
Proof. Due to the tightness of {Unq)n>u (Vnq)n>i and (50.2), {Xn)
satisfies (47.3). Using the following easy inequalities:
u{6,x,N) <2sup|x(t)|,
t<N
J'{6, x + y,N)< u>'(6, x, N) + lj{26, y, TV),
uj{6, x, TV) < 2u'(6, x, N) + sup | Aar(t)|,
t<N
we get
o/(«, Xn, N) < Jr(<5, U7^ + V™*, N) + w(2, Wnq, N)
< u'(6, Unq, N) + 2u'(26, Vnq, N) + sup |AVf«| + 2 sup |W^|.
t<N t<N
For any e > 0, rj > 0, from (50.2) there is a number q such that aq < rj
and
ImP(sup \W?*\ >r))<e.
71 t<N
Now by virtue of assumptions i) and ii) we may choose no and 6 > 0 such
that while n > no we have
P(uj'(6, Unq, N)>7])< e, P(lj'(26, Vnq, N) > rj) < e,
P(sup \AVnq\ >r))<e, P(sup \W?q\ >r))< 2e.
t<N t<N
Thus P(uj'(6,Xn,N) > 677) < he and (Xn) satisfies (47.4), therefore {Xn)
is tight. □
Remark. If for each q> (Vnq) is C-tight, then ii) holds.
15.51 Corollary. Assume that (Yn) and (Zn) are two sequences of
Rd-valued cadlag processes. If (Yn) is C-tight and (Zn) is tight (resp.
C-tight), then (Yn + Zn), (Yn,Zn) are tight (resp. C-tight).
Proof. For (Yn + Zn), it suffices to apply Lemma 15.50 with Unq = Zn,
ynq = yn ^ ^nq __ q ^^ a^ = _ usjng the same technique as in Theorem
15.27, we get the conclusion about (Yn, Zn). D
15.52 Lemma. Suppose that Xn admits a decomposition Xn = Ynq+
Znq satisfying

§3. Weak Convergence and Tightness 465
i) _
lim Inn" P(sup \Z?q\ > 7?) = 0, ViV G N, 77 > 0, (52.1)
q—*oon—*oo £<#
ii) y«« _£• wi, as n -» 00, q > 1 and
W^W as ?^oo. (52.2)
Then Xn -£ W.
Proo/. For x,y € Dd, (7.1) yields p(x,y) < \\x-y\\N + 2~N+1. For any
£ > 0, take N satisfying 2~N+1 < e. Thus for any closed set F we have
P(Xn € F) < P(Ynq G F2e) + P(sup \Zn*\ > e),
t<N
where F2e = {y : p(y,F) < 2e}. Applying Theorem 15.37, we get
]imP{Xn € F) < Bm"P(Fn« G F2e) + Iim"P(sup |Zn«| > e)
n n n t<^
< P(W G F2F) + Ikn"P(sup |Zn9| > e).
n t<N
Letting q —► 00, by (52.1) and (52.2) we obtain
Iim"P(Jrn G F) < JhHP(Wq G F5?) < P(W € F5").
n q
Since F = f| F£, letting e j 0, we have
ImP(Xn G F) < P(W e F).
Now the claim is deduced from Theorem 15.37. □
15.53 Definition. Let A, B be two increasing processes. We say that
A strongly majorizes B, and denote it by B -< A, if the process A — B
itself is increasing.
15.54 Theorem. 1) Suppose that (Xn), (Yn) are two sequences of
increasing processes and Xn -< Yn, n > 1. If(Yn) is tight (resp. C-tight),
then so is {Xn).
2) Suppose that (X71) is a sequence of real processes with finite
variation, Yn = Vai(Xn) is the variation process of Xn. If (Yn) is tight (resp.
C-tight), then so is (Xn).
Proof. 1) Since
\X?-X?\<\Y?-Y?\, X?<Ytn, (54.1)
we have sup |Xf| < sup |Y"tn|, u'(6,Xn,N) < u'(6,Yn,N), u(6,Xn,N) <
t<N t<N
w(6,Yn,N). Applying Theorem 15.47 and Lemma 15.49, the tightness

466 Chapter XV Weak Convergence for Cadlag Processes
(resp. C-tightness) of (Xn) may be deduced from the tightness (resp.
C-tightness) of (Yn).
2) Since (54.1) is also true, the proof of 1) remains available. □
The following criterion for tightness is due to D. Aldous.
15.55 Theorem. Suppose that (Xn) is a sequence of adapted cadlag
R -valued processes on a filtered probability space (Vl,T,F = (^),P).
Then (Xn) is tight if the following conditions hold:
lim fiEP(sup \X?\ > a) = 0, V7V G N, (55.1)
a—oo n t<N
limlmi sup P(\X% - Xg| >e) = 0, V7V e N,e > 0, (55.2)
$—0 n s,TeTN,S<T<S+6
where Tjy is the collection of all stopping times bounded by N.
Proof Since (55.1) is just (47.3), it suffices to prove that (47.4) may
be deduced from (55.2).
For fixed N G N, e > 0 and 77 > 0, due to (55.2), for each r > 0 there
are 6(r) > 0 and n(r) G N such that
n > n(r),S,Te TNlS<T< S + 6(p) =» P(|X£-X£| > rj) < r. (55.3)
Set
S? = 0, SJ+1=inf{t>SJ?:|Xtn-X3n !>!;}.
Applying (55.3) to r = e, S = S% A TV, T = S%+1 A (SJ + S(t)) A N and
noticing that \Xgn — X§n\ > r] while 5^+1 < oo, we have
n > n(e), Jk > 1 =► P(Sjfc+i < TV, S£+1 < SJ + «(e)) < e.
We choose q £ N with g<$(£) > 27V. Then applying (55.3) entails that
n>n0 = n(e)Vn(-),ifc>0=»P(S£fl <7V,S£+1 <S£+ *(-)) < -.
(55.4)
Since S% = El=i(sk ~ Sfc-i), we have
6(e)q
P(S%<N)> NP(S% <N)> E(SZI[S„<N])
= JB(E(^-5fe«_1)/[s„<N])
2
I
'Jfc=l
jk=i
> £ S(e)[P(S2 <N)- P(S- <N,S%- S£_x < 6(e))}
k=l
> 6(e)qP(S? <N)~ 6(e)qe, n > n(e).

§3. Weak Convergence and Tightness 467
Hence
P(S% <N)<2e, as n > n{e). (55.5)
Next, set An = [S% > N] n ( ft [fijf - S^ > «(-)])• By (55.4) and
(55.5) we obtain
P{An) > 1 - 3e, as n > no- (55.6)
Now if a; € An, we take r = inf(z : Sf > TV), t* = Sf, i < r - 1
and consider the partition of [0,7V]: 0 = to < • • • < tr = TV. Prom the
definitions of An and SJ1 we get
uJ{[ti-UtilXn)<2r]J l<t<r,
*i-*i-i >*(")> l<»<r-l,
hence "'(«(-), -X"nM, ^) < 2??. Thus (55.6) implies
P(u/(£(-),Xn,^ > 2t?) < 3e, as n > n0.
Therefore (47.4) holds and {Xn) is tight. D
15.56 Theorem. Let {Xn) be a sequence of locally square integrable
martingales. If ((Xn)) is C-tight, then {Xn) is tight
Proof Since (X71)2 is dominated by predictable increasing process
(X71), for a, b > 0 Lenglart's inequality (the Corollary 9.24) implies
P(sup \X?\ > a) < \ + P((Xn)N > 6),
t<N <*>'
ImP(sup \X?\ > a) < 4- + HS P{(Xn)N > b).
n £<7V CL n—>oo
Letting a —♦ oo and 6 —♦ oo successively yields (55.1).
Next, for S,T G TN, 5 < T < S + 6, consider Nn = Xn - {Xnf.
(Nn)2 is dominated by (Xn) - {Xn)s. For e > 0 and rj > 0, again
applying Lenglart's inequality, we have
P(\X$ - X§\ > e) < 1 + P((Xn)T - (Xn)s > /?)
< l + P(u,{6,(X»),N)>r,).
By virtue of the C-tightness of ({Xn)), (49.3) implies
lim Em" sup P{\X% - Xg| > e) < \.
«—0 n s,TeTtf,S<T<S+6 £

468 Chapter XV Weak Convergence for Cadlag Processes
Since 77 may be arbitrary positive number, (55.2) holds and the tightness
of {Xn) follows from Theorem 15.55. □
§4. Weak Convergence of Step Processes
Before discussion the general conditions for weak convergence of semi-
martingales in the next chapter, we give the conditions for weak
convergence of step processes in this paragraph, since in this case we can
characterize the weak convergence in terms of jump times and jump sizes.
The priorities of this appoach he in that one may avoid the verification of
tightness and obtain the necessary conditions at the same time. Markov
step processes and the approximation of Markov step processes via Markov
sequences are also discussed.
p
15.57 Lemma. Let X,Xn,n > 1 be step processes, p(Xn,X)—>0
(resp. Xn-+X) and
lim Urn P(0 < \AX$nI[Tn<N]\ < e) = 0, V7V > 0, (57.1)
where Tf is the first jump time of Xn. Then for /(£, x) = (£, 2~l arctanx)
we have
f(T?,X%) ^{resp. £)f(TltXTl). (57.2)
Proof. By virtue of Theorem 15.42, it suffices to prove the conclusion
for the case of convergence in probability. Take uk j 0, Uk € U(X). Then
Theorem 15.30 implies Xft ^ X0,
T1(Xn,uk)^T1(X,uk), V*>1, (57.3)
and on [Ti(X, uk) < oo],
AX-(T1(^,ufc))-AX(T1(X,^))1\ Vfc> 1. (57.4)
Next, for given e > 0, 7? > 0, by (57.1) there are £, k such that for all
n we have
P(t < n < oo) + P(0 < lAXfTOI/p.^, < tifc)
+P(0 < \AXn(T?)\I[T?<t] < uk) < T}.
2) For convenience of typesetting, we write AX(t) instead of AXt.

§4. Weak Convergence of Step Processes
469
Thus
P{\T? -Ti\>e or \AXn(T?) - AX(Tx)\ > <r,7i < oo)
< P(t < Tx < oo) + P(0 < \AXn{T^)\I[Tu<t] < uk)
+P(0 < |AA-(Ii)|JIri<t, < uk)
+P(T1 < i,7? > t, lAA-^OI/^^, > uk)
+P(\Tln - Til > c or \AXn{J\n)-AX(T1)\>e,
\AXn(T?)\I[Tr<t] > uk, lAA-m)!/^^ > Ufc)
< 77+ P(77>*,T1 (*,«*)<*)
+P(\T1(Xn,uk)-T1(X,uk)\ > e,Tx(X,uk) < t)
+P(\AXn(T1(Xn,uk)) - AX(T!(X,Ufc))|
>£,ri(X,«fc)<t). (57.5)
By (57.3) we get
P(7? > *,Ti(X,tifc) < 0 < ^Ti^.m) > «, ^(X.Ufc) < 0 - 0.
(57.6)
According to (57.3), (57.4) and (57.6), letting n -» oo and rj -» 0
successively on the right-hand side of (57.5) yields
7lim)P(|rjl-ri|>e or \AXn(T?)-AX(T1)\>e,T1<oo) = 0. (57.7)
On the other hand,
P(7? < t,Tx = oo)
<P(0<|AXn(Tr)|/p?<t,<«fc)
+ P(|AXn(rr)|/[7T.<t) > uk,Tx = oo)
< 7? + P(T1(Xn,Ufc) < t.T^X.Ufc) = oo).
Letting n —► oo and 77 —♦ 0 successively also yields
Jiim^ P(77l < t,Ti = 00) = 1. (57.8)
(57.7), (57.8) and XJ^^o mean /(Tf ,Xn(7T)) Zf{TuX{T{)). D
15.58 Theorem. Suppose that X,Xn,n > 1, are step processes,
{Tj), (Tj1) are the jump times of X, Xn respectively. Then the following
statements are equivalent:
1) For f(t,x) = (t,2-'arctanx),
(/(7-,X^), j>0)±(f(Tj,XTj), j>0). (58.1)
2) Xn±X and
lim fim P(0 < |AX^»|/rr»<tl < e) = 0 Vj > l,t > 0. (58.2)

470 Chapter XV Weak Convergence for Cadlag Processes
Proof. 1) => 2). By Theorem 15.42, without loss of generality, we may
suppose that
f(Tf,X^)^f(Tj,XTi) a.s., Vj>0.
Then Theorem 15.33 implies p(Xn,X) -► 0, a.s. and
inf{|AX£n| :0<77<t,n>l}>0 a.s., V* > 0.
Hence 2) is valid.
2) =>• 1). By virtue of Theorem 15.42, we may assume that p{Xn, X) -*
0 a.s.. Then Lemma 15.57 entails f{T?,X$n)£ f{Ti,XTl).
Put Xn = Xn - X1?lp?M,X = X - Xnlp-Lool- Then ^
Theorem 15.30 p(Xn,X)£o. Hence /(T2n, AX^)4/(T2, AXTi) and conse-
p
quently /(T^Xjin) —♦ /(T2,Xr2). By induction it is easy to know that
f{T?,X%?)^f{Tj,XTj), Vj>1,
and therefore (58.1) holds (cf. Problem 15.8). D
15.59 Corollary. Suppose that X,Xn,n > \, are counting processes,
Xo = Xq = 0, (Tj), (Tj1) are the successive jump times of X, Xn respec-
tively, then the following statements are equivalent:
1) Xn-^X,
2)(T?J>l)MTjJ>l),
3) for a dense subset D of R+
X^l^Xu teD. (59.1)
Proof 1) <=> 2) is trivial, since for counting processes X, Xn we have
X?fy = j on [Tf < oo], XTj = j on [Tj < oo].
1) => 3) is apparent. Conversely, for tj G D, j G N
P{T?<tj, l<j<k) = P(X?j >j, 1< j<k)
= P(X?. > j - ^,1 < j < k) - P(Xtj > j - i,l < j < k)
= P(Xtj >j,l<j<k) = P(Tj < tj, 1 < j < fc),
hence 2) and also 1) are true. □
In order to discuss the weak convergence of Markov step processes,
we state some elementary results about the weak convergence of Markov
sequences in advance.

§4. Weak Convergence of Step Processes
471
15.60 Definition. Suppose that S is a Polish space, £ is the Borel a-
field on S. Let N(x, A), Nn(x, A), n > 1, be transition probability kernels
on (5,5), and g,gn,n > 1 be functions on S.
If for all / G Cb(S), #(-, /) G C6(5), then TV is called Fellerian.
If for all / G Ch(S) and compact /fcS,
lim sxip\Nn(x,f)-N{xJ)\ = 0,
then (TV71) is called uniformly convergent to N on compact and denoted
by Nn —♦ N. If ((7n) uniformly converges to g on compact we also denote
it by gn^>g>
If for all / G Cb(S) and sequence (xn) C S with xn —► x,
lim Nn(xnJ) = N(xJ),
n—►oo
we denote it by Nn=tN. If for all (xn) C S with xn -» x lim ^n(^n) =
n—►oo
j(x) holds, then denote it by gn=*g.
The following lemmas are easy and their proofs are left to the reader.
15.61 Lemma. Suppose that N, Nn, n > 1, are transition probability
kernels, and g, gn, n > 1, are functions on a Polish space S. Then
1) Nn=*N if and only if N is Fellerian and Nn ™ N.
2) g-nr^g if and onty tf gn—+ g and g is continuous.
15.62 Lemma. Suppose that X = {Xk,k > 0), Xn = (X£,fc > 0),
n > 1, are S-valued Markov sequences, p, pn are £/ie initial distributions,
andp, pn are the one step transition probability kernels of X, Xn
respectively. Then the following statements are equivalent:
1) Vn^P,
2) for all pn with pn ^ py Xn -£ X holds.
Moreover, in this case, if pn —► p, then for every continuous mapping f
from S to S
(/(*£),*><)) £(/(**), *>0).
Let X = (Xt) be a real Markov step process, (T)t, k > 1) be its sequence
of jump times, To = 0. Then by the strong Markov property it is not hard
to know that (Tjt, Xxk)k>o is an R+ x R-valued temporarily homogeneous
Markov sequence. It is also called the jump chain of X and the distribution
of X is uniquely determined by the initial distribution p of X and the one
step transition probability R(s,x;dt,dy) of (Tj^At^), so we also write
C(X)~(n,R).
15.63 Lemma. Suppose that X, Xn, n > 1, are Markov step
processes, C{Xn) ~ (Lin,Rn), C(X) ~ (n,R) and (Tk,XTk, k > 0) is the

472 Chapter XV Weak Convergence for Cadlag Processes
jump chain of X. If fj,n'—► \i, for f(t,x) = (t, 2~* arctanx), sn —♦ s < oo,
xn —» x and all g G C&(i2+ x it)
j/jf fl o f(t,y)Rn(snxn]dt,dy)^ jjgo f(t,y)R(s,x;eft,dy), (63.1)
then{f(T£,XT»), k > 0) -£>{f{Tk,XTk), k > 0) andXn±X.
Proof At first, we prove that (63.1) holds for 5 = oo as well. In fact,
from the definition of R we have
h
\f
g o f{t, y)i?(oo, x; dt, dy) = g(oo, 0).
while sn -» oo, for e > 0 there is TV > 0 such that
$ o f(t, y)Rn{sn, xn; dt, dy) - g(oo, 0)|
= \[ J [g(t,2-1 aictamy) - g{oo,0)]Rn(sn,xn;dt,dy)\ <e, n>N.
Hence (63.1) holds for s = oo. Now Lemma 15.62 implies
(f(7Z,Xfy,k>0)£(f(Tk,XTk), *>0),
and Xn -£ X is deduced by Theorem 15.58. D
Let X = (Xt) be a temporarily homogeneous real Markov step
process with transition probability (pt(x, A)) and corresponding infinitesimal
characteristics:
Q(X,A) = ^-lA^+tPtiX,A\ <««>
q(x) = -Q(x,{x}), (64.2)
N(x,A) = \ q(x) ' lf<^°> (64.3)
I /^(x), if g(x) = 0.
In this case, the one step transition probability R for the jump chain
(Tj,Xtj) of X may be written as follows (cf. He and Wang [3])
R<< T-dt A\ - \e-q(x)(t-s)<l{x)N{x,A)dtI[t>s], q(x) >0,s < oo,
[ ' ' ' '" \«oo(<ft)*,(i4), otherwise.
(64.4)
Meanwhile, it is not hard to compute that the Levy system v of X
v(dt, dx) = q(Xt-)N{Xt-,Xt- + dx)dt. (64.5)
Recall that the distribution of X is uniquely determined by its initial
distribution (i and g, N. So we write C{X) ~ (/z, q, N).

§4. Weak Convergence of Step Processes 473
15.64 Lemma. Let (q,N), (qn,Nn),n > 1, be the infinitesimal
characteristics of temporally homogeneous Markov step processes X, Xn,
xi > 1, and R,Rn be connected with {q,N), (q,Nn) by (64.4) respectively.
If Qn==^(l and for xn —► x G {y : q(y) > 0} we have
Nn(xn,g)^N(x,g), V5 € Cb(R),
then for sn —► s, xn —► x and all g G C\>{R+ x R)
jjgo f{t, y)Rn{sn, xn', dt, dy)^ jj go f(t, y)R{s, x; dt, dy), (64.6)
where f{t,x) = (£,2~* arctanx).
Proof. For g G Cb{R+ x it), g o f is a bounded continuous function
on R+ x R and for any to G .R+
lim sup |0 o /(£, y)-go f(t0, y)\ = 0.
t->U) y
If q(x) > 0, then
II go f{t,y)Rn(snyxn',dt,dy)
= | AT(*n> dy) ^°° <j o /(t, y)g»(sn)e^n<*»><'-*»> tf
= yV(xn,dy)My), (64-7)
/oo
-n
= / 0°/(5n + -T7—r,y)e~'cft
70 \ qn{Xn) J
"-* / <7°/(5H—rr^je"^ (uniformly in y as n -» oo)
Vo v ?(#) y
= y°° 5 o /(t, y)g(x)e-^)^-5)dt = /i(y).
Thus
\f WixnMhnly)- JN(x,dy)h(y)\
< sup |/in(y) - /i(y)| + IJV1^, /i) - N(x, h)\ -> 0.
y
So (64.6) holds.
Assume q(x) = 0. While qn(xn) = 0, we have
Rn(sn> xn;goh)=go /(oo, xn) = 5(00,0) = R(s, x; 5 o /).
While qn{xn) > 0, (64.7) is still valid and as n —► 00
roo , t \
K(y)= 9°f(sn + —,—r,y)e~*<ft-» 5(00,0) (uniformly in y).
JO v 9 \xn) '

474 Chapter XV Weak Convergence for Cadlag Processes
Thus
Rn{sn, xn;gof) = J hn{y)Nn(xn, dy) -* (/(oo, 0) = R(s, x;go h).
In sum, (64.6) holds. D
15.65 Theorem. Let X, Xn, n > 1, be temporally homogeneous
Markov step processes, C(X) ~ (n,q,N), C(Xn) ~ (vn, qn, Nn). Then
the following statements are equivalent:
1) i) qn=*q,
ii) for all xn -♦ x G {y : q(y) > 0},
J 9(y)Nn(xn,dy) - Jg(y)N(x,dy), V<? G Cb(R). (65.1)
2) for all iin^»,Xn^X holds and
Urn JS^P(0 < \AX?nI[Tn<N]\ < e) = 0, V7V G N. (65.2)
Proof 1) => 2). In view of Lemma 15.64, (64.6) is valid. Thus Lemma
15.63 implies Xn±X and (f(T£,Xfy), k > 0)±(f{Tk,XTk), k > 0),
hence (65.2) is deduced from Theorem 15.58.
2) => 1). Assume xn —+ x, take /zn = <5Xn, // = <5X. Then /Z1^//.
By the assumption, Xn —>X and (65.2) hold. Hence by Lemma 15.57
Tf^Ti. But
PiT1 >t)= exp[-qn{xn)t], P{T > t) = exp[-g(x)*],
thus qn(xn) -» q(x). If g(x) > 0, then T\ < oo a.s.. Lemma 15.57 yields
also Xj>n -» X^ and for g G Cb{R),
Jg(y)Nn(xn,dy) = E[g(XZn)\ - £?b(XTl)] = Jg(y)N(x,dy).
So (65.1) holds. D
Remark. The following example explains that (65.2) may not be
relaxed for 2) => 1).
Let X, Y be two independent homogeneous Poisson processes with
intensity 1 and Xn = X + -Y, then X, Xn, n > 1, are temporally
n
homogeneous Markov step processes and
q{x) = 1, N(x,dy) = <5x+i(dy),
qn(x) = 2, Nn(x,dy) = Ux^{dy) + ±6x+1/n(dy).
Obviously, Xn-+A', but 1) does not hold.

§4. Weak Convergence of Step Processes 475
15.66 Corollary. Let X, Xn, n > 1, be temporally homogeneous
Markov step processes with state space {0,1,2, •••}. The Q-matrices of
X, Xn are (qij), (q™j) respectively. Then the following statements are
equivalent:
1) for alli,j, qfj -♦ qiJ9'
2) for all fin % jx, Xn^X;
3) for all i,ifnn = n = 6U then Xn -£ X.
15.67 Theorem. Suppose that for each n, Yn = (Y£)k>o is a
temporally homogeneous real Markov sequences with initial distribution /zn and
transition probability pn(x, dy). Let en [ 0. Define
oo
*" = Y"t = £ Y?I(ken <t<(k + IK), t > 0. (66.1)
lej A;=0
Let X be a temporally homogeneous Markov step process, C(X) ~ (/z, g, TV).
Then the following statements are equivalent:
1) for all xn -* x, i)
— (Pn(xn, {Xn}) ~ 1) - -?(*), (66.2)
£n
ii) // g(x) > 0, /or a// g € Cb( Jl)
— /9(y)l[y*xn]Pn(xn,dy) -► g(x) f g(y)N(x,dy); (66.3)
2) /or a// jxn -> //, X71 -£ X and (65.2) Aofcb.
Proo/. Notice that X71 is a Markov step process, but may not be
temporally homogeneous. The transition probability for the jump chain
ofXnis
([p(z, {x})]l-k-il[yMp"{x,dy), pn(x, {x}) * 1,
Rn(ken,x;{len},dy) = l k<l
{I[i=oo] f>x{dy), otherwise.
1) => 2). Similarly to the proof of Theorem 15.65, it suffices to verify
(63.1). Assume that sn = ken —► s < oo, xn —► x, # G C&(.R+ x ii) and
f(t,x) = (£, 2~*arctanx), we have

476 Chapter XV Weak Convergence for Cadlag Processes
II 9° f{t,y)Rn{sn,xn;dt,dy)
= / £ 9°f(^n + k£n,y){pn(xn,{xn}))k-1I[y¥:x]pn(xn,dy)
= I hn(y)—I[y^Xn]pn(xn,dy),
J £n
90f{Sn+\ -\en + £n,y)(pn(Xn,{xn})) £n dt.
If q(x) > 0, then
/OO
0 ° /(«, y) exp[-q(x)(t - s)]dt,
and the above convergence is uniform in y. Hence
II 9° f{t,y)Rn{sn,xn;dt,dy)
-* J J 9° f(ty y) exp[-q(x)(t - s)]dtq{x)I[y¥:x]N{x, dy)
= J J 9° ^'' y^R(S'X; dt>dy^'
If q(x) = 0, then
l-pw(gw,{gw}) .
hn(y) ► fl(oo,0), umformly in y.
Thus
// fl ° /(*' y)Rn(sn, Xn; dt, dy)
-/^)1"ytW),-,fe'w)^-*>
-> 0(00, 0)= II go f(t, y)R{s, x; dt, dy).
2) => 1). Assume xn —► xo- Take tf1 = <5Xn, /z = <5Xo, u > 0. Write
Pn = Pn(zn, {^n})- If (If) and (Tj) are the jump times of Xn and X
respectively, Lemma 15.57 implies Tf—► Ti and
E[exp(-ul?)] = . ^—^ rexp(-u£n)
1 -pnexp(-u£n)
-> JBIexpt-nT!)] = 9(X)
g(x) + u"
It is not hard from this relation to conclude that pn —♦ 1 and —(1 — pn)
£71
q{x). Hence (66.2) is true.

Problems and Complements
477
If q(x) > 0, then T\ < oo a.s.. Lemma 15.57 also yields Xjm —>Xtx
and for all g G Cb{R)
E[g{X%)\ = —L- f g(y)I[y?Xn]pn(xn,dy)
1 ± ~ Pn J
-> E[g(XTl )} = l 9(y)N(x, dy). (66.4)
(66.4) and (66.2) imply (66.3). □
15.67 Corollary. Assume that for each n, Yn is a temporarily
homogeneous Markov sequence with state space {0,1,2, •} and one step
transition probability (pfj) and X is a temporarily homogeneous Markov
step process with state space {0,1, 2, • • •} and Q-matrix (qij). Let en [ 0.
If Xn is defined by (66.1), then the following statements are equivalent:
1) for all i,j, (p£ - 6ij)/en -► g^;
2) for all p,n ^ H, Xn±X;
3) for all i, if p?1 = p = Sif then Xn^X.
Problems and Complements
15.1 Consider the following functions:
xn(t) = l[t>l-2/n]» Vn{t) = l[t>l-l/n]-
Explain that D is not a topological vector space and D2 is not the product
topological space of D by D.
15.2 Consider the elements xn(t) = I[\<t<i+i/n] °f D. Explain that D
is not complete under p defined by (10.5).
15.3 Let pu(x, y) = sup \x(t) - y{t)\. Explain that Dd is not separable
t
under pu, but in Dd the a-field generated by pu-open balls coincides with
cr(x(u) : u < a).
15.4 Prove that under p defined by (7.1) Cd is a closed subset of Dd.
15.5 Let x, En, n ^ 1> be increasing cadlag functions on .R+, null at
zero. Then the following statements are equivalent:
1) xn —► x under the Skorokhod topology.
2) There is a dense subset D of il+ such that i) xn(t) —► x(t), t G D, ii)
for all t > 0 there is a sequence (tn) such that Axn(tn) —> Ax(t), tn —► t,
and moreover, tn < t if t G D.
3) There is a dense subset D of ii+ such that for t G D xn(t) —► x(t),
Zs<t(**n(s))2 - £s<t(A*(*))2.

478
Chapter XV Weak Convergence for Cadlag Processes
15.6 Let S be a Polish space. 1) If TV is a Fellerian transition
probability kernel on (S,B(S)), then for each e > 0 and compact K C S there
is a compact C£ik C S such that supx€^ N(x, C\K) < e.
2) If TV, TV71, n > 1, are transition probability kernels and Nn=tN,
then for all e > 0 and compact K C S there is a compact C£ik C 5 such
that Umrisup:E€A-7Vri(x,C,g^) < e.
15.7 Suppose that M,N,Mn,Nn,n > 1, are transition probability
kernels on a Polish space S. If Mn=*M, Nn=*N, then Mn * Nn=*M * TV
where M * TV(x, /) = fs M(x, dy)N(y, /).
15.8 Let ii00 = n^j Ri, Ri = R and equip R°° with the product
topology. Prove that i) R°° is a Polish space, ii) Let B°° be the Borel
a-field of R°°, 7Tfc be the projection of R°° to Rk = Oi=i ft* A^Mn be
probability measures on R°°. Then tin-*V> if and only if for all k > 1,
-l w -l
15.9 Let P, Pn be probability measures on (ii, B(i2)), Pn^+P. Set
Gn(ar) = inf {y G ii; Pn(] - oo, y]) > x},
G(x) = inf{y G R : P(] - oo, y]) > x).
Let £ be r.v uniformly distributed on [0,1]. Prove that Pn (resp. P) is
the law of Gn(0 (resp. G(0) and Um^Gn(0 = G(£) a.s..
15.10 Assume that S is a Polish space, ,4 C B(S), A is closed under
the formation of finite intersection and each open set in S may be expressed
as a countable union of sets in A. If P, Pn, n > 1, are probability
measures with lim Pn(A) = P{A), VA G A, then Pn -^ P.
n—►oo
15.11 Assume that (? is a family of real continuous functions on Rd,
for any different a,b,c G Rd. There is / G <? such that f(a),f(b),f(c)
are different. Prove that a sequence {Xn) of iid-valued cadlag processes
is tight if and only if i) for all e > 0, t > 0 there is a compact Ket C R?
such that
P(X? G /fCft,a < t)>l-e, Vn > 1,
ii)V/€0,(/(Xn), n > 1) is tight.
15.12 A sequence (Xn) of cadlag processes is tight if and only if
i) lima-.00Kn_00 P(\X?\ > a) = 0, Vt > 0,
ii) lim^o limn_oo P(u/(6, Xn, TV) > r/) = 0, Vt? > 0, TV G AT.
15.13 Suppose that a sequence {Xn) of cadlag processes satisfies
i) (XS) is tight,
ii) sup supP(\X?+h - X?\ > rjN(h)) < £N(h), V/i > 0, TV > 0,
0<t<Nn>l
where ejv(/i), t?n(^) satisfy / -^-—d/i < oo, / J^J-dh < oo for
</]0,a] Al 7]0,a] /I22

problems and Complements 479
some a > 0. Then (Xn) is C-tight. In particular, if {Xn) satisfies i) and
P(\X? -X?\>\)< ±(F(t) - F(s))1+a, 0<s<t,X>0,
where r > 0, a > 0 and F is a nondecreasing continuous function, then
{Xn) is C-tight.
15.14 Prove that a sequence (Xn) of measurable cadlad processes is
C-tight if and only if i) (X$) tight, ii)
lim IS sup P(|X?-Xg|>e) = 0,
6->0n->oo s,T£KNS<T<S+6
where Hjy is the collection of all non-negative r.v. bounded by N.
15.15 Assume that Xn is a temporally homogeneous right-continuous
Markov process with initial distibution fj,n and Fellerian transition
probability function Pt(x, A). If (fin) is tight and
limsupp?(x,O*(x))=0,
*-*0 x,n
where O^(x) = {y : \y — x\ > £:}, then (Xn) is tight. Furthermore, if
limsuPPr(^Oec(x))A = 0,
t—►() x,n
r then (X71) is C-tight.
15.16 Let (Tji)j>i be the sequence of jump times of a counting process
Xn, Wf = TJ1 - Tf_x,j > 1, T0n = 0. Characterize the tightness of {Xn)
in terms of (W?J > l,n > 1).
15.17 Assume that X, X71, n > 1, are renewal processes with renewal
functions m(t) = E[Xt], mn{t) = E[X?] respectively, F, Fn are the
(sub-) distributions of intervals between successive jump times of X, Xn
respectively. Prove that the following conditions are equivalent:
i) Xn±X,
ii) mn —> m, i.e., fR f(t)dmn(t) —► fR f(t)dm(t) for every continuous
function / with compact support,
iii) Fn^F.
15.18 For x e Dd, set Sa(x) = inf{t : \x(t-)\ > a or |x(t)| > a}.
1) If a ^ V(x) = {& : Sb(x) < 5j>+(x)}, then x »—> Sa(x) is a continuous
function on Dd.
2) Let V'(x) = {a > 0 : Sa(x) G J(x) and \x(Sa{x))\ = a}, x5°(*) =
x(t A Sa). Then x »—► (x,x5°) is continuous from Dd into D2d at each x
such that a £ V(x) U F'(x).
15.19. Assume that X, Xn, n > 1, are adapted cadlag iid-valued
c
processes. Then Xn —► X if and only if there is at most a countable subset
A C R+ such that

480
Chapter XV Weak Convergence for Cadlag Processes
where S(a,X) = inf{i : \Xt\ > a or \Xt-\ > a}, Xf = AW-
15.20 Assume that X, Xn, an, n > 1, are the cadlag processes, a
is a deterministic continuous function. If Xn —>X, sup3<t |a™ — as\ —»0,
V* > 0, then Xn + an £ X + a.

Chapter XVI
Weak Convergence for Semimartingales
In this chapter we will discuss the conditions of weak convergence for
semimartingales and some of their applications. The general sufficient
conditions of weak convergence to quasi-left-continuous (abbreviated as
q. 1. c.) semimartingales will be given in §1. In §2 and §3 the general
results will be applied to the case, where the limit process is a Levy process
whether continuous or not. In particular, it includes the weak convergence
for processes with independent increments. Finally, the results will be
applied to the case, where the limit process is a generalized diffusion process
in §4. The classical results of weak convergence for empirical processes
are also given in §4.
For the sake of simplicity, in this chapter we consider the real-valued
processes only, most of the results still hold for Rd-valued processes. The
basic setting is a filtered probability space $ = (fi,.F, F = (Tt)t>o,P),
unless otherwise stated, all semimartingales are defined on $.
§1. Convergence to a Quasi-left-continuous
Semimartingale
16.1 Definition. Let h be a bounded real function. If for some a > 0
it satisfies
h(x) = { "' '"' ""'"" \h(x)\<a, (1.1)
f x, \x\ < 1/a,
(x) = <
{ 0, |x| > a,
then h is called a truncation function. Z is the collection of all truncation
functions and Zc is the collection of all continuous truncation functions.
Set hi(x) = x/[|x|<i]. Then hi e Z.

482 Chapter XVI Weak Convergence for Semimartingales
16.2 Definition. Suppose that X is a semimartingale, (a,0,v) is
its predictable triplet, /z is the jump measure of X. By using a
truncation function h, we can get its integral representation, similar to that in
Theorem 11.25. If h satisfies (1.1) then x - h(x) = 0 as \x\ < \/a. Put
X(h) = E(AX - h(AX)) = {x- h{x)) * /*, (2.1)
X(h)=X-X(h). (2.2)
Since \AX(h)\ = \AX - AX(h)\ = \h(AX)\ < a, X{h) e Sp. It has the
following canonical decomposition
X(h) =X0 + M(h) + a(h), M(h) e Mioc,o, <*(h) G?flV0.
On the other hand, Theorem 11.25 gives
X(h) = X - X(h)
= X0 + a + Xc + (x/[|x|<i]) * (fi - v) + (x/[|x|>i]) * m - (x - h) * fi
= X0 + Xc + h * (fi - v) + a + (h(x) - xl[\x\<i}) * "-
Therefore
M(h)d = h*(»-v), (2.3)
a(h) = a + (h(x) - ar/[|*|<i]) * ^, (2.4)
X = X0 + Xc + h * (jx - u) + a(h) + (x - fc(rc)) * /z. (2.5)
If h(x) = hi(x) = z/^i^!], then a(h\) = a. Denote h = Aa(h) and
0(h) = (M(h)). Prom (2.4) and (2.3) we have
ht= f h(x)Vt(dx), (2.6)
(M(h)) = 0 + h2*v- E(Aa{h))2. (2.7)
Later we will see that it is more convenient to state the conditions
of weak convergence for semimartingales in terms of (a(/i), 0(h), is) for
h e Zc. Since (a(h), 0(h),u) and (a,0,v) are determined by each other,
(a(h), 0, v) or (a(/i), 0(h), v) are also called the predictable characteristics
(or predictable triplet) of X, and a or a(h) the ^rst characteristic of X, /?
or /3(/i) the second characteristic.
It is easy to know that for h,g 6 Z,
a(h) - a(g) = (h - g) * v, (2.8)
flfc) - 0(9) = (h2 -g2)*v- E(( Aa(/i))2 - (Aa(<j))2)
= (/i2 - <72) * i/ - (hh -gg)*v (2.9)

§1. Convergence to a Quasi-left-continuous semimartingale 483
Suppose that X is a locally square integrable semimartingale (see
Theorem 11.31), its canonical decomposition is
X = X0 + M' + a', M' G MJUt0, ol G V0 n V.
Then
a' = a(h) + (x- h(x)) * v, (2.10)
(M'> = 0(h) + (x2 - h\x)) * v - £((£)2 - (h)2)
= (3 + x2*v-X(x)2. (2.11)
(M') is also denoted by /?'.
16.3 Theorem. Suppose that (Xn)n>i is a sequence of semimartin-
gales, and for each n (an(h),f3n(h),vn) is the predictable triplet of Xn.
1) If the following conditions are satisfied:
i) (Xfi) is tight,
ii) VAT > 0, e > 0
lim Em P(i/n([0, N] x {x : \x\ > a}) > e) = 0, (3.1)
iii) (an(/i))n>i, (^n(/i))n>i are C-tight and (gp*vn)n>i is C-tight for
allp G N, gp(x) = (p\x\ - 1)+ A 1, then (Xn)n>i is tight.
2) Conversely, if (Xn)n>i is tight, then 1) i) and ii) hold.
In order to prove the theorem we need the following two lemmas.
16.4. Lemma. If conditions 1) iii) in Theorem 16.3 holds for a
certain h G Z, then it holds for all h G Z.
Proof. If /i, h G Z, then they satisfy (1.1) for some a > 0. Take p G N,
p > 2a,then
\h(x) - h(x)\ < 2agp(x), \h2(x) - h2(x)\ < a2gp(x),
Var[an(/i) - an(h)] -< \h - h\ * vn < 2agp * i/n,
Var[E((Aan(/i))2 - {Aan(h))2)} -< S|Aan(/i) + Aan(fc)||Aan(fe - fc)|
-< 4a|/i -h\*un ^ 8a2gp * i/n,
Var[^(/i) - (3n(h)} = Var[(/i2 - h2) * i/» - S((Aan(/i))2 - {Aan(h))2)]
-< a2gp *vn + 8a2gp * vn = 9a2#p * i/n,
hence from Theorem 15.54 and Corollary 15.51 the assertion follows. D

484 Chapter XVI Weak Convergence for Semimartingales
16.5 Lemma. 1) For N > 0, a > 0, the following two conditions are
equivalent:
i) limP(sup|AX?| >a) = 0,
71 s<N
ii) limP(i/n([0, JV] x {x : \x\ > a}) > e) = 0, to > 0.
n
2) For N > 0, £/ie following two conditions are equivalent:
i) Urn IimP(sup |AX?| > a) = 0,
a-oo n s<^
ii) Urn IhaP(i/n([0, N]x{x: \x\ > a}) >e)=0, Ve > 0.
a—>oo n
Proo/. 1) Set
A" = ^[|x|>a] * Mn = £ J[|AX»|>a],
A? = I[\x\>a) *«tf = «^([0,t] x {x : |x| > a}).
Then An is the compensator of An, and An,An are dominated by each
other. By Lenglart's inequality
P(sup |AXan| > a) < P(AnN > 1) < e + P{AnN > e),
s<N
whence 1) ii) => 1) i) and 2) ii) => 2) i) are deduced.
On the other hand, again by Lenglart's inequality,
P{AnN >e)<r) + -E{sup AA?) + P(AnN > erj).
£ s<N
Since |A>ln| < 1 and [sup AA£ > 0] = [AnN >erjAl] = [sup | AX?| > a],
3<N 3<N
P(AnN >e)<V+(- + l)P(sup |AX;| > a).
V£ J s<N
Now letting n —+ oo (a —+ oc) and 77 —► 0 successively, 1) i) =>> 1) ii) and
2) i) => 2) ii) are deduced. D
16.6 The proof of Theorem 16.3. For fixed h G Z. Let hq(x) =
qh(x/q)1 then hq G Z. We will prove the tightness of (Xn) by making use
of Lemma 15.50 and (2.5). For each n we have
Xn = XS + Mn{hq) + an(hq) + Xn(hq)
= jjnQ + vnq + Wnq
where Unq = X£ + Mn(/i9), Fn« = an(fe,), Wn<? = Xn(/ig). By Lemma
16.4 {Vnq)n>i is C-tight for all q. Expression (XV.50.2) holds for aq = l/q.
Also by Lemma 16.4 0n(hq) = {Mn(hq)))n>i is C-tight for all q. Hence

§1. Convergence to a Quasi-left-continuous semimartingale
485
by Corollary 15.51 {U^n^i is tight for all q. Finally, if h G Z satisfies
(1.1) for some a, then hq(x) = x as \x\ < aq and
AXn{hq) = AXn - hq{AXn) = 0, on [\AXn\ < aq].
Therefore
P(sup \W?«\ > 0) < P(sup \AX?\ > aq).
s<N s<N
Now by Lemma 16.5.2) and (3.1), (XV. 50.2) holds. Hence all conditions
of Lemma 15.50 are satisfied and {Xn)n>i is tight.
2) Conversely, if {Xn)n>i is tight, 1) i) holds apparently. Meanwhile,
sup |AXtn| < 2 sup \X?\, hence from (XV. 47.3) we have
t<N t<N
lim IELP(sup | AX?\ > a) = 0.
a—oo n t<N
Now it is easy from Lemma 16.5.2) to know that (3.1) is necessary. D
16.7 Definition. Suppose that $£> = (D1, V, D, Pq) is the canonical
filtered probability space and the filtration D = (Dt)t>o satisfies the usual
conditions. Let X be the canonical process. Assume that a is a predictable
process with finite variation and ao = 0 on $£> and a(h) is connected with
a by (2.4); (3 is a continuous increasing process with /?o = 0 on $£>, and
/3(h) is connected with (3 by (2.7); v is a predictable random measure on
$£>. In what follows, the triplet (a, /?, v) (or (a(/i), (3(h), u)) satisfying the
above conditions is also called a predictable triplet on $£>.
The main topic in this paragraph is the following problem: if {Xn) is
a sequence of semimartingales with predictable triplet (a71,/?71,*/1) on $,
in terms of (a71,/?71, vn) and (a,/?, v), what are the sufficient conditions
to guarantee that Xn —► X and X is a semimartingale with predictable
triplet (a,/?, v).
Suppose that Y is a r.v. on $£>, and Xn is a semimartingale on $,
then Y o Xn is a r.v. on $.
Before stating the main results, we introduce some conditions and
notations. Let D C il+ and set
[a-D] : a?{h) - at(h) o Xn ^ 0, V* G D, for some /i G 2C-
[/?-£>] : ^(/i)-A(/i)o^^0, VtGJ), for some /i G ZC. (7.1)
[*/-£>]: ^*^n-(^*^)°^ri^0, VteD,geJi. (7.2)
[sup a] : sup \a?{h) - at(h) o Xn| 4 0, V7V > 0, for some h G 2C-
*<7V

486 Chapter XVI Weak Convergence for Semimartingales
[sup 0\ : sup \ffl(h) - 0t(h) or|^0, V7V >0, for some h G ZC.
t<N
[sup i/] : sup \g * i," - (g * i/t) o X»| 4 0, V7V >0, <j G Ji, (7.3)
where
{lim /(rr) exists and is finite, 1
/€C6(fl+): ~~ l . ' (7.4)
/(x) is zero in a neighbour of 0 J
[C]: For # G J\, h G £c, ^CO? A <7 * ^ are continuous processes.
It is easy to see that for /i, <7 G £c, h — g G Ji and h2 — g2 € J\. Hence
it is easy to deduce from (2.8), (2.9) that under [v-D] (resp. [sup v\) and
[C], if [a-D] or \J3-D] (resp. [sup a] or [sup 0\) hold for some h G 2C,
then they hold for all h G Zc.
If we deal with locally square integrable semimartingales, we also need
the following conditions:
[sup a'] : sup \a[n - olt o Xn\ 4 0, V7V > 0. (7.5)
t<N
[P'-D]: \0*-fioXn\£o, VteD. (7.6)
Similarly, we can also define [a/-/}] and [sup /?'].
16.8 Lemma. Suppose that Gn G V0(*), G G V0(*d), Hn G V+(*),
i/ G V+($d) and F is a deterministic continuous function, Var(Gn) -<
/F\ Var(G) -< H < F. If D is a dense subset of R+ and
G?-GtoXn^0, VteD, (8.1)
H?-HtoXn^0, VteD, (8.2)
then
sup|G?-GaoXn|4o, Vt>0. (8.3)
3<t
Proof For e > 0, take {ti} C D such that *o = 0, t» < U+i, ti —► oo
and F(£j+i) - ^(^i) < e. Now for 5 G [£i,*i+i] we have
|G? - Gsor| < |G? - GJJI + \Gnti - GtioXn\ + \Gti oXn-Gso Xn\
<H^-H?% + \Gl-GtioX»\+e
< |fl*+1 - Htt+l oXn\ + e + \HU or-^| + \Gl - Gtl o Xn\ + e,
sup|G? - G9 oXn\ <2e+ sup {2| J5ff - Hti oXn\ + |G? - Gu oXn\},
s<t i'-U-i<t %
Therefore (8.3) is derived from (8.1) and (8.2). D

§1. Convergence to a Quasi-left-continuous semimartingale 487
16.9 Corollary. If D is a dense subset of ii+ and the following
strong majoration condition holds: there exists an increasing continuous
[deterministic) function F such that
Var(a) + ^ + (1Ax2)*mF, (9.1)
then \fi-D] «=*► [sup J3], [u-D] => [sup i/].
Proof It suffices to take Gn = Hn = fin (resp. g * i/n), G = H = 0
(resp. g * v) in Lemma 16.8. □
16.10 Lemma. Suppose that Gn G V0(*), G G V0(*d), Var(G) -< F,
and F is an increasing (deterministic) right-continuous function.
1) // p(Gn, GoXn)Z 0, then (Gn) is fc^/i*.
2) // sup |GJ — G3 o Xn| —♦ 0 for all t > 0 and F 25 continuous, then
s<t
(Gn) 25 C-tight.
Proof. Since Var(G) -< F, Theorem 15.54 implies that the sequence
(G o Xn) is tight. Moreover, if F is continuous, then (G o Xn) is C-tight.
Now, suppose (G o Xn ) is a subsequence of (G o Xn) and
the assumption in 1) implies that Gn —+ Y as well, this also means (Gn)
is tight. Moreover, if F is continuous, then (Gn) is C-tight. □
16.11 Theorem. Suppose that [Xn) is a sequence of semimartingales
on $ and (a,/3,v) is a predictable triplet on $£>. //
i) (X$) is tight;
ii) For a dense subset D of R+ [sup a], [0-D], [u-D] holds;
iii) Strong majoration condition holds: there exists an increasing
continuous function F such that:
Var(a) +^ + (x2Al)*MF, (11.1)
iv) Condition on big jumps holds:
lim sup i/(y, [0, t] x {x : \x\ > a}) = 0; (11.2)
a—►oo y
then (Xn) is tight.
Proof. We will verify that (Xn) satisfies the conditions of Theorem
16.3.1). The assumption i) is just the same as i) of Theorem 16.3.1).
For p > 0, consider
gp(x) = (p\x\-l)+Ale Ji.

488 Chapter XVI Weak Convergence for Semimartingales
Thenar*;) < (p2 Vl)(x2 A 1). For fixed t GD,£>0,?|>0, (11.2) implies
that there exists a > 0 such that
*vP92/a*vt(v) < suput{y,[0,t] x {x : |x| > -}) < -. (11.3)
Due to [v-D], if n is large enough, we have
P(\92/a * *? ~ (92/a * *t) O X»\ > |) < 7/. (11.4)
Since i/n([0,i] x {x : \x\ > a}) < g2/a * i/J1, (11.3) and (11.4) imply
P(vn([0,t] x {x : \x\ > a}) >e)<V-
Thus condition ii) of Theorem 16.3.1) holds.
For h e Z, using (2.4) and (11.1) we have
Var(a(/i)) -< kF, (11.5)
where A; is a constant. From (11.5), [sup a] and Lemma 16.10.2) the
C-tightness of (an(h)) is deduced. Due to Corollary 16.9, [sup (3] and
[sup v\ hold. Thus for g G J\ the C-tightness of (J3n(h)) and (g * vn)
is also deduced from Lemma 16.10.2). In sum, condition iii) in Theorem
16.3.1) holds and the tightness of (Xn) is a consequence of Theorem 16.3.
□
Remark. From (2.8) it is easy to see that for any h G Z, if a is
replaced by a(/i), F is replaced by kF (k is a constant depending on h)
in (11.1), then (11.1) is still true.
16.12 Lemma. Suppose that a family of r.v. (Zf, i G /, n > 1)
satisfies the following conditions:
i) (Z™, i G /, n > 1) is uniformly integrable,
ii) Vi G / Z? -£ Zi? as n -♦ oo,
£/ien (Zj, i e I) is uniformly integrable and
\miQE[Z?]=E[Zi], iel. (12.1)
Proof Due to the uniform integrability of (Z™,i G 7,n > 1) for any
e > 0 there exists an TV such that
E[\Z?\I(\Z?\>N)]<e, »€/, n>l.
In fact, by Skorokhod's representation theorem we may assume Z™ —>
Zj, a.s.. Then by Fatou's lamma
E[\Zi\I(\Zi\ > JV)] < limf;[|Zf |/(|2TI > N)] < e, » € /.

§1. Convergence to a Quasi-left-continuous semimartingale 489
Therefore (Zi,i G I) is uniformly integrable and (12.1) holds. D
16.13 Lemma. Suppose that (TV71), (Yn) are two sequences of adapted
cadlag processes on $, TV, Y are two adapted cadlag processes on $£>,
and G = (Gt) is the usual natural filtration of (TV, Y). If the following
conditions are satisfied:
i) (TV71, n > 1) is a sequence of martingales and for each t > 0 (TV™, s <
t,n> 1) is uniformly integrable,
ii) D is a dense subset of R+ and {Nn,Yn)C^{N,Y),
then TV is a G-martingale.
Proof Take u\ < • • • < uk < s < t, ui,s,t G D and / G Cb(R2k).
Since TV71 is a martingale,
E[(Nt-N?)f(Y?1,---tYZk,N21,---,N:k)] = 0. (13.1)
Owing to uniform integrability of {N",s < t,n > 1), letting n —► oo in
(13.1), by Lemma 16.12 we have
E[(Nt - Ns)f(YUl,---,YUk,NUl,---,NUk)} = 0.
Now by the monotone class theorem for £ G bQs- we get
E[{Nt - Ns)£] =0, t>s,t,seD,
where Gt = v{Ns,Ys,s < t}. For arbitrary s < t, take Sk < tk such that
Sk,tk G D, Sk || 5, tk || £• Since G®+ C ^fc_5 TV is right continuous, and
(TVS, s < t) is uniformly integrable, for £ G &^J+ we have
E[(Nt - N.)t] = 0.
Therefore TV is a G-martingale. □
16.14 Lemma. Suppose that (Xn) and (Mn) are two sequences of
adapted cadlag processes on $, for each n Mn is a martingale, X is the
canonical process on $£>, M is a cadlag process on $d, D is a dense
subset of R+ and D C R\J(X). If the following conditions are satisfied:
i) for each t > 0, (Mj\ s < t, n > 1) is uniformly integrable;
ii) Xn±X;
iii) for each t G D, x »—► Mt(x) is an C(X)-a.s. continuous mapping
on D;
iv) for each t G D, M? -MtoXn±> 0;
then M is a martingale on $£>.

490 Chapter XVI Weak Convergence for Semimartingales
Proof If Nn,Yn,N and Y are replaced by Mn,Xn, Maudlin
Lemma 16.13 respectively, then condition i) of Lemma 16.13 holds and Qt
is just Vt. Now conditions ii) and iii) yield
(MtjoXn,X^)1<j<kMMtjoX,Xtj)i<j<k = (Mt^XtjhKjKk, tj e D.
Moreover, from condition iv) we get
(M^X^j^MMt^Xt^j^tj € D.
Thus condition ii) of Lemma 16.13 also holds and we know that M is a
martingale on $£>. □
16.15 Lemma. Suppose that M G -M2oc0, |AM| < a, then there exist
two constants &i,&2, irrelevant to a and M, such that
E[M;4} < kia2(E{(M)2t})^2 + k2E[(M)2t}. (15.1)
Proof At first, we suppose that M and (M) are bounded. Write
N=[M]- (M) G Mioco- Then \&N\ = l(AM)2 - ^(AM)2)| < a2 and
[N] = £(A7V)2 -< a2Var(iV) -< a2{[M] + (M)).
Hence (TV) -< 2a2 (M). Now by B-D-G inequality (Theorem 10.36) we get
E[M?4} < kE[M]2t < 2kE[{M)2] + 2kE[N2)
= 2kE[(M)2} + 2kE[(N)t] < 2kE[{M)2} + 4a2fc(£[(M)2])1/2.
Thus (15.1) holds with ki = 4k and fc2 = 2k. Finally, for any M G X?oc0,
set
Tn = inf{t : \Mt\ > n or (M)t > n},
then from |AM| < a we have |MTn| < n + a, (M)Tn <n + a2. Therefore
(15.1) holds for MTn. Letting n —► oo, we know that (15.1) also holds for
M. □
16.16 Theorem. Suppose that for each n Xn is a semimartingale
with predictable triplet (an,/3n,vn). Let X be the canonical process on
$£>. If Xn—>X, there exists a continuous {in t) triplet (a,/?, v) on $£>
and a dense subset D of R+, D C R+ \ J{X) such that the following
conditions are satisfied:
i) [a-Dl\J3-D],[v-D] hold;
ii) for some h G ZCj
sup|/?t(y,/i)| < oo, sup|^*i/t(y)| < oo,* > 0,# G J\\ (16.1)

§1. Convergence to a Quasi-left-continuous semimartingale 491
iii) continuity condition holds: For each t G D, g G J\ and some
h G ZCl the following mappings on D are C(X)-a.s. continuous under the
Skorokhod topology:
y^at(y,h), y^0t(y,h), y^>g*vt(y), (16.2)
then X is a semimartingle on $£> with predictable characteristics (a, /3, v).
Remark. Due to the continuity of (a,(3,v), (2.8) and (2.9) become
a(h) - a(g) = (h - g) * v,
0(h)-0(g) = (h2-g2)*v.
If hyg G Zc, then h- g, h2 - g2 G J\\ and if (16.1) and (16.2) hold for
some h e Zc, then they hold for all h G Zc as well.
Proof For fixed g G Ji, h G 2C, set
Xn(/i) = X71 - E(AXn - /i(AXn)), X(fe) = X - £(AX - /i(AX)),
Vn = Xn(h) - an(h) - X£, V = X(/i) - a(h) - X0,
Zn = yn2 _ pn(h^ Z = V2 - 0(h),
Nn9 = Xg(AXn) -g*vn, W = Xg(AX) -g*v,
then
Xn(h) = X(h)oXn. (16.3)
By virture of (2.4) and (2.7), for every n, Vn, Zn, Nng are local martingales
on $ and we need to prove that V, Z, N9 are local martimgales on $#. To
this end, we will use Lemma 16.14.
a) Take T G D. If sup#r(?/, /i) < AT, set
y
Tn=inf{«:ifl51(fc)>iif + l}l M? = VtnATnAT, M = FT
We shall prove that M = V"T is a martingale. If |/i| < a, then
£7[sup|Mtn|2] < 4E[{M^,)2} < iE\ffin{h)\ < 4(K + 1 + 4a2).
Thus condition i) of Lemma 16.14 holds. The assumption that Xn—>X
guarantees the condition ii) of Lemma 16.14. For t G D C R \ J(X),
Corollary 15.31 yields that the mapping x *—► Xt(h) is continuous on D.
Furthermore, x »—► Vt(x), x i—► Mt(x) = VtAT(x) are also continuous on D
and 16.14.iii) is met. Finally, due to (16.3)
Vtn -VtoXn = at(h) oXn- a?(h),

492 Chapter XVI Weak Convergence for Semimartingales
P(\M?- Mt o Xn\ >e) = P(\V&TnAT - Vt^T o Xn\ > e)
< P(Tn <T) + P(\atAT(h) o Xn - a^T(h)\ > e). (16.4)
Since 0(h) oXn<K, jfy(h) - 0r(h) orio,
P(Tn <T) = P(j%(h) >K + 1)
< P(\0Z{h) - 0r{h) o Xn\ > 1) ^ 0. (16.5)
Therefore from [a-D] and (16.4) we know that 16.14. iv) holds and Lemma
16.14 yields that M = VT is a martingale and V € A^ioc-
b) Let T, Tn be the same as in a) and
M? = Z?ATATn, M = ZT.
By Lemma 16.15, there exists a constant k' depending only on h and K
such that
E[snp\VtlTn\4}<k'.
Hence {Z?aTaT , t > 0,n > 1) is uniformly integrable and 16.14. i) holds.
Conditions 16.14. ii) and 16.14. iii) may be verified the same as in a).
Moreover,
M/1 - Mt o Xn = (V£TATn)2 - (VtAT o Xn)2 - (ftATATn - AAr o Xn)
= iYJATATn - VtAT o Xn)(VtnATATn + VtAT o Xn)
~ (PtATATn ~ fit AT °XU),
we have seen above that (VtATAr„, Vt/\T ° Xn, n > 1) are uniformly
p
integrable and that V£TATn — VtAT°Xn —► 0. Then, similarly to a), 16.14.
iv) can be easily deduced from [/3-D] and (16.5). Thus M = ZT is a
martingale and Z G M\oc.
c) Take T G D. If supg * vT(y) < K, let
y
Tn = ud{t:g*u?{y)>K + l}, M/1 = N%TATn, Mt = N^T.
Since
(n < 92 * »&r- <K' = (K + 1 + \\g\\)\\9l
we have
£[sup|Mtn|2] < 4£[(Mn)00] < 4K'.
t
Hence {M^t > 0, n > 1) is uniformly integrable and 16.4.i) is met.
Conditions 16.14.ii) and 16.14.iii) may be verified the same as in a). Finally,
M?-MtoXn=g* v?ATATn - (g * utAT) o Xn.

§1. Convergence to a Quasi-left-continuous semimartingale
493
Similarly to a), one may deduces from [v-D] that 16.14. iv) holds also.
Thus M = (N9)T is a martingale and N9 G M\oc-
Since for each g G Ji, V, Z, N9 are local martingales, therefore X is a
semimartingale with predictable characteristics (a, /?, i/). D
16.17 Theorem. Suppose that for each n Xn is a semimartingale on
$ with predictable triplet (an,/?n, i/n). Let X 6e £/ie canonical process on
$£>. If there exists a predictable triplet (a,/?, */) on $£> and a dense subset
D of il+ suc/i £/ia£ £/ie following conditions are satisfied:
i)C(XS)Z\0;
ii) [sup a], [/?-/?], [i/-JD] AoW;
iii) strong majoration condition holds: there exists an increasing
(deterministic) continuous function such that:
Var(a) +^ + (x2Al)*MF; (17.1)
iv) condition on big jumps holds:
lim sup i/(y, [0, t] x {x : |x| > a}) = 0, V* > 0; (17.2)
v) Continuity condition holds: for each t G D, g G J\ and some
h G Zc, the following mappings are continuous on D:
x t—► ctt(x, /i), x h-* /?t(x, ^),xh^* ^t(x). (17.3)
vi) P# 25 £/ie unique solution of the semimartingale problem (X,D°;
Remark, vi) means X is a semimartingale with predictable triplet
(a, (3, v) and (17.1) guarantees that X is q.l.c.
Proof. At first, we observe that due to (17.1), if g G Ji, the processes
a> A <7 * ^ are sdl continuous and for any h G 2 there exists a constant fc
such that
Var(a(/i)) + /?(/i) + (1M2)*M kF.
Due to conditions i)-iv), all conditions of Theorem 16.11 axe satisfied
and Theorem 16.11 implies that (Xn) is tight.
Suppose that there is a convergent subsequence of (C(Xn)), for
simplicity, we denote it still by (C(Xn)) and assume
C(Xn)^P'. (17.4)
P' is a distribution on (D,Z>). We will prove that
J\X) = {t > 0 : P'{AXt ? 0) > 0} = 0.

494
Chapter XVI Weak Convergence for Semimartingales
For t > 0, e > 0, by (17.1) there exist s < t < s' such that
92/e * «V(V) ~ 92/e * Mv) < ^ V € D> (17.5)
where #p(x) = (p\x\ — 1)+ A 1. Then there are also r,r' G R+ \ Jf{X) and
s<r<t<r'<sf. Now similarly to Lemma 15.20, we have
C( sup \AXZ\)^C( sup |A*tt||P'),
r<u<r' t<u<t'
where C(-\P') is the distribution under P'. Moreover,
P'(\AXt\ > e) < P'{ sup |AXU| > e) < limP( sup |AX£| > e)
t<u<t' n t<u<t'
<iiSP(sup |AX-|>£)<iI^P( £ j2/e(AX»)>l).
Since (£s<u<. 02/e(AX£))P = (f]a,oo[ff2/e) * vn * Lenglart's inequality
implies that
P'(\AXt\ >e)<2e + EP((/£/2 * vns, - fle/2 * vns > 2e).
But from [v-D] we have
02/e * I# - (fl2/e * "a') ° *" ^ °>
P (17-6)
92/e *"7- (92/e * "a) ° Xn -> 0.
Hence (17.6) and (17.5) entail
P'{\AXt\ >e)< 2e.
Since e may be an arbitrary positive number, P'(\AXt\ ^ 0) = 0 and
J'(X) = 0.
Now it remains to verify that the conditions of Theorem 16.16 are
satisfied. Since J'(X) = 0, for C(X\P') the subset D in the assumption
satisfies the requirements of Theorem 16.16 and condition i) in Theorem
16.16 is valid. Condition ii) in Theorem 16.16 is deduced from condition
iii) in Theorem 16.17 and condition iii) in Theorem 16.16 is just condition
v) in Theorem 16.17. Hence Theorem 16.16 implies that X is a semi-
martingale on (D, V,D,P') with predictable triplet (a,/?, v), i.e. P' is
a solution of the semimartingale problem (X, D°; Ao,a,/3, v). Now from
condition vi) we get P' = Pp. This means that the limit point of (C(Xn))
is unique and
C(Xn)^PD = C{X). □
Conditions iii)—vi) in Theorem 16.17 are imposed upon the predic-
table triplet (a,/3, v) on $p and condition i) is necessary for Xn —>X.

§1. Convergence to a Quasi-left-continuous semimartingale 495
Condition ii) requires that the triplets (a71,/?71, vn) of Xn converge in
probability to the triplet (a,/?, v) in a special way. It is not a natural
requirement, because conclusion is only concerned with convergence in
distribution. Hence it is reasonable to expect that condition ii) should
be replaced by (an,/3n,g * vn) —►(a,/?, # * v). But the following example
explains that even for counting processes, the convergence in
distribution of compensators cannot guarantee the convergence in distribution of
processes.
16.18 Example. Suppose the TV = (Nt) is a Poisson process on $
with parameter 1, 6 G To is a r.v. independent of TV, and P(9 = 0) =
p(0 = 1) = 1/2. Let
,4< = log2(*-0(*-l)+),
Xt = NAt, Qt = Tt\og2,
Yt = XtI[0ii](t) + /[X1>0]^log2^]l,oo[(0-
Then X and Y are counting processes. Since A = (At) is continuous and
(^)-adapted, the compensators of X and Y w.r.t. (Gt) me
X? = At,
Ytp = A} + I[Xl>0](t - 1) log2/]li0o[ = ln2(* - I[Xl=0](t - 1)+).
Because Ai = In 2, X\ is independent of 0 and A,
P(Xt = 0) = e_ln2 = 1/2, P(X!>0) = l/2.
Hence C{X^) = C{Y*>). But
P(X2 = 0) = P(X2 = 0,6 = 0) + P(X2 = 0,6 = 1)
= e-2log2/2 + e-lo62/2 = 3/8)
P(V2 = 0) = P(Xi = 0) = e-'°82 = 1/2.
Thus X and Y have different distributions.
Now we will give a theorem about weak convergence for square inte-
grable semimartingales and express the conditions in terms of u and a', /3'
denned by (2.10), (2.11).
16.19 Theorem. Suppose that for each n Xn is a locally square
integrable semimartingale with predictable characteristics (an,0n,un) and
alirnoHmP((x2/[|l|>a]) * v? > e) = 0, V* > 0. (19.1)

496 Chapter XVI Weak Convergence for Semimartingales
Let X be the canonical process on $£>. // there exist a predictable triplet
(a, /?, v) on $d and & dense subset D of R+ such that the following
conditions are satisfied:
i)C(XS)^\0,
ii) [sup a'], [/?'-£>], [*/-£>] hold;
iii) there exists an increasing (deterministic) continuous function F
such that
Var(a') + 0 + x2 * v < F; (19.2)
iv)
lim sup(x2/|x.>a) * ut(y) = 0, V* > 0; (19.3)
v) for each t G D and g G J\, the followig mappings are continuous
on D:
x i—► a[(x), x h j8j(x), x i-^ g * vt(x);
vi) Po is the unique solution of the semimartingale problem (X,D°;
Ao, OLj 0, v); then Xn —♦ X.
Proof. We will verify that the assumptions here imply all assumptions
of Theorem 16.17. Comparing these assumptions, it suffices to validate
(17.1)-(17.3), [sup a] and [0-D].
For /ie2, from (2.11) we know that there is a constant k such that
\a(h) - a'\ = \h(x) - x\ * v < kx2 * v,
Var(a(/i)) < Var(a') + kx2 * v < kF.
Therefore (17.1) is valid and for g € J\, a(h),(3,g * v are continuous in t.
For h G 2C, set
ka{x) = (h(x) - x)(l - gi/a(x)) G Ji,
ka(x) = (h2{x) - x)(l - 9i/a(x)) G JU
where gp(x) = (p\x\ - 1)+ A 1. From (2.10) and (2.11) ensure
a(h) - a' = (h(x) - x) * v = ka(x) * v + ({h(x) - x)gl/a(x)) * v, (19.4)
0(h)-0 = x2*v-h2(x)*v = ka(x)*v + ({h2(x)-x2)g1/a{x))*i>. (19.5)
There is a constant k such that
\h(x) - x\g1/a(x) < kx2I[lx>a], (19.6)
\h2(x) - x2\g1/a(x) < kx2I[M>a].

§2. Convergence to a Levy Process 497
Due to (19.3), if a is large enough, the second term in (19.4) and (19.5) may
be less than any given positive number. Therefore the continuity condition
of Theorem 16.17 may be deduced from condition v) in Theorem 16.19.
Similarly to (19.4), we have
an(h) - a'n = ka(x) * i/B + flj,
\n»\ < kx2i[]x]>a] * un.
Hence [sup a] may be deduced from (19.1)-(19.3), [v-D] and [sup a'].
Finally, by (2.12)
0n(h) - 0n = ka(x) * vn + if (a) + S7n, (19.7)
|7f (a)| = \(h2(x) - x2) * un\ < k(x2I[]x]>a]) * un, (19.8)
7? = (Aa«(h))2 - (Aa'P)2.
P ~
Therefore, if we can prove ^7^->0, then [0-D] may be deduced from
s<t
(19.8), (19.1), (19.3) and [fi'-D]. But
E 7?| < E |Aa?(fc) - Aa?|(|Aa?(fc)| + |A<|), (19.9)
s<t
s<t
£ |Aa?(fc) - AaT\ < ka(x) * v? + k(x2I[M>a]) * i/?.
S<t
For fixed 'i, by (19.2), [v-D] and (19.1), (ka(x) * i/J1 + k(x21{\x\>a]) * v?,
n > 1) is tight. Owing to [sup a] and [sup a'],
sup |Aa?(/i)| = sup |AaJ(fc) - Aas(/i) o Xn| 4 0, (19.10)
8<t 8<t
sup | Aa'sn| = sup \Aa'sn - Aa's o Xn\ 4 0. (19.11)
s<t s<t
p
Now from (19.9)—(19.11) we get E 7? ~>0 an(i aU assumptions of Theo-
s<t
rem 16.17 hold. Therefore Xn -£ X. D
§2. Convergence to a Levy Process
16.20 Lemma. Assume that X is a semimartingale (resp. process
with independent increments) with predictable triplet (a,/?, v), a is a
predictable process with finite variation (resp. deterministics right-continuous
function), then X = X — 5 is also a semimartingale (resp. process with

498 Chapter XVI Weak Convergence for Semimartingales
independent increments) and its predictable triplet (a, /?, V) may be written
as follows:
at(h) = at(h) -5t+Z I V{s, x)u({s} x dx) + £ V(s, 0)(1 - a3),
s<t JR s<t
0 = 0,
V([0,t] xA) = JJ IA{x - Aas)I[x¥:A~t]u(ds,dx)
+ E (1H- as)I[A~aji0]IA(-^as) (20.1)
8<t
where as = v({s} x R),
V(t, x) = Aat + h(x - Aat) - h(x). (20.2)
In particular, if 5 is continuous, then
a{h) = a(h) - 5, /? = /?, 17 = i/. (20.3)
Proof At first, assume that X is a semimartingale, X = Xc implies
(3 = /?. Suppose that /r*, /z* are the jump measures of X, X respectively,
W e V+ and W'(s,x) = W{s,x- A5s)/[x/A~a]. Then
W * Mf = £ W(a, AXS - A5s)/[AXa/A-s]
= W> * tf + £ W(s, -A53)/[aSj#0AXj=0].
By virtue of Theorem 5.42, there is a sequence (Tn) of predictable times
with disjoint graphs such that (JnPn] is the predictable support of the
thin set [AX ^ 0] U [A5 ^ 0]. Write D = [AX ± 0]. Then o = \lD) and
E[W*nl]
= E(W'*n*)+Y: EmTP,-AaTp)I[A~{TpmIDc(Tp)I[Tp<oo]}
= E(W * i£) + E £[W(TP, -A5Tp)/[A5(:rp)5,0](l - aTp)/[rp<oo]]
P>1
= E[W*I700],
where F is defined by (20.1). Thus (fix)p = V.
Next, let h € Zc. By (2.2) and direct calculation we have
X(h) = X- E[AX - h{AX)}
= X0 + M(h) + a(h) -a + V*nx + Y: V{-, 0)IDc

§2. Convergence to a Levy Process 499
Since h(x) = x while |x| < c, V(t,x) = 0 while \x\ + \Aat\ < c. It is easy
to know that
v*tixeAloc, (v*nx)p = v*v,
E v(; o)iDc e Aloc, (E v(., o)/Dc)p = £ v(-, o)(i - a).
Hence
X(/i)- (V * i/ + EV(-, 0)(1 - a) + a(/i) - 5)
= X0 + M(/i) + V * (m - i/) + £V(., 0)(/Dc - (1 - a)) G Moc-
Therefore a(/i), given by (20.1), is the first characteristic of X.
Finally, if X is a process with independent increments, calculating the
characteristic functions of X yields (20.1) and (20.2). □
Remark. For convenience of calculation, introduce the following
predictable random measures on R+ x Ri
i/*([0,t] xA) = u([0tt] xA)+£(l- a3)60(A)I[aa>0]u[~a>0](s),
V*([0, t]xA)= V([0, t) x A) + £ (1 - a,)*o(>l)/t5s>o]uP,>o](s)'
where as = v{{s} x R). Then for non-negative /(s, x) with /(s, 0) = 0 we
have
f * [/* = / * v, / * |7* = / * 17.
Using these notations, (20.1) may be written as follows:
at(fc) = at(h)-at + E / ^(s,*K(W x dx), (20.4)
£ = /?,
/ [ W(s,x)u*(ds,dx)= f f W(s,x-Aas)v*(ds,dx), (20.5)
JO JH JO Jil
where W € V+. Moreover, from (2.7) we obtain
0(h) = (M(h))=0 + (h-h)*v*t (20.6)
where hs = I h(x)u({s} x dx) = I h(x)v*({s} x dx). The corresponding
characteristic /?(/i) for X is
^(h) = /3 + (h-k)2*V*, (20.7)
ks = I h(x)u* ({s} x dx) = / h(x - Aas)i>*({s} x dx)
= I h(x- Aas)v{{s} x dx) + (1 - a3)h{-Aas). (20.8)

500 Chapter XVI Weak Convergence for Semimartingales
In particular,
J3(/i) - J3(h) =(h- k)2 * 17* - (h - h) * i/*
= [(/i(x - A5) - A:)2 - (h(x) - h)2} * v\ (20.9)
16.21 Lemma. Let h e Zc with \h(x)\ < K {constant), h(x) = x
while \x\ < c. Ifa = a(h) and
sup \Aas(h)\ < £ < c/2 a.s., (21.1)
s<t
then fora,f3,T/ given by (20.1), (20.7) and (20.8) we have
Vart[a(/i)] < [e + w(e, fc)]i/([0, t] x {x : |x| > c/2}), (21.2)
sup |^(fc) - &(/i)| < 4tf[£ + 3w(e, fc)]i/([0, t] x {x : |x| > c/2}), (21.3)
s<t
s\ip\g*Vs-g*vs\ <Lj(e,g)v([0yt] x {x : |x| > 2e}), (21.4)
where u(6,h) = sup |/i(x) —/i(x')|, g e J\ defined by (7.1) and g(x) = 0
|x-x'|<£
as |x| < 2e.
Proo/. If |x| < c/2, then (21.1) implies |x| + |A5s| < c and V(s,x) = 0.
Meanwhile, \V(s, x)| = |A5S + h(x - A5S) - h(x)\ < £ + v(£, h). Hence
Vart(a(fc)) < sup £ I / V(ryx)u({r} x dx)\
s<t r<s ' J[\x\>c] '
< [e + Lj(e,h)]v([Q,t] x {x : |x| > c/2}),
sup \g * u3 - g * vs\ = sup / / [g(x - Aar) - g(x)]u(dr x dx)\
s<t s<t ' JO ^[|x|>2e] '
<«>(e,g)i/([0,t]x{x:\x\>2e}).
If |x| < c/2, for h and k given by (20.8), we have
\h(x - Aas(h)) -ks- h(x) + hs\
= | - Aas(h) + ( I + f ) [h(x) - h(x - Aa,(ft)]/({«} x dx)|
[|x|<c/2] [|x|>c/2]
<[e+u(eth)]u({s}x{\x\>c/2});
if |x| > c/2,
\h(x - Aas) -ks- h(x) + hs\
< u>(e, h) + \ f [h(x) - h(x - Aas(h))]v* ({s} x dx) < 2w(e, h).

§2. Convergence to a LeVy Process
501
Thus
sup|/i>)-to|
s<t
< E / \h(x - A5S) - ks)2 - (h(x) - hs)2\u*({s} x dx)
s<tJRl '
< 4AT £ / r + / r IM* " A5S) - k3 - h(x) + ha\v*({8} x dx)
s<tJ[\x\<%] J[|x|>f]
<4K[e + 3u(e,h)]vn{[Q,t] x {\x\ > §}).
So (21.2)-(21.4) hold. D
16.22 Theorem. Suppose that for each n Xn is a semimartingale
on $ with predictable triplet (an,/?n, un) and X is a Levy process with
predictable triplet (a,f3,v). If the following conditions are satisfied: i)
Xg^Xo, ii) [sup a], \ji-D], [i/-JD] (cf. (7.1)-(7.3)) hold for a dense
subset D of R+, then Xn —>X.
Proof. Since X is a Levy process, without loss of generality, we may
assume that X is a Levy process on canonical filtered probability space
$£> with deterministic triplet (a,/?, v), which is continuous in t.
At first, assume a(h) = 0, for some h G 2C, then X is a
semimartingale. We will verify that all assumptions of Theorem 16.17 are satisfied.
Conditions i) and ii) in Theorem 16.17 are just conditons i) and ii) in
Theorem 16.22 respectively. Take F = i8 + (x2Al)*i/, then condition iii)
holds. Since a,f3,v are deterministic, (17.2) and condition v) hold, and
Corollary 11.37 yields condition vi). Therefore by Theorem 16.17 we have
Xn^X.
Next, if a(h) ^ 0, set
Y = X - a(/i), Yn = Xn - an(h).
By Lemma 16.20, Y is a Levy process with predictable triplet (a,/3,V):
a{h) = 0, /? = /?, ]5(h) = 0(h), V = u. (22.1)
Yn is a semimartingale with predictable triplet (a12,/? ,1^):
*t(h) = £ / V(s, x)vn*({s}, dx), (22.2)
3<tJR
T{h) = (3n + (h- kn)2 * Vn\ (22.3)
A£ = J h(x)Vn*({s},dx),

502 Chapter XVI Weak Convergence for Semimartingales
0*17?*=/ f g(x-Aa^(h))un*(dsxdx). (22.4)
Jo Jr
We will verify that all assumptions of this theorem also hold for Yn
and Y. Since C(Y<?) = C(X$)^C(X0) = C(Y0), Condition i) holds for
Yn and Y. Due to [sup a] and the continuity of a(/i), for all t > 0.
sup|Aa£(/i)| —>0 as n —> oo.
3<t
Meanwhile for h G 2C, there are c > 0 and K such that \h(x)\ < K and
h(x) = x while \x\ < c. By virtue of [i/-D], (^n([(M] x {x : |x| > c/2}))n>!
is tight. Therefore Lemma 16.21 and condition ii) in Theorem 16.22 imply
that
suP|a?(/i)|£o, j%(h)-0t(h)Zo, w>o,
s<t
g*v?-g*vt£o, V*>0, ge J.
So all assumptions of this theorem also hold for Yn and Y. The result
proved for the case of a(h) = 0 implies Yn —> Y. Now, due to [sup a], the
continuity and non-randomness of a(/i), we get (cf. Problem 15.20)
Xn = Yn + an{h) ^Y + a{h) = X. D
16.23 Corollary. Suppose that for each n, Xn is a cadlag process with
independent increments on $, the predictable triplet of Xn is (an,f3n,vn),
and X is a Levy process with predictable triplet (a,/?, v). If the following
conditions are satisfied: i) Xft —> Xq, ii) for a dense subset D of R+,
[sup a], [(3-D] and [v-D] hold, then Xn^X.
Proof. Since Xn and X are processes with independent increments,
(an, f3n, vn) and (a,/?, u) are deterministic.
If for each n, Xn is a semimartingale (i.e., an(h) is a function with
finite variation), then Theorem 16.22 implies Xn —+X. In general, set
Yn = Xn - an(/i), Y = X-a.
Then Y is a Levy process with predictable triplet (0, /?, */), yn is a process
with independent increments, its predictable triplet (a71,/? ,I7n) is given
by (22.2)-(22.4). By (21.2), a^/i) is a function with finite variation, and
hence Yn is a semimartingale. Now by the same argument as in Theorem
16.22 we have Yn £ Y and Xn -£ X. D

§2. Convergence to a Levy Process
503
16.24 Theorem Suppose that for each n, Xn is a semimartingale
on $ with predictable triplet (an,/?n,i/n), and X is a Levy process with
predictable triplet (a,/?,*/). If
lim iiS P((x2JN>a] * i/tn > rj) = 0, Vt? > 0, t > 0, (24.1)
and i) Xq-^Xqj ii) /or a dense subset D of R+, [sup a7], [/3'-.D] and
[v-D] (cf. (7.5) (7.6) (7.2)) hold, thenXn^X.
Proof From the proof of Theorem 16.19 we know that (24.1) and
[sup a7], [/3'-D] [v-D] guarantee that [sup a], [/3-D] hold. Hence Xn £ X
is deduced from Theorem 16.22. □
16.25 Corollary. Suppose that for each n, Xn is a process with
independent increments, its predictable triplet is (a71,/J71,*/1), and X is a
Levy process with predictable triplet (a,/?, v). If
lim Bm(x2/ixi>a) * v? = 0, V* > 0, (25.1)
/•
and i) XJ —>Xo, ii) /or a dense subset D of JR+, [sup a'], [/3'-.D] and
[i/-D] AoW, thenXn^X.
Proof Prom the proof of Theorem 16.19 we know that (25.1) and
[sup a'],[f3'-D],[v-D] guarantee that [sup a],[/?-D] hold. Hence Xn -£ X
is deduced from Corollary 16.23. □
Remark. Theorems 16.22 and 16.24 give sufficient conditions of
convergence in law for semimartingales to a Levy process. Comparing with
Theorems 16.17 and 16.19, the assumptions of Theorems 16.22 and 16.24
are rather simple and natural. Even if they are not necessary (cf. the
remark after Theorem 16.30), when Xn is a process with independent
increments, conditions i) and ii) in Theorem 16.23 (and conditions i) and ii)
in Theorem 16.24 under assumption (25.1)) are necessary (cf. Theorem
3.4 and Theorem 3.7 of Chapter VII in Jacod and Shiryaev [1]).
16.26. Assume that $ = (fi,^",F = (Tk)k>o,P) is a probability
space with a filtration of discrete time. Recall that r = {rt)t>o is a time
change on $ if each path of r is an N-valued cadlag function and for each
£, rt is an ^-stopping time.
For each n let Un = (U£)k>i be an adapted sequence of r.v. on $,
rn = (t/1) be a time change on $. Set

504 Chapter XVI Weak Convergence for Semimartingales
then Gn = (G?) is a filtration on {fl,?, P). Put
*t= E tffc, *>0, (26.1)
fe<T,n
then Xn is a semimartingale on $n = (ft, Jr,Gn,P). For liG2, the
predictable triplet (an, /?n, i/n) of Xn are:
a?(h)= £ ^fc-i[fc(t^)], (26.2)
Jfc<r(n
ft{h)= E i?fc-i[fc(^)], /3T = 0, (26.3)
fc<Tt"
vn(dt,dx) = ZIlx*o,k<T?(t)]Pk-i[Uk € rfx]fffc(A),
<7*^ = E £fc-lfo(tf?)/[tf»*>]], (26.4)
where E^fc] = £Kl^ib-i], Dfc-ifc] = ^fc-iK]2 - (£k-i[£])2. Moreover,
if JE?[(£/£)2] < oo, then Xn is a locally square integrable semimartingale
and
x?= £*>?=£ (u?-JBfc_i[u?])+ £ JB*-i[w?],
tt;»= £ «nK], (26.5)
k<T?
P'tn= E Ofc-il^]- (26.6)
k<T?
16.27 Theorem. Assume that for each n e N', Un = (U%,k > 1)
is an adapted sequence of r.v. on $, rn = (rtn)t>o is a time change on $
with to = 0, and
x?= z u%. (27.1)
k<T?
Suppose that X is a Levy process with predictable triplet (a, /?, f) and
X0 = 0.
1) If for a dense subset D of R+ the following conditions hold:
[sup a]: sup I £ Ek-i[h(U%)] - as(h)\ -£o, V« > 0 and
- * /or some ft € Zc,
\fl-D] : E ^fc-itM^fc)] "^ A(h). V< € I> and /or somen € £c,
[i/-D] : E £*-ifo(tffc )1 -% * *, VteD, g£ Jx,
fc<T(n
ften Xn -4 X.

§2. Convergence to a Levy Process
2) If (U%) satisfies the following conditions:
lim IiEp( £ «fc-i[(^)2/[|w|>a]] > v) = 0 Vij > 0,* > 0,
a—►OO 71 \ j^<Tn u fc i J /
and for a dense subset D of R+
505
[sup a']: sup £ £*_![[/£]-a's
► 0, Vt>0,
k<T»
[u-D]: Z Ek-MUSyZgtvu VteD,geJ,
k<T?
thenXn±X.
Proof Since for each semimartingale Xn defined by (27.1), its
predictable triplet is given by (26.2)-(26.6), 1) and 2) follow directly from
Theorems 16.22 and 16.24 respectively. □
In particular, applying the previous results to a sequence of
independent r.v., we get the following corollary.
16.28 Corollary. Suppose that for each n, Un = (U%, k > I) is a
sequence of F-independent r.v. on $, rn — (r/1) is a time change on $
with Tq = 0 and (X™) is defined by (27.1). Let X be a Levy process with
predictable triplet (a,/3, v).
1) If for a dense subset D of R+ the following conditions hold:
[sup a] : sup £ E[h(U%)]—as(h)\ —► 0, Vt > 0 and for some h G Zc,
\J3-D] : £ D[h(UE)] — A(/i), VtGD and for some h G Zc,
Jfc<rtn
[u-D]: £ E\g(U2)]->g*vu Vt G D, g G Ju
k<r?
then X —> X.
2) // (U£) satisfies the following conditions:
lim IimP( £ E[(U2)2Im]>a] > 7/) = 0, Vt? > 0, < > 0,
and for a dense subset D of ii+
[sup a']: sup £ £[[/£] - a'.
0, Vt > 0,
["-£] : £ £[<7(C£)] - 5 * *u Vt € D, 9 e Ji,

506 Chapter XVI Weak Convergence for Semimartingales
thenXn±X.
16.29 Lemma. Assume that for each n, Xn is a step process on $, /in
is the jump measure of Xn, vn = (/zn)p. Let X be both a step process and
a Levy process, and v be the dual predictable projection of jump measure
ft of X. If the following conditions are satisfied:
i)xs±x,
ii) for a dense subset D of Jt+
g*u?Zg*vt, VteD, geJ2, (29.1)
where J2 = {g : g(x) and g{x)/x are bounded continuous on R\ {0}},
thenXn^X.
Proof Let (an,(3n,vn) be the predictable triplet of Xn, then
an(h) = h(x) * i/n, f3n = 0, f3n(h) = h2*vn- £(Aan(/i))2.
And the predictable triplet (a, /?, v) of X is
a(h) = h*v, f3 = 0, 0(h) = h2*v.
Because J2 D Ji, (29.1) imphes [v-D). For each / G J2, f*vt is continuous
in t. Thus Lemma 16.8 entails
sup|/*i/sn-/*i/s|^0, V*>0, feJ2.
s<t
Since Zc C J2, (29.1) also implies [sup a]. If h G £c, then /i2 G J2, and
owing to the continuity of a(h) and [sup a] we have sup | Aa£| —♦ 0, V* > 0,
s<t
\ft(h) - (3t(h)\ < \h2 * i/? - fc2 * i/tl + Es<t |AaJ(fc)|2
< |/i2 * i/? - /i2 * i*| + suPs<, |A<(/i)||/i * i/?| £ 0,
i.e., [/3-JD] holds. Therefore Theorem 16.22 implies Xn^X. D
16.30 Theorem. Assume that for each n, Xn is an adapted counting
process, (Xn)p = A71, X is a Poisson process, Xp = A is continuous. If
for a dense subset D of ii+
A?ZAU VtGD, (30.1)
thenXn^X.
Proof. Xn is a semimartingale and its predictable triplet (a71,/?71,*/1)
satisfies
a?(k) = fc(lK", ft(h) = h2(l)(A? - E(A^)2), g*v?=g(l)A?.

§3. Convergence to a Continuous Levy Process
507
Similarly, the predictable triplet (a, /?, v) of X satisfies
a?(h) = h{l)Au A{h) = h2(l)Au g*v? = g(l)At.
Since A is continuous and nondecreasing in t, it is easy to deduce [sup a],
[/?-/?], [v-D] from (30.1). Thus Theorem 16.22 implies Xn^X. D
Remark. The following example explains that (30.1) is not necessary
for convergence of counting processes to a Poisson process.
Suppose that X is a homogeneous Poisson process with 25[-X$] = A£,
(Tk) is the sequence of jump times of X. F is the complete natural
filtration of X. Set
X? = S I[t>Tk+l/nh
then Xn is a counting process. Since (7* + l/n)k>i is a sequence of
predictable times, Xn e V and (Xn)p = Xn. Because Tk + 1/n -♦ Tk as
n -> oo, we have Xn -£ X by Corollary 15.59. But (Xn)f = X? ~h \t =
(*)?■
§3. Convergence to a Continuous Levy Process
In this paragraph we will apply the general results for convergence
in law of semimartingales to a special case, where the limit process is
a continuous Levy process. In particular, we will give sufficient
conditions for convergence in law of locally square integrable martingales and
semimartingales to a continuous Levy process. By the way, necessary
conditions for these cases are also discussed.
16.31 Lemma. If for each n, Xn is a semimariingale with jump
measure iin and predictable triplet (a71,/?71,*/1), then the following statements
are equivalent:
1) (AXn)*t = sups<t \AX?\ £ 0, as n - oo, Vt > 0,
2) (z2/[|*|>£]) * /*? £ 0, <w n^oo.VOCoO,
3) [A0] : /[W>e) * v? Ao, as n - oo, Vt > 0, e > 0,
4) / * i/tn 4 0, as n -+ oo, Vt > 0, / € Ji-
In this case, for any h,h' € Z we have
sup \a?(h) - a?(fe')l £ 0, sup |#(/i) - #(fe')| 4 0, (31.1)

508 Chapter XVI Weak Convergence for Semimartingales
(Xn(h)Yt=sup\X?(h)\-?>0, (31.2)
(AMn(h))l = sup \AM?(h))\ - 0, (31.3)
where Xn(h) = £(AXn - h(AXn)), Xn(h) = Xn - Xn(h) = X$ +
Mn(h) + an(h), and Mn(h) € MocO-
Proof. Notice that for e > 0 and 0 < 6 < 1
[(AX»); >e}= [ EtI[\AX?\>e) >S}= [/[w>£) * tf > 6]
= [(*2/N>*])*M?>e2]. (31.4)
Hence 1) <=>■ 2). Because (I[\x\>e]*iin)p = J[|*|>e]*"B and A(^[|x|>e]*/*B) <
1, Lenglart's inequality implies the equivalence of 2) and 3).
For / € Ji, there is a constant a such that /(x) = 0 while |x| < 1/a
and |/(x)| < a. Put g(x) = (f|x| - 1)+A 1 € Jj. Then
1/ * v?\ < a/[|«|>i/«] * «f, 7[|x|>e] *v?<9* ""•
Hence 3) and 4) are equivalent. If h, h! € Z, there exists a constant a such
that
h{x) = h'(x) = x, |x| < l/o, |fc(a:)| < a, \h'(x)\ < a.
Thus \h(x) - h'(x)\ < 2a/[|s|>1/o). By (2.8) and (2.9) we get
|a?(fc) - a?(V)| = |(/i - /»') * if| < 2a/(|l|>1/a] * if,
Iff (fc) - #(fc')| = |(/>2 - /i'2) * if - £.<t[(Aa»(fc))2 - (A<(/i'))2]|
< 8a2/t)l|>1/a] * if.
Therefore 3) implies (31.1).
Finally, since [(Xn(h))*t ? 0] C [(AIn(/i))(' > 1/a], (31.2) holds.
Meanwhile, if 0 < e < 1/a, we have
\AM?(h)\=\AX?(h)-Aa?(h)\
< |AXB| + |AAf (/i)| + I / /i(x)i/n({f}, dx)
1 ^flil>el
+ e,
'[M>«1
(AM»(/»))? < (AI")J + 2(*»(/i))? + a/[N>£, * if + e.
Hence (31.3) is true. □
16.32 Lemma. Assume Mn e M\oc and t > 0 is fixed.
1) If((Mn)t, n > 1) is tight, then ([Mn]u n > 1), (sup|Msn|, n > 1)
3<t
are tight.

§3. Convergence to a Continuous Levy Process 509
2) If Es\ip\AM?\2 < C < oo, then the tightness of {[Mn]t, n > 1),
s<t
(sup|Mp|, n > 1) and ((Mn)t, n > 1) are equivalent.
s<t
Proof 1) Since (Mn) = [Mn]?, (M71)2 - (Mn) e M{Qc and (Mn) is
predictable, 1) may be deduced by Lenglart's inequality.
2) Since (M71) is dominated by [Mn] and (A[Mn])* = (AM71);2 G L\
the tightness of ((Mn)t) may be deduced from the tightness of ([M71]*)
by Lenglart's inequality. Meanwhile, by Davis inequality, ^[M71] is
dominated by k(Mn)*, where k > 0 is a constant, and A(Mn*) < (AM71)*.
Hence the tightness of ([M71]*, n > 1) may also be deduced from that of
((Mn)l n > 1). D
16.33 Lemma. Assume that Mn € M^, |AMn| < c and for all
t>0 (AMn)^0.
1) We have
(A(Mn))t*-^0, Vf>0. (33.1)
2) // ([Mn]t) n > 1) is tight, then
sup|[Mn]s-(Mn)s|£o, Vf>0. (33.2)
3<t
Proof. 1) Since A[Mn] = (AM")2 < c2, E(A[Mn])f -» 0 by the
dominated convergence theorem. But A(Mn) = fl(A[Mn]) and by Doob's
inequality (Theorem 2.15)
E[A((Mn))f] < 4£7[(A[M"]);2] -> 0,
hence (A(Mn))t*£o.
2) Set Yn = [Mn] - (Mn>, then |Arn| < c2, Yn € M*£ and
[Yn]t = E(A[M"]S - A{Mn)a)2 < 2 £[(A[M"]3)2 + (A(M")a)2]
< 2{A[Mn])*t[Mn]t + 2{A{Mn)Yt(Mn)t.
Owing to the tightness of ((Mn)t,n > 1), (33.1) and the assumptions,
we get [Yn]t -+0. Furthermore, by the boimdness of |Ay| and Lenglart's
inequality we obtain
sup\[Mn}s-(Mn)a\ = (Ynrt^0.
(33.2) holds. D

510 Chapter XVI Weak Convergence for Semimartingales
16.34 Corollary. If Mn € M?^, \AMn\ < c, p is a (deterministic)
continuous function, and D is a dense subset of R+, then the following
statements are equivalent:
2) {AMn)*t-^0, W>0 and
(Mn)t^pt, VteD. (34.1)
Proof. 1) => 2). By Lemma 16.8, 1) is equivalent to
sup\[Mn)s-0s\Zo.
3<t
Then {{AMn)2)*t = (A[Mn])£^0, and (34.1) is deduced from (33.2). •
2) => 1). By Lemma 16.32.1) we know {[Mn]u n > 1) is tight. Now 1)
may also be obtained from (33.2). D
16.35 Lemma. Let Mn e M?oc, \AMn\ < c and M € M^. If
Mn±M, (35.1)
then
[Mn]±(M). (35.2)
Furthermore, if (M) is deterministic, then
sup|[Mn]a-(M)a|^0, V*>0. (35.3)
3<t
Proof. Let 0 = £§ < *i < * * * be a partition of R+ for each k such that
lim j
k—>oo
Urn sup|^-^_!| =0.
3
By Ito formula,
(M§ ~ M^_f = 2 j^ {Ml. - M?u)dMZ + ([M"]t* - [M^J,
[n=E(^At-^ J2-2E/t k (M2--M?k )dMZ
= Ytnk + Zf, (35.4)
(Znk)t < 4[u;(max(^ - tJ.J, M", *)]2(M")t. (35.5)
j J J
By the assumptions, (Mn) is C-tight, hence Lemma 15.49 implies
lim IImP(u>(max(^ - tk,),Mn,t) > rj) = 0,Vt? > 0,t > 0. (35.6)
Jfc—>oo n j

§3. Convergence to a Continuous Levy Process 511
By virtue of the assumptions and Lemma 16.32, ((Mn)t,n > 1) is tight
for each t > 0. Hence (35.5), (35.6) and Lenglart's inequality imply
lim KmP({Znk)*t > t?) = 0, VO0,r|>0. (35.7)
k—kx) n
Next, write Ytk = £ (MtkM - Mtk M)2. Prom (35.1) and the remark
of Theorem 19.33 we have
Ynk±Yk, asn^oo, Vfc > 1, (35.8)
Yk±{M), as Jfc-oo. (35.9)
Finally, due to (35.4), (35.7)-(35.9), applying Lemma 15.52, we get
(35.2). If (M) is deterministic and continuous, then (35.2) implies (35.3).
D
16.36 Theorem. Assume that for each n, Xn is a semimartingale
on $ with predictable triplet (a71, f3n, vn), and X is a continuous Levy
process with predictable triplet (a,/?,0). If [sup a] holds, then the following
statements are equivalent:
1) Xn^X,
2)i)XS±X0,
ii) for a dense subset D of R+, [Mn(h)]t^> 0t, V* € D,
iii) [A0] : JN>e] * i/tn ^ 0, V* > 0, e > 0,
3)i)X0"^X0.,
ii) for a dense subset D of R+, [(3-D] holds,
iii) /*i/tn4o,V«>0,/6Ji.
Proof. 1) => 2). 2) i) is apparent. Since {Xn) is C-tight, Lemma 15.49
implies (AXn)t*-^0. Hence by Lemma 16.31, 2)iii) holds and (Xn(h))*t^0.
Furthermore, by [sup a] we get
Mn(h) = Xn - X$ - Xn(h) - an(h)
= Xn - X$ - Xn(h) -a- {an(h) - a)
±X-X0-a = Me M^0.
Notice that (M) = /? is deterministic and M is Gaussian. Therefore (35.2)
entails 2)ii).
2) ««=> 3). By Lemma 16.31, 2) iii) and 3) iii) are equivalent and they
imply (31.4). Hence under 2) iii) or 3) iii) the equivalence of 2) ii) and 3) ii)
follows from Corollary 16.34.

512 Chapter XVI Weak Convergence for Semimartingales
3) => 1). This is just the conclusion of Theorem 16.22. D
16.37 Example Assume that W is a Brownian motion on $, b = (bt)
is an adapted process with |6| < 1. Let
-/
Jo
bsds + Wu
and G be the usual natural filtration of Y. If °b is the G-optional
projection of 6, take
X?=Xt= [\bs-°b9)ds + Wu
Jo
then the predictable triplet (a71,0n, vn) of Xn on $ is:
a? = f\bs - %)ds, 0? = t, vn = 0.
Jo
But Xn is a G-Brownian motion (cf. Problem 16.4), hence C(Xn) =
£(W).
This example illustrates that [sup a] is not necessary for Xn —► X.
16.38 Lemma. Assume for each n, Xn is a semimartingale on 3>
with predictable triplet (a71, /3n, vn). For 6 > 0, write
[A6]:(\x\6IlM>e])*v?£o, as n^oc,V*>0, e > 0.
1) If 6 > 0 and for every t > 0, ((AXn)f, n > 1) is uniformly
integrable, then
Urn nEP((|x|*/[|x|>a]) * i/» > rj) = 0, V* > 0,7? > 0. (38.1)
2) For 6 > 0, [A$] /io/ds z/ and only if [Ao] and (38.1) hold.
3) J/Xn € Alloc and [Ai] ho/ds, </ien /or every h € Z
Vai(an(h))t ^ 0, Vt > 0. (38.2)
4) 7/(38.2) holds, anc(h) = an{h) - EAan(/i), then
(<*nc(h)yt + £ |Aa?(/»)| -0, V< > 0. (38.3)
5) 7/ (38.3) /io/ds, then
{an{h))*t-?>0, V*>0. (38.4)
Proo/. 1) Write Vt"(a) = (|*|*/N>a]) * tf = £ |A^|*/[|AXn|>o].
Then AVt"(a) = |AXtf/,|Ax.|>a], (^»)p = (M'/[|*|>a]) * ""• By

§3. Convergence to a Continuous Levy Process 513
Lenglart's inequality we have
P((\x\°I[{xl>a])*v?)>V)
< i(e + E[sup |AX?\sI[lAXnl>a]]) + P(Vtn(a) > e)
Letting n—► oo, a —► oo, e —► 0 successively we get (38.1).
2) For a > e > 0
(W'/[w>e]) * v? < (\x\6I[\x\>a]) * *? + a6I[lxl>£] * i/?.
Letting n—► oo, a —► oo, £ —► 0 successively, [A$] follows from [Ao] and
(38.1). The reverse implication is apparent.
3) If Xn G Alloc, then an(h) = (h(x) - x) * vn. For h G Z there is a
constant a such that /i(x) = x while |x| < 1/a and
Var(ari(/i)), < |/i(x) - x\ * i/» < a(|x|/N>1/a]) * v? £ 0.
Hence (38.2) is true.
4) and 5) are obvious, since {an{h))*t < {anc{h))*t + £ |Aa£(/i)| <
s<t
Var(an(/i))*. D
16.39 Lemma. Let Xn G M\oc$ with |AXn| < K(a constant),
X G A^f^o ^^ deterministic (X) = (3, and D be a dense subset of R+.
Then the following statements are equivalent:
1) Xn^>X,
2) [Xn]t£pt, VteD,
3) [Ao] and(Xn)t^(3t, V* G D.
Proof Notice that 2) implies (AX71)?2 = (A[Xn])*t-?>0. If take h e Zc
such that /i(x) = x while |x| < K, then Xn(/i) = 0, an(h) = 0, Mn(/i) =
Xn. Hence the equivalence is just a consequence of Theorem 16.36. □
16.40 Theorem. Let Xn G M\oc,o with predictable triplet (a71,07\ vn),
X G M\oc$ with deterministic (X) = /?, and D be a dense subset of R+.
If for some h G Z
(<*nc(h)yt + £ |Aa?(fe)|4o, Vt > 0, (40.1)
Wien £/ie following assertions are equivalent:
1) Xn±X,

514 Chapter XVI Weak Convergence for Semimartingales
2)[Xn]tZ0t,Vt€D,
3) [A0] and for some(every) heZ, [Mn(/t)]t-^/3t,Vf € D,
4) [Ao] and [/3-D] holds for some(every) h € Z :
0?(h) = (Mn(h))tZ(3t, Vt€D.
Proof. At first, notice that each of 1) and 2) implies [Ao]. By Lemma
16.38.5), (40.1) entails [sup a]. Then by Theorem 16.36, 1), 3) and 4) are
equivalent to each other.
If h G Z satisfies
{x,
o,
\x\ < 1/a,
h(x) = { \h(x)\ < a, (40.2)
|x| > a,
then
\[X"]t- [Mn(h)]t\ = £(AX?)2 - £(fc(AX?) - A<(fc))2
1s<t s<t
< E l(AX«)2 - (h(AX?))*\ + 3a £ |A<(/>)|,
s<t s<t
[\[X%- [M"(h)]t\ >e]c [(AX")*t > i] U [ £ |AaJ(fc)| > ^
|A<WI> ^
Due to [Ao] and (40.1), 2) and 3) are equivalent. □
16.41 Theorem. Let Xn G M\oc0, X G M\£c0 with deterministic
(X) = (3 and D be a dense subset of R+. Recall
[A2]: (x2/[|I|>el)*^n^0, VO0, e>0.
Then the following assertions are equivalent:
1) [A2] and Xn -£ X,
2) [A2] and{Xn)tZ/3t,Vt<=D,
3) [A2] and[Xn]t^l3t, Vt <= D,
4) [A2] and (Mn(h))t -£ A, V* € £> and for some h€Z,
5) [A2] and [Mn(h)]f^(3t, Vt € D and for some h G Z,
6)(Xn)Zf3t,[Xn}tZl3t,VteD,
7) Xn-^X and {Xn)t-^0t, V* € D.
Proof. Due to Lemma 16.38, [A2] implies [Ao] and (40.1). So by
Theorem 16.40, we know 1), 3), 4) and 5) are equivalent mutually. If [A2]

< (a4 + l)(x2/ i ) * i/? + a E |Ao?(fc)| £ 0.
< \(Xn)t(Mn(h))t\ + a Vax(an(h))t £0,
§3. Convergence to a Continuous Levy Process 515
holds and h e Z satisfies (40.2), then
\{X»)t - (Mn(h))t\ =\(x2- h2(x)) * v? + £ (Aa?(/i))2|
1 S<t '
Hence 2) and 4) are equivalent.
It is obvious that 6) is deduced from 2) and 3). Conversely, if 6) is
true, then [Ao] holds and
(|z|/[W>a]) * ^ < \(x%x>ii) * i/? < l(Xn)t.
Hence (38.1) holds for 6 = 1. Now Lemma 38.1 implies (38.3). Thus for
h satisfying (40.2) by Theorem 16.40 we have
(X»)t - (Mn(h))t = (Xn)t - A - ((Mn(h))t - A) £ 0, V* > 0,
|(z2 - h2(x)) * .tf | = \(X»)t - (M"(h))t - £s<<(A<(/i))2
<\(X»)t(M»(h))t\ + a
(x2/[|x|>^l) * "t < (*2/[|x|>a]) * »t + °>2I[\x\>e] * "t
<\(x2- h2(x)) * i/»| + 2a2/[|x|>eA(1/a)] * i/? 4 0.
[A2] holds and therefore 6) is equivalent to 2).
If 7) holds, [Ao] is also true. By the same argument, [A2] may be
deduced from [Ao] and {Xn)t —► 0t,Vt G D. Hence 7) is also equivalent to
2). D
16.42 Recall the notations in 16.26. For each n let Un = (U£,k > 1)
be an adapted sequence of r.v. on $ = (fi, T, F = (^fc)fc>o, P), t71 = (rtn)
be a time change on $. Write
X? = £ ^• (42.1)
fe<Tt"
Then the following statements are obtained, while applying Lemma 16.38
toXn:
1) maxn |C/£ | -£ 0 is equivalent to [Ao]: EjLi P*-i(|t/£| > 0 ^ 0,
2) If £/n = (Ug, k > 1) is a sequence of martingale differences, i.e.,
.Ejfc_i[[/£] = 0, [Ao] holds and (maxi<jfc<T« |£/£|,n > 1) is uniformly inte-
grable, then ££i |Efc-i[tf?J|i/f|<.)]| £<>.

516 Chapter XVI Weak Convergence for Semimartingales
3) If [A2] holds: T,ZiEk-i[(Uj:)2I[\un\>e]]^0, Ve > 0, then [A„] is
valid.
Applying these results to Xn in (42.1), we may get diverse conditions
for the sums in row of array (U£,k > l,n > 1). The next theorem is an
example.
16.43 Theorem. Suppose that for each n, (U£,k > 1) is a sequence
of martingale differences on $, rn = (r/1) is a time change on $. Let
X G -M1(^o with deterministic (X) = /? and D be a dense subset of R+.
If one of the following conditions holds:
^^[max^fc^nl^ll^O^^i^)2-^, as n^oc,Vt€D,
2) ^maxJt/^0, £Li(^)2-A, ELi l*fc-iMP%tfl>i]H -°> as
n —► oo, V< € D,
3) Y,tiEk-i[(U2)2I[]u;:\>e]}Zo, ZtiEk-iWZnZpt, asn^oc,
Vt € D, e > 0, tfien A"71 -£ X, where Xn is defined by (42.1).
The following corollary is usual called Donsker's invariance principle.
16.44 Corollary. Suppose (Yk,k > 1) is an i.i.d. sequence of r.v.,
E[Yk]=0,D[Yk] = l. Let
1 Int]
Then Xn —♦ W, where W is a standard Brownian motion.
Proof Take Tk = a{YjJ < k}, U% = -j=Yk, t? = [nt], then
y 71
Z E[{UZ)2Im>e]\ = ^[(Yia/|,yil>V&]] "> °« ««-«,•
k=l 4
Hence [A2] is valid. Meanwhile,
E«)] = M^i, asn-oc.
So the condition 16.43.3) holds and Xn -£ W by Theorem 16.43. D
§4. Convergence to a Generalized Diffusion
16.45 Definition. If X is a semimartingale on a filtered probability
space $ = (fi, T, F, P) and for some h G Zc its predictable triplet (a, /?, */)

§4. Convergence to a Generalized Diffusion 517
can be expressed by
at(h) = / b(s,Xs)ds, fit = a(s,Xs)ds, (45.1)
Jo Jo
v(dt, dx) = K{t, Xt, dx)dt, (45.2)
where a > 0 and b are Borel functions on R+ x R, K is a transition kernel
from R+ x R to R satisfying
K{t, x, {0}) = 0, /(l A y2)K{t, x, dy) < oo, V<> 0,
then X is called a generalized diffusion or a diffusion with jumps. (b,a,K)
is also called the infinitesimal characteristics of X. In particular, if v = 0,
X is called a diffusion and its trajectories are continuous almost surely.
If b(s,x), a(s,x) and K(s,x,dy) do not depend upon s, X is called a
homogeneous (generalized) diffusion.
If Ao = C(Xo), then P is a solution of the semimartingale problem
r,(X,F;Ao, a,/?,!/).
From (2.8) it is easy to know that for every h € Z, a(h) has the same
expression as (45.1) with a different b(s,x) depending upon h and
J3t(h) = / a(s,Xs,h)ds,
Jo (45.3)
a(s, x, h) = a(s, x) + K(s, x, dy)h2(y).
Let X be a homogeneous generalized diffusion with infinitesimal
characteristics (b,a,K). For / G C2(R) put
Aft*) = b(x)f'(x) + ±a(x)f"(x) + J K(x,dy)[f(x+y)-f(x)-h(y)f'(y)}.
Then by Ito formula it is easy to known that
Yt = f(Xt) - f(X0) - f Af(Xs)ds, t > 0,
Jo
is a local martingale.
16.46 Definition. Assume that X is a homogeneous generalized
diffusion with predictable triplet (a,/?, v). If
i) for each x 6 R, TS(X, F',6x,a,/3, v) has a unique solution Px,
ii) for each A6f, ih Px(j4) is a Borel function,
then we say that the uniqueness-measurability hypothesis holds for X.
If X satisfies this hypothesis, for every distribution A on R the
semimartingale problem TS(X, F; A, a, /?, */) has a unique solution.

518
Chapter XVI Weak Convergence for Semimartingales
The conditions for validity of the uniqueness-measurability hypothesis
can be found in §111 2.c of Jacod and Shiryaev [l].
It is easy to verify that the uniqueness-measurability hypothesis holds
for Brownian motion and Ornstein-Uhlenbeck processes.
16.47 Theorem. Suppose for each n, Xn is a homogeneous
generalized diffusion on $ with infinitesimal characteristics (6n, a71, Kn) for some
h G Zc, and X is a homogeneous generalized diffusion on the
canonical probability space $d with infinitesimal characteristics (6, a,, K) for
h G Zc. Ifi) X$±X0, ii) &n=>6, an=*a, Kn=tK{-,g), Vg G Ji, iii) the
uniqueness-measurability hypothesis is met for X, then Xn —* X.
Proof. For simplicity, we only prove this theorem under the following
additional assumptions of uniformity:
sup \bn{x) - b(x)\ -> 0, sup \an{x) - a(x)\ -> 0, (47.1)
X X
sup\Kn(x,g) -K(x,g)\ -> 0, as n -♦ oo, V# G Ji, (47.2)
sup [\b(x)\ + a(x) + f K(x, dy)(l A y2)] < L < oo, (47.3)
lira supK(x, {y : \y\ > b}) = 0. (47.4)
b—>oo x
For the general case, the proof needs to use the stopping technique and is
left to the reader (cf. Problem 16.2 or Jacod and Shiryaev [1]).
Now we will verify that all conditions of Theorem 16.17 are met.
Condition i) in Theorem 16.17 is just the condition i) in Theorem 16.47. From
(45.1) and (47.1) we have
\a?(h)-at(h)oXn\ = \ f bn(X?)-b(X2)ds\ <tsup\bn(x)-b(x)\ - 0.
Hence [sup a] holds. Similarly, [/3-i2+], |y-it+] also hold and the condition
ii) in Theorem 16.17 is satisfied. If take F(t) = Lt, then (47.3) implies
(17.3). The condition iv) in Theorem 16.17 follows from (47.4).
By virtue of the condition ii) in Theorem 16.47 and Lemma 15.61, 6, a
and K(-,g) are continuous in x and by (47.3) they are bounded. Hence
by (45.1)-(45.4), the condition v) in Theorem 16.17 takes place.
Finally, the conditiion vi) in Theorem 16.17 follows from the condition
iii) in Theorem 16.47. Therefore from Theorem 16.17 we know Xn —> X.
Let X be a homogeneous generalized diffusion with infinitesimal cha-

§4. Convergence to a Generalized Diffusion 519
racteristics (6, a, K). If / y2K(x,dy) < oo, then X is a locally square
integrable semimartingale. Let
b'(x) = b(x) + JK(x,dy)(y-h(y)),
a'(x) = a(x) + / K(x,dy)y2,
a't = f b'(Xs)ds, ft = f a'(X3)ds.
Jo Jo
then
16.48 Theorem. Assume that for each n, Xn is a homogeneous
generalized diffusion on $ with infinitesimal characteristics (bn, an, Kn) for
some h G Z,
Urn IIE sup / Kn(x, dy)y2IM>h] = 0, (48.1)
and X is a homogeneous generalized diffusion on $£> with infinitesimal
characteristics (b,a,v) for h G Z. If i) Xo—>Xq, ii) b'n=tb,a'n=*a',
Kn('i9)=*K('i9)> ^9 € Ji> iii) *Ae uniqueness-measurability hypothesis
is satisfied for X, then Xn —► X.
Proof. It is easy to verify that (48.1) and b'n=tb', a!n=*a! imply bn=tb,
an=ta. So Xn -£ X follows from Theorem 16.47. D
16.49 Corollary. Suppose the infinitesimal characteristics of a
homogeneous generalized diffusion Xn is (6n,0, Kn),
lim sup / Kn{x, dy)y2Il{v{>£] = 0, to > 0, a > 0, (49.1)
n->°°\x\<aJ l J
and X is a homogeneous diffusion on $£> with infinitesimal characteristics
(6, a,0). // i) XJ —+Xo, ii) b'n=tb,a'n=ta, iii) £/ie uniqueness-measurability
hypothesis is satisfied for X, then Xn —► X.
16.50 Example. Assume that yn is a homogeneous simple birth-
death process with state space Z, birth rate \n and death rate /zn, i.e., Yn
is a Markov step process with infinitesimal characteristic (see (XV.64.1))
Qn{x, A) = \n6x+i(A) + Mx-i(il).
Hence Yn is also a homogeneous diffusion with jumps. Let X™ = hnYp,
where hn is a real. Then Xn is also a homogeneous diffusion with jumps,

520 Chapter XVI Weak Convergence for Semimartingales
its infinitesimal charactersitics are
Kn{x,dy) = \n6hn(dy) + im&-hn{dy),
b'n{x) = I yKn(x, dy) = (\n - im)hn, (50.1)
an(x) = Jy2Kn(x,dy) = (Xn + »n)h2n.
If An, fMn and hn obey the following conditions:
hn I 0, (Xn - iin)hn -* m, (\n + Hn)hn -► a2, as n -► 00, (50.2)
then Kn in (50.1) satisfies (49.1). Let X be a continuous semimartingale
with predictable triplet (mt, a2t, 0), i.e., X is a Brownian motion with drift
coefficient mt. If Xft -» Xq, then by Corollary 16.49 we have Xn —► X. In
particular, take
hn = 2~n, \n = 22n-\ tin = 22n-la}'2\
1 c
then (An - fin)hn —► --loga, (An + /in)^n ""> 1 an(^ Xn —>X. X is a
standard Brownian motion while a = 1.
Now we discuss the problem of approximating a diffusion by Markov
sequences.
15.51 Theorem. Assume that for each n, Yn = (Y£, k > 0) is a
temporally homogeneous Markov sequence with transition probability pn(x, A),
and X is a homogeneous diffusion on $d with infinitesimal characteristics
(6, a, 0). Let en j 0 and put
bn(x) = - f(y- x)pn(x, dy), an(x) = - f(y- x)2pn(x, dy).
£71 J £n J
If i) X% = yon -£ X0, ii) bn=tb, an=ta and
lim sup — / (y - x)2h\y_x\>s]pn{x, dy) = 0, V<5 > 0, (51.1)
iii) the uniqueness-measurability hypothesis is satisfied for X, then Xn —► X.
Proof. For simplicity, we also prove this theorem under the following
additional assumptions of uniformity:
sup \bn{x) - b(x)\ -> 0, sup \an{x) - a{x)\ -♦ 0, as n -► 00(51.2)
X X
lim sup — (y- x)2Lly_xl>6]pn(x, dy) = 0, V<5 > 0, (51.3)
rc-><» x en J "* ■- J
sup[|6(x)| + a(x)] < L < 00. (51.4)

§4. Convergence to a Generalized Diffusion 521
Set UZ = Yp-Y?_v then
Y"fl Yn V71 V71 V^ IT71
A* - A0 - r[*/en] ~ r0 - 2-, ^Jfc •
Xn is a locally square integrable semimartingale, its predictable triplet
(an,/3n,vn) is
oo
vn(dt,dx) = e ^„(^)pn(nn-i>n-i+dx)i[XJt0\
k=l
OO
= E^en(^)pnW-^r-+^)^0],
k=l
1 f£n[t/£n)
(*2/[W>*]) * i/» = 1 Tn £n / z2/[|x|>,]P"(X;\ X? + dz)ds,
*tn = E *WI^-i] = / 6n(X?)cfa
jt=i
= f [an(X?)-en(bn(X?)f]ds.
Jo
Then (51.2)-(51.4) imply [sup a'], \ff-R+] and [i/-fi+]. Similarly to the
proof of Theorem 16.47, we can verify that all assumptions of Theorem
16.19 are satisfied. Hence Xn —> X □
16.52 Example. Assume that for each n, rf1 = (77^, k > 1) is an
i.i.d. sequence of r.v. on $ such that
P(l?2 = l)=Pn, P(7?£ = -l) = <7n, Pn + ?n = L
Set
Then 6n, a71 and the transition probability pn of yn are:
pn(x, dy) = pn6x+hn(dy) + qn6x-hn(dy),
bn{x) = e~l I {y- x)pn{x, dy) = hn{pn - qn)/en,
an(x) = e-1 J(y - x)2pn(x, dy) = h2Jen,
and
J(y - x)2I[ly_xl>hn]pn(x,dy) = 0.

522 Chapter XVI Weak Convergence for Semimartingales
If en,hn,pn, and qn obey the following conditions:
£n I 0, h2n/en -► a2, {pn - qn)/hn -> ra/a2,
then 6n=>ra, an=*a2 and Xn ±> X by Theorem 16.51, where X is a
continuous Levy process with predictable triplet (mt, a2t, 0) and Xo = 0.
16.53 Example. If for each n, tyn = (ty£, A; > 0) is an Ehrenfest
model, i.e., a Markov sequence with transition probability
pn{x,dy) = -(l - p)6x+x{dy) + ^(l + ^)tfx-i(dy), M < /n.
Set Y^71 = /inf?jt, Jf* = ^r?/c]> t^ien fen> fl71 ^d t^ie transition probability of
(Ykn) are
pn(x,dy) = -(l - ^-)tx+hn{dy) + -(l + — )sx_hn(dy), \x\ < lnhn,
t>n(x) = £nl f(y ~ x)pn{x,dy) = -j^-I[\x\<hnin)>
an(x) = e-1 f(y - x)2pn(x, dy) = ^/t|x|<^/„,,
J On
and
J(y - x)2I[ly_x]>hn]pn(x,dy) = 0.
If h>n, £n, and ln obey the following conditions:
en 1 0, —r -> fc, -^ -> a2,
then 6n(x)=* — fcx, an=ta2. Let X be a continuous semimartingale with
at = —kXt, fit = a2t, v = 0 and Xo = 0, i.e., X is an Ornstein-Uhlenbeck
process, then Xn —>X by Theorem 16.51.
We conclude this paragraph by studying the weak convergence of
empirical processes.
16.54 Definition. Let {Z^i > 1) be an i.i.d. sequence of r.v. .
Fn(t) = - £ I[Zi<t], t € R, (54.1)
is called the empirical process of size n for (Z{).
16.55 Lemma. Let (Zi,i > 1) be an i.i.d. sequence of non-negative
r.v.. Assume that P(Zi < t) = F(t) is continuous and Fn(t) is the
empirical process of size n for {Z{). Then

§4. Convergence to a Generalized Diffusion 523
1) Ytn = nFn(t), t > 0, is a counting process, its compensator w.r.t
its natural filtration is
2) Vtn = y/n(Fn(t) - F{t)), t > 0, is a semimartingale. If h G Z,
h(x) = x while \x\ < 1, then the predictable characteristics (a71,/?71,*/1) of
Vn are
^-jft1-^!^)))^^ (55-3)
un(dt, dx) = [n- v^^JdF(t)61/n(dx). (55.4)
Proof. 1) For every i, At = I[t>Zi) is a single step process, its
compensator w.r.t. its natural filtration is
AP-I*F <* dF{8) Al A \ dF^
(n Kp
J2 I[Zi<t]) h^ the form of
(55.1).
2) Since h(x) = x while |z| < 1 and AVn < 1/y/n, we have h(AVn) =
AVn. Because
V? = ±=.{Y? - (Ynft) + {-^{Ynft ~ V^F(t)),
hence
Let /in be the jump measure of Vn and g G Ji, then
« . rf = E 5(Ay-/Vi) = S »(^)/|AV?-.l = 9(^)l-e".
' • * - '(^)C">f = »(£) / [» " ^T^))dFW-
Thus (55.4) holds.
Finally, (yn - (Yn)p)/y/n is a martingale with locally integrable
variation. Hence f3n = 0 and

524
Chapter XVI Weak Convergence for Semimartingales
Therefore (55.2)-(55.4) are all .valid. □
16.56 Definition. X = (Xt,0 < t < 1) is a Brownian bridge if it is
a centered continuous Gaussian process with covariance function c(s,t) =
s At(l - s\/ t), s,t e [0,1]. (cf. Problem 2.16.) A Brownian bridge is
also a continuous semimartingale w.r.t. its usual natural filtration with
predictable triplet (a,/?,v) as follows (cf. Problem 16.10):
at = - -^ds, (3t = t, u = 0. (56.1)
Jo 1 - s
16.57 Theorem. Let (Zi,i > 1) be an i.i.d. sequence of r.v.,
uniformly distributed on (0,1), Fn(t) be the empirical process defined by
(54.1), Vtn = >Jn{Fn(t) -t),Q<t<\. Then
Vn±>X,
where X is a Brownian bridge.
Proof. Let T G (0,1) be fixed. Consider the stopped processes of X
and Vn at T:
Xt(T) = XtAT, Vtn(T) = VtnAT.
By (56.1), the predictable triplet (a(r),/3(T),i/(T)) of X(T) is
rt/\T V
a{T)t = - -^ds, 0(T)t = tAT, u(T)=0.
Jo 1 — s
It is easy to verify that they meet the localized conditions iii), iv) in
Theorem 16.17 (cf. Problem 16.3). Meanwhile, by Lemma 16.55 the
predictable triplet (an(T),0n(T),vn(T)) of Vn(T) is:
an(h,T)t= f^ Vsn(T)-^—ds,
JO 1 — 5
rtAT f yn
9^(T)t = j^9{^)[n-^-V:(T)]d,
For g G Ji, g(—=) = 0 while n is large enough. Hence g * un(T) = 0 and
g*vn(T)-(g*v(T))oVn(T) = 0,
an(h, T) - a(h, T) o Vn(T) = 0,
/•tAT yn
~ ~ rtAi yn
(3n(h, T) - 0(h, T) oVn(T) = - -=-f ,ds.
Jo v^(l -

Problems and Complements
525
Since
ftAT yn . rtAT /jg/yn\2\l/2
E
,«Ai yn ^ rtAl (g^)')*/
Jo y/n(l - s) I ~ Jo y/n(l - s)
ds
\l-s)
TM
i r1" r^~,
= —pz I w ds —► 0, as n->oo,
we have
(3n(h, T) - 0(h, T) o Vn{T) £ 0.
Therefore for each T G (0,1), Vn(T) ± X{T) (cf. Problem 16.3).
Let U? = Vflt, Un(T) = U?AT = Vf_tAT, t G (0,1). Because Z{ and
1 - Zi are identically distributed, so are Vn and £/n. Thus Un(T) -£ X(T).
Now we extend Vn and X to R+ such that V71 = Xt = 0 while * > 1.
For TV > 1 we have
. - ^)l+suphw
s<N s<Nl x^>/l a<7V'
sup \V3n\ < sup |W-)| + sup |[/;(-)|,
^U^^)<^(^^(^)^)+^(^^(^)^), for 6<^.
<3J7 ' \ 7 V3>
Hence from Theorem 15.47 we obtain the tightness of (Vn).
Finally, for 0 < £1 < t<i < • • • < tp, if U-\ < 1 < U for some i,
^(ti-O^Xfe-i) entails (V,-, • • •, V£_J £(Xtl, • • • ,**,_,). Moreover,
Vi? = --- = V5 = -Yti =... = X,p = 0, sowehave^,...,^)^^,
• • •, Xtp). Since (Vn) is tight, we get Vn-$X. □
Problems and Complements
16.1 On the canonical measurable space (D,P°,£>°), let T° = {T : T
is a D°-stopping time}. We say that the local uniqueness holds for the
semimartingale problem TS(X,D°; A,a,/?,v) if for every T G T°, any
two solutions P,Pf of the "stopped" semimartingale problem TS(X, Z?°;
) coincide on the a-field V\, where vT = l[o,r]^- Prove that
if the semimartingale problem TS(X, D°; A, a, /?, */) has a solution P and
local uniqueness holds, then for every T G T°, rs(X, 15°, A,aT,/?T,i/T)
has a unique solution P o (XT)_1.
16.2 Assume that for every n e N, Xn is a semimartingale on $
with predictable triplet (an,/3n, i/n), X is the canonical process on $£>,
(a,P,v) is a predictable triplet on $#, D is a dense subset of ii+ and

526 Chapter XVI Weak Convergence for Semimartingales
S(a) = iuf{t : \Xt\ > a or \Xt\ > a}, Sn{a) = inf{t : \X?\ > a or
IX™_\ > a}. If for some h G Zc the following conditions hold:
")
[sup aioc] : supa<t |aJAgB(a)(fc) - aaAS(a)(fc) o A"n| -»0, Vi, a > 0,
[Aoc-£>] : fasn[a)(h) - 0tAS(a)(h) o Xn 4 0, V* € £>, a > 0,
[i/-D] : g * ^nASn(a) - <7 * ^AS(a) o X» £ 0, Vt € D, a >0, $ € J;
iii) Va > 0 there is a deterministic increasing continuous function F(a)
such that
Var(as(a)) + /35(a> + (x2 A 1) * z/5(a) -< F(a),
iv) lim sup v(y, [0, i A S(a)] x {x : |x| > 6}) = 0, Va, <> 0;
v) Vi € £>, 5 € J\, x i-> at(x,/i), x •-» (3t(x,h), x *-> g * vt{x) are
continuous functions on D equipped with the Skorokhod topology;
vi) Po is a locally unique solution of rs(X, D°; Ao, a,/?, i^);
then Xn -£ X.
16.3 Assume that for every n € iV, Xn is a locally square integrable
semimartingale on $ with predictable triplet (an,/3n, i/n), X, S(a), Sn(a),
(a, /3, i/), D are the same as in the previous problem. Suppose that
lim BmP((x2/tw>6]) * i/tnA5„(a) > Ji) = 0, Vi, a, i) > 0,
and the following conditions hold:
i)£(X0")^A0,
ii)
[sup a\oc] : sup |o>5„(a) - a'^s(a) o Xn\ £0, Vi, a > 0,
^]:^S»(a)-4S(a)^n'0- V<€A a>0,
[*/-£)] : the same as in the previous problem,
iii), v), vi): the same as in the previous problem.
Prove Xn^X.
16.4 Prove that the Xn in Example 16.37 is a G-Brownian motion.
16.5 Let Mn G M\oc, |AMn| < c. Prove
1) If (Mn) is C-tight, then {[M71]) is tight,
2) If Mn -4 M and M G Mfoc, then (Mn, [M71]) -£(M, (M)).
16.6 Suppose for each n, Un = (£/£, A; > 1) is an adapted sequence of
r.v. on $, rn = (t/1) is a time change on $, X" = Sjb=i ^T> ^d ^ *s a

Problems and Complements
527
continuous Levy processes with predictable triplet (a, /3,0) and Xo = 0.
Assume
sup £ «fc-i[0T/[|i/»|<ii] - aJ £ 0, Vt > 0,
and D is a dense subset of R+. Prove the equivalence of the following
assertions:
l)Xn±>X,
2) E£i Pk-i[\V£\ > e] - 0, V* > 0, e > 0,
£i<Jfc<Tr(^nVfc"l<i " ^-i[Ufc^|uri<i]])2-^ A, V« € L>,
3) Ei<jfc<r4" *ViM > e] £ 0, Vt > 0, e > 0,
Ei<fc<rr ^*-i[^ntei<i]] ^A. vt € d.
16.7 Suppose that for each n, J7n = (U£) is a sequence of martingale
differences on $, and X is a Levy process with predictable triplet (0, /?, 0).
Let £), X71 be the same as in the previous problem. Assume
E |£?fc-i[f/fcnVnl<a]]|4o, v*>o.
l<A:<Tt" *
Prove the equivalence of the following statements:
1) Xn-^X,
2)Ei<fc<rrW)2^A. V'e£)'
3)El<Jfc<re«i,fc-l[lC7fcl>£]^0' V*>°> e>0'
El<fc<r«(^)2V„"l<l]^A, Vt€P,
4) Ei<jfc<rt« Pk-i[\UJt\ > e] Z 0, Vt > 0, e > 0,
£>fc-i[t/fcnV«|<i]]^A, V«€D.
16.8 Use the notations of the previous problem. Write
[A2]:£^fc-i[(^)2%?|>e]^<), Vf>0, e>0.
fc=i
Prove the equivalence of the following statements:
1) [A2] and Xn -£ X,
2)[A2]andE£i(^n)2-A, V* € D,
3) [A2] and ElU I>fc-i[tf£] - A, Vt € 0,
4)E?=iW)2-A, E£i^-i[^n]-A, vteD,
5) ElLi £>Jfe-i [U?] ^ A, Vt € I> and X" £ X.
16.9 Let W" be a standard Brownian motion. Then the process
- t)W"(«/(l - t)), 0 < t < 1,
-tt

528
Chapter XVI Weak Convergence for Semimartingales
is a Brownian bridge.
16.10 Let X = (Xt, 0 < t < 1) be a Brownian bridge. Prove 1) X is a
semimartingale w.r.t. its usual natural filtration with predictable triplet:
at = - / -——ds, (3t = t, v = 0,
Jo 1 - s
x
2) X is a diffusion with infinitesimal characteristics: b(t,x) = ,
a(t,x) = 1,
3) X satisfies the following stochastic differential equation:
dXt = ~YZ~t^ + dWt, X0 = 0,
where W is a standard Brownian motion.
16.11 Let (Zi,i > 1), be an i.i.d. sequence of r.v. with distribution F,
and Vtn = ^n(s E?=i I[Zi<t] ~ F{t)). Prove that Vn £ X, where X is a
cadlag Gaussian process with E[Xt] = 0, E[XsXt] = F(sAt)(l-F(sVt)).
16.12 Let {Zi)i>\ be an i.i.d. sequence of r.v. with distribution F
and continuous density function /, (Fn(t), t > 0) be the empirical process
defined by (54.1), for each t P™ = nFn(t/n). If X is a homogeneous
Poisson process with rate /(0), then Pn —> X.

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Index
I Up, 1.0
| || for function, 15.7
| || for measure, 14.3
I ||ba<o, 10.6
I II*p, 10.37
I j|s, Problem 8.20
I \\Sp, Problem 10.12
I Ha, 15.6
II ll«, 15-7
±, 6.12
JL, 6.12, 7.33
loc
<, 12.1
<J, 14.23
A, 14.23
{•}\ 6.16
[•][.,•], 6.27, 7.29, 8.2
(•)(•,•), 6.24, 7.29, 8.2
®, 1.0
-±>, 15.41
C,(D)
V
', 15.45
S 2.18
,1.11
►, 1.11
-^*, 15.60
-^, 15.35
=», 15.60
-<, 15.53
[a-D], 16.7
(a, /?, t/), 11.25
(a', /3», 16.2
(a(h),P(h),v), 16.2
[0-Z>], 16.7
[AQ], 16.38
r(M), 13.10
re(M), 13.10
rm(X,F°), 12.38
r,(A-,F°), 12.38
r,(X,.F0;F,a,/?,i/), 12.38
AX, 2.41.0
A—class, 1.1
A, 15.6
Ao, 15.6
H(f), 15.34
[i> - £>], 16.7
/i-a.s. continuous, 15.36
0t(dx), 11.16
7T—class, 1.1
p(x,y) in Od, 15.7
p(i,j/)inDd, 15.10
<r—integrable, 1.15
EX, 7.39
$D, 15.19
*, 16.0
$, 16.26
uj(A,x), 15.2
w(*,x,o), 15.2
w'(6,x,a), 15.2
u/'(£,x,a), 15.23
a, 11.14
A, 6.0,
.4+, 6.0
A\oc, 7.8
AL, 819
^oc- 7.8
A{F), 1.25
1,6.0
Ad, 3.41
Ac, 3.41
Ad«, 4.25

540
Index
Ada, 4.25
accessible process, 3.37
accessible time, 3.34
accessible part of a stopping time,
4.20
accessible a-field, 3.37
adapted stochastic sequence, 2.1.0
adapted process, 2.41.0
Aldous' theorem, 15.55
angle bracket process, 6.24
B
bT, 1.0
6.F+, 1.0
BMO, 10.6
BAlO-martingale, 10.6
6(H), 1.0
B(R+), 1.0
B-D-G inequality, 10.36
Brownian bridge, 16.56, Problem
2.16
Browian motion, 2.71
Burkholder-Davis-Gundy inequality,
10.36
Burkholder's inequality, 10.36
C
C(H+), 2.41.0
Cb(S), 15.34
CU(S), 15.34
CJ, 15.1
Ca, 15.1
cadlag process, 2.41.0
canonical
decomposition of a special semi-
martingale, 8.5
filtered probability space, 15.19
measurable space, 15.19
predictable decomposition of a
predictable random
measure, 13.30
probability space with
filtration, 15.19
process, 2.41.0
capacitable, 1.33
change of time, 3.47
Choquet ^"-capacity, 1.33
Choquet's theorem, 1.35
class(D), 5.44
compensated
Poisson process, 7.37
stochastic integral, 9.7, 9.9
sum of jumps, 7.15
compensation, 6.0
compensator, 5.21, 11.7
complete continuity of filtration,
5.37
complete filtration, 2.63
complete natural filtration, 2.63
completion of a filtration, 2.63
conditional expectation, 1.17
conjugate convex function, 10.30
contiguity of measures, 14.23
continuous local martingale, 7.21
continuous martingale part, 6.18,
7.25, 8.1
continuous part of a process with
finite variation, 3.41
continuous process, 2.41.0
convergence in distribution, 15.41
convergence in law, 15.41
convergence theorem of martingales,
2.^7
coordinate process, 2.41.0
counting process, 2.77
C-tight, 15.48
D
D(Jt+), 2.41.0
£>[£], 2.40
D, 15.1
Dd, 15.1
Da, 15.1
Df, 15.1
P°, 15.19
Vu 15.19
2>°, 15.19
2>, 15.19
D°, 15.19

Index
541
D, 15.19
Davis inequality, 10.24, 10.28
debut, 4.1
density process, 12.4
diffusion, 16.45
diffusion with jumps, 16.45
distribution law, 4.1
Doleans-Dade exponential formula,
9.39
dominated by an increasing process,
9.20
Doob decomposition of supermartin-
gales, 2.28
Doob measurability theorem, 1.5
Doob's inequality, 2.15, 2.49
Doob's stopping theorem, 2.10, 2.35,
2.38
Doob-Meyer decomposition of
supermartingales, 5.48
dual optional projection, 5.21
dual predictable projection, 5.21,
11.7
£
m i.o
Emery topology, Problem 8.20
empirical process, 16.54
entire separation of measures, 14.23
essential infimum, 1.12, 1.14
essential supremum, 1.12, 1.14
essinf, 1.12
ess sup, 1.12
evanescent set, 4.9
evanescent process, 4.9
exponential semimartingale, 9.39
exponential of a semimartingale,
9.39
F
^"-analytic set, 1.25
^"-capacity, 1.33
.F+, 1.0
TT, 2.5, 2.57, 3.3
TT+, 2.57, 3.3
TT-, 3.3
^i+, 2.41.0
Tt-, 2.41.0
Tb, 1.0
Ta, 1.0
f, 11.1
F, 2.0, 2.41.0
F(X), 2.63
F°(X), 2.1, 2.41.0
family of finite dimentional
distributions, 2.41.0
Fefferman's inequality, 10.17
filtered probability space, 5.0
filtration, 2.0, 2.41.0
of discrete type, 5.51
finite dimensional distributions,
2.41.0
F-martingale, 2.1, 2.41
F-submartingale, 2.1, 2.41
F-supermartingale, 2.1, 2.41
Fellerian transition probability
kernel, 15.60
Follmer's lemma, 2.44
formula of integration by parts, 1.39,
9.33
foretellable, 3.26
foretellable a.s., 4.14
fundamental couple sequence, 8.19
fundamental sequence for a
predictable set of interval type,
8.18
G
G{li), 11.16
Gi(n), n.21
&(/*), H.21
Garsia's lemma, 10.35
Gaussian process, 2.72
generalized diffusion, 16.45
Girsanov's Theorem for
local martingales, 12.13, 12.20
random measures, 12.26
semimartingales, 12.14, 12.18
stochastic integrals, 12.21,
12.22

542
Index
graph of a stopping time, 3.14
H
MP,-**), 14.1
H-decomposition, 9.13
H.X, 3.45, 9.1, 9.6, 9.13
HbX, 9.7, 9.9
HhX, 5.1
H(a), 14.7
H\ 10.1
W, 10.37
W1-martingale, 10.1
Wp-martingale, 10.37
Hellinger integral, 14.3
Hellinger-Kakutani metric, 14.3
Hellinger process, 14.7
homogeneous
diffusion, 16.45
Poisson process, 2.75
process with independent
increments, 2.64
Ia, 1.0
I-capacible, 1.33
increasing process, 3.41
increasing stochastic sequence, 2.27
indistinguishable processes, 2.45, 4.9
infinitesimal characteristics of
diffusion, 16.45
inhomogeneous Poisson process,
11.42
integer-valued random measure,
11.12
integrable
increasing process, 5.18
stochastic sequence, 2.27
random measure, 11.3
w.r.t. an increasing process,
3.45
integral representation of semimar-
tingale, 11.25
intensity of
counting processes, 11.50
step processes, 11.50
Ito equation, 9.54
Ito formula, 9.35
J, 11.14
Ji, 16.7
J(x), 15.29
J(X), 15.43
John-Nirenberg inequality, 10.42
jump chain, 15.62
jump measure of a process, 11.15
jump process, 2.41.0
jump time of a process, 4.22
jump times of step functions, 15.32
K
£(//), 13.13
K, 11.14
Kolmogorov inequality, 6.7
Krickeberg decompositon, 2.32
Krickeberg-Kazamaki
decomposition, Problem 8.13
Kunita-Watanabe inequality, 1.40,
6.33, 6.34, 8.3
L(X), 9.14
Lm(X), 9.1
£(X), 15.41
law, 2.41.0
left-continuous process, 2.41.0
Lebesgue's lemma, 1.37
Lenglart's inequality, 9.23
Levy prcess, 2.64
Levy system, 11.15
Levy's theorem, 2.19, 2.23, 11.39
Levy-Ito decomposition, 11.45
local characteristics of semimartin-
gale, 11.25
local martingale, 7.11
on a set of interval type, 8.19

Index
543
with locally integrable
variation, 7.11
local time, 9.43
localized class, 7.1
localizing sequence, 7.1
locally
absolutely continuous, 12.1
bounded martingale, 7.17
bounded process, 7.5
integrable increasing process,
5.18, 7.8
square integrable martingale,
7.11
square integrable semimartin-
gale, 11.31
M
Mda, 6.22, 7.25
Mdi, 6.22, 7.25
Af,6.0
Afd, 7.21
Af0, 6.0
M2, 6.6
Af2c, 6.10
M2d, 6.17
Alloc, 7.11
Algc. 819
A<foc, 7.21
(A*foc)B, 8.19
AC 7.21
(Mfoc)B, 8.19
Af2[T], 6.20
marked point process, 11.55
martingale, 2.1, 2.41
right-closed, 2.33
with integrable variation, 6.1
martingale measure w.r.t. (X, F°),
12.38
martingale problem (X,F°), 12.38
maximal inequality of supermartin-
gales, 2.10
measurable process, 3.10
measure generated by an increasing
process, 5.16
measure generated by //, 11.3
mesh of a partition, 9.28
moderate convex function, 10.32
modification of process, 2.45
monotone class, 1.1
multivariate point process, 11.55
N
TV, 1.0
TV, 1.0
A", 2.63
natural filtration, 2.1, 2.41.0
normal process, 2.72
O
0, 3.15
0,11.1
optional
function, 11.1
measure, 5.12
process, 3.15
projection of a measure, 5.17
projection of a process, 5.1
random measure, 11.3
section theorem, 4.7
set, 3.15
set of interval type, 8.17
a-field, 3.15
in fi, 11.1
time, 2.5, 2.57
optionally a-integrable, 11.3
Ornstein-Uhlenbeck process, 9.55
orthogonality for square integrable
martingale, 6.12
orthogonality for local martingale,
7.33
Ottaviani's inequality, 2.67
P
V, 3.15
V, 11.1
V{S), 15.34
path of a process, 2.41.0
paving, 1.24

544
Index
paved set, 1.24
point process, 2.77
Poisson arrivals see time average,
Problem 9.15
Poisson process, 2.75
potential, 2.29, 2.55
generated by a predictable
integrable increasing process,
5.45
predictable
characteristics of a semimartin-
gale, 11.25, 16.2
decomposition of a predictable
random measure, 13.30
function, 11.1
measure, 5.12
process, 3.15
projection of a measure, 5.17
projection of a process, 5.2
quadratic variation
(covariation) of
square integrable martingale,
6.24
local martingales, 7.29
semimartingales, 8.2
random measure, 11.3
section theorem, 4.8
set, 3.15
of interval type, 8.18
a-field, 3.15
in fi, 11.1
stochastic sequence, 2.27
support of a random set, 5.39
time, 3.25
triplet
on $£>, 16.7
of semimartingale, 11.25,16.2
predictably a-integrable, 11.3
prelocally integrable increasing
process, 5.18
process, 2.41.0
with finite variation, 3.41
with independent increments, 2.64
with integrable variation, 5.18
with locally integrable
variation, 5.18, 7.8
with prelocally integrable
variation, 5.18
with stationary increments,
2.64
progressive a-field, 3.13
progressively measurable process
(progressive process), 3.10
Prokhorov's theorem, 15.39
purely discontinuous
local martingale, 7.21
local martingale part, 7.25
part (or jump part) of a pocess
with finite variation, 3.41
process with finite variation,
3.41
square integrable martingale,
6.18
square integrable martingale
part, 6.18
0,1.0
Q+, 1.0
Q, Qc, Problem 8.12
quadratic variation (covariation) of
local martingales, 7.29
semimartingales, 8.2
square integrable martingales,
6.27
quasi-left-continuous process, 4.22
quasi-left-continuity of a filtration,
3.39
quasimartingale, 8.12
R
H, 1.0
H+, 1.0
R, 1.0
Jt+, 1.0
random measure, 11.3
Rao's decomposition of a
quasimartingale, 8.13
regular supermartingale, 5.49
restriction of a stopping time, 3.8

Index
545
Riesz decomposition, 2.30, 2.55
right-closable, 2.33
right-closed element, 2.33
right-continuous filtration, 2.41.0
right-continuous process, 2.41.0
right-inverse function, 1.37
S
5,8.1
Sp, 8.4
5B, 8.19
5pfl, 8.19
sample function of a process, 2.41.0
section, 4.3
section lemma, 4.3
section theorem, 4.7, 4.8
semimartingale, 8.1
on a set of interval type, 8.19
semimartingale measure w.r.t.
(X,F°), 12.38
semimartingale problem (X, F°),
12.38
semimartingale problem (X, F°;
F, a,/?,!/), 12.38
single step process, 10.3
Skorokhod topology, 15.10
Skorokhod's representation
theorem, 15.42
special semimartingale, 8.4
square bracket process, 6.27
standard
Brownian motion, 2.71
sequence of stopping times
exhausting the jumps of
a cadlag adapted process,
4.21
Wiener process, 2.71
square integrable martingale, 6.6
stable family of processes, 6.15
stable subspace, 6.15
step process, 11.48
stochastic
integral of a progressive process
w.r.t. a continuous local
martingale, 9.6
integral of a predictable process
w.r.t. a local martingale, 9.1
w.r.t. a semimartingale, 9.13
w.r.t. a compensated random
measure, 11.16
interval, 3.14
partition of interval, 9.28
process, 2.41.0
sequence, 2.0
set, 3.13
stochastically continuous process,
2.64
stopped process, Problem 2.5, 3.24
stopping time, 2.5, 2.57, 3.1
in wide sense, 2.57, 3.1
Stratonovich integral, Problem 9.13
strong law of large number, 9.37
strong majoration, 15.53
strong property of predictable
representation for
martingale, 13.1
filtration, 13.39
submartingale, 2.1, 2.41
summation process of a thin process,
7.39
supermartingale, 2.1, 2.41
support of an integer-valued random
measure, 11.13
[sup a], 16.7
[sup a'], 16.7
[sup /?], 16.7
[sup i/], 16.7
T
ip(x,u), 15.29
Tp(X,u), 15.43
T, 3.17
Tanaka-Meyer formula, 9.43
thin set, 3.18
thin process, 7.39
tight, 14.23, 15.38, 15.41
totally inaccessible part of a
stopping time, 4.20
totally inaccessible time, 4.19
trajectory of a process, 2.41.0

546
Index
truncation function, 16.1
U
t/(x), 15.29
U(X), 15.43
uniformly integrable family, 1.6
uniqueness-measurability
hypothesis, 16.46
universal completion, 1.35
universally measurable set, 1.35
upcrossing inequality of supermar-
tingales, 2.17, 2.42
usual augmentation of nitrations,
2.63
usual conditions for filtrations, 2.63
usual natural filtration, 2.63
V
V, 6.0
V+, 6.0
VB, 8.19
Var(X), 8.12
W
W.(ii-v), 11.16
W, 6.1
Wioc, 7.11
Wald's equation, 2.40
Watanabe's theorem, 11.42
weak convergence of measures, 15.35
weak orthogonality, 6.12
weak property of predictable
representation for
semimartingales, 13.13
filtration, 13.39
Wiener process, 2.71
X
[X ->], 8.27
°X, 5.1
pX, 5.2
X-integrable, 9.13
Y
Yceurp's lemma, 9.4
Young inequality, 10.31
Z
W./i, 11.3
W*v, 11.3
Z, 16.1
Zc, 16.1

Ho - Wang - Yan
Semimartingale Theory cm
Semimartingale Theory and Stochastic Calculus presents a
systematic and detailed account of the general theory of
stochastic processes, the semimartingale theory and related
stochastic calculus. The emphasis is upon stochastic integration for
semimartingales, characteristics of semimartingales, predictable
representation properties and weak convergence of
semimartingales. Also included is a concise treatment of absolute
continuity and singularity, contiguity and entire separation of measures
by semimartingale approach. Special attention is presented for
two basic types of processes frequently encountered in applied
probability and statistics: processes with independent
increments and marked point processes encountered frequently in
applied probability and statistics. This self-contained and
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