Brownian Motion, Martingales, and Stochastic Calculus

Graduate Texts in Mathematics

Jean-François Le Gall

Brownian Motion,

Martingales, and

Stochastic Calculus

Graduate Texts in Mathematics 274

Graduate Texts in Mathematics

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More information about this series at http://www.springer.com/series/136

Jean-François Le Gall

Brownian Motion,

Martingales, and Stochastic

Calculus

123

Jean-François Le Gall

Département de Mathématiques

Université Paris-Sud

Orsay Cedex, France

Translated from the French language edition:

‘Mouvement brownien, martingales et calcul stochastique’ by Jean-François Le Gall

Copyright © Springer-Verlag Berlin Heidelberg 2013

Springer International Publishing is part of Springer Science+Business Media

All Rights Reserved.

ISSN 0072-5285 ISSN 2197-5612 (electronic)

Graduate Texts in Mathematics

ISBN 978-3-319-31088-6 ISBN 978-3-319-31089-3 (eBook)

DOI 10.1007/978-3-319-31089-3

Library of Congress Control Number: 2016938909

Mathematics Subject Classification (2010): 60H05, 60G44, 60J65, 60H10, 60J55, 60J25

© Springer International Publishing Switzerland 2016

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Preface

This book originates from lecture notes for an introductory course on stochastic

calculus taught as part of the master’s program in probability and statistics at

Université Pierre et Marie Curie and then at Université Paris-Sud. The aim of this

course was to provide a concise but rigorous introduction to the theory of stochastic

calculus for continuous semimartingales, putting a special emphasis on Brownian

motion. This book is intended for students who already have a good knowledge

of advanced probability theory, including tools of measure theory and the basic

properties of conditional expectation. We also assume some familiarity with the

notion of uniform integrability (see, for instance, Chapter VII in Grimmett and

Stirzaker [30]). For the reader’s convenience, we record in Appendix A2 those

results concerning discrete time martingales that we use in our study of continuous

time martingales.

The first chapter is a brief presentation of Gaussian vectors and processes. The

main goal is to arrive at the notion of a Gaussian white noise, which allows us to give

a simple construction of Brownian motion in Chap. 2. In this chapter, we discuss

the basic properties of sample paths of Brownian motion and the strong Markov

property with its classical application to the reflection principle. Chapter 2 also gives

us the opportunity to introduce, in the relatively simple setting of Brownian motion,

the important notions of filtrations and stopping times, which are studied in a more

systematic and abstract way in Chap. 3. The latter chapter discusses continuous time

martingales and supermartingales and investigates the regularity properties of their

sample paths. Special attention is given to the optional stopping theorem, which

in connection with stochastic calculus yields a powerful tool for lots of explicit

calculations. Chapter 4 introduces continuous semimartingales, starting with a

detailed discussion of finite variation functions and processes. We then discuss local

martingales, but as in most of the remaining part of the course, we restrict our

attention to the case of continuous sample paths. We provide a detailed proof of

the key theorem on the existence of the quadratic variation of a local martingale.

Chapter 5 is at the core of this book, with the construction of the stochastic

integral with respect to a continuous semimartingale, the proof in this setting of the

celebrated Itô formula, and several important applications (Lévy’s characterization

v

vi Preface

theorem for Brownian motion, the Dambis–Dubins–Schwarz representation of

a continuous martingale as a time-changed Brownian motion, the Burkholder–

Davis–Gundy inequalities, the representation of Brownian martingales as stochastic

integrals, Girsanov’s theorem and the Cameron–Martin formula, etc.). Chapter 6,

which presents the fundamental ideas of the theory of Markov processes with

emphasis on the case of Feller semigroups, may appear as a digression to our main

topic. The results of this chapter, however, play an important role in Chap. 7, where

we combine tools of the theory of Markov processes with techniques of stochastic

calculus to investigate connections of Brownian motion with partial differential

equations, including the probabilistic solution of the classical Dirichlet problem.

Chapter 7 also derives the conformal invariance of planar Brownian motion and

applies this property to the skew-product decomposition, which in turn leads to

asymptotic laws such as the celebrated Spitzer theorem on Brownian windings.

Stochastic differential equations, which are another very important application of

stochastic calculus and in fact motivated Itô’s invention of this theory, are studied

in detail in Chap. 8, in the case of Lipschitz continuous coefficients. Here again

the general theory developed in Chap. 6 is used in our study of the Markovian

properties of solutions of stochastic differential equations, which play a crucial

role in many applications. Finally, Chap. 9 is devoted to local times of continuous

semimartingales. The construction of local times in this setting, the study of their

regularity properties, and the proof of the density of occupation formula are very

convincing illustrations of the power of stochastic calculus techniques. We conclude

Chap. 9 with a brief discussion of Brownian local times, including Trotter’s theorem

and the famous Lévy theorem identifying the law of the local time process at level 0.

A number of exercises are listed at the end of every chapter, and we strongly

advise the reader to try them. These exercises are especially numerous at the end

of Chap. 5, because stochastic calculus is primarily a technique, which can only

be mastered by treating a sufficient number of explicit calculations. Most of the

exercises come from exam problems for courses taught at Université Pierre et Marie

Curie and at Université Paris-Sud or from exercise sessions accompanying these

courses.

Although we say almost nothing about applications of stochastic calculus in

other fields, such as mathematical finance, we hope that this book will provide a

strong theoretical background to the reader interested in such applications. While

presenting all tools of stochastic calculus in the general setting of continuous

semimartingales, together with some of the most important developments of the

theory, we have tried to keep this text to a reasonable size, without making any

concession to mathematical rigor. The reader who wishes to go further in the theory

and applications of stochastic calculus may consult the classical books of Karatzas

and Shreve [49], Revuz and Yor [70], or Rogers and Williams [72]. For a historical

perspective on the development of the theory, we recommend Itô’s original papers

[41] and McKean’s book [57], which greatly helped to popularize Itô’s work. More

information about stochastic differential equations can be found in the books by

Stroock and Varadhan [77], Ikeda and Watanabe [35], and Øksendal [66]. Stochastic

calculus for semimartingales with jumps, which we do not present in this book, is

Preface vii

treated in Jacod and Shiryaev [44] or Protter [63] and in the classical treatise of

Dellacherie and Meyer [13, 14]. Many other references for further reading appear in

the notes and comments at the end of every chapter.

I wish to thank all those who attended my stochastic calculus lectures in the last

20 years and who contributed to this book through their questions and comments.

I am especially indebted to Marc Yor, who left us too soon. Marc taught me most

of what I know about stochastic calculus, and his numerous remarks helped me to

improve this text.

Orsay, France Jean-François Le Gall

January 2016

Interconnections between chapters

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

The original version of this book was revised. An erratum can be found at DOI

10.1007/978-3-319-31089-3_10

Contents

1 Gaussian Variables and Gaussian Processes .............................. 1

1.1 Gaussian Random Variables ............................................ 1

1.2 Gaussian Vectors ........................................................ 4

1.3 Gaussian Processes and Gaussian Spaces.............................. 7

1.4 Gaussian White Noise .................................................. 11

2 Brownian Motion ............................................................ 19

2.1 Pre-Brownian Motion ................................................... 19

2.2 The Continuity of Sample Paths........................................ 22

2.3 Properties of Brownian Sample Paths.................................. 29

2.4 The Strong Markov Property of Brownian Motion .................... 33

3 Filtrations and Martingales................................................. 41

3.1 Filtrations and Processes................................................ 41

3.2 Stopping Times and Associated -Fields .............................. 44

3.3 Continuous Time Martingales and Supermartingales ................. 49

3.4 Optional Stopping Theorems ........................................... 58

4 Continuous Semimartingales ............................................... 69

4.1 Finite Variation Processes .............................................. 69

4.1.1 Functions with Finite Variation ................................ 69

4.1.2 Finite Variation Processes ...................................... 73

4.2 Continuous Local Martingales ......................................... 75

4.3 The Quadratic Variation of a Continuous Local Martingale .......... 79

4.4 The Bracket of Two Continuous Local Martingales................... 87

4.5 Continuous Semimartingales ........................................... 90

5 Stochastic Integration ....................................................... 97

5.1 The Construction of Stochastic Integrals .............................. 97

5.1.1 Stochastic Integrals for Martingales Bounded in L2 .......... 98

5.1.2 Stochastic Integrals for Local Martingales .................... 106

xi

xii Contents

5.1.3 Stochastic Integrals for Semimartingales...................... 109

5.1.4 Convergence of Stochastic Integrals ........................... 111

5.2 Itô’s Formula ............................................................ 113

5.3 A Few Consequences of Itô’s Formula................................. 118

5.3.1 Lévy’s Characterization of Brownian Motion ................. 119

5.3.2 Continuous Martingales as Time-Changed

Brownian Motions .............................................. 120

5.3.3 The Burkholder–Davis–Gundy Inequalities ................... 124

5.4 The Representation of Martingales as Stochastic Integrals ........... 127

5.5 Girsanov’s Theorem .................................................... 132

5.6 A Few Applications of Girsanov’s Theorem........................... 138

6 General Theory of Markov Processes ..................................... 151

6.1 General Definitions and the Problem of Existence .................... 151

6.2 Feller Semigroups....................................................... 158

6.3 The Regularity of Sample Paths........................................ 164

6.4 The Strong Markov Property ........................................... 167

6.5 Three Important Classes of Feller Processes .......................... 170

6.5.1 Jump Processes on a Finite State Space ....................... 170

6.5.2 Lévy Processes.................................................. 175

6.5.3 Continuous-State Branching Processes ........................ 177

7 Brownian Motion and Partial Differential Equations ................... 185

7.1 Brownian Motion and the Heat Equation .............................. 185

7.2 Brownian Motion and Harmonic Functions ........................... 187

7.3 Harmonic Functions in a Ball and the Poisson Kernel ................ 193

7.4 Transience and Recurrence of Brownian Motion...................... 196

7.5 Planar Brownian Motion and Holomorphic Functions................ 198

7.6 Asymptotic Laws of Planar Brownian Motion ........................ 201

8 Stochastic Differential Equations .......................................... 209

8.1 Motivation and General Definitions.................................... 209

8.2 The Lipschitz Case ...................................................... 212

8.3 Solutions of Stochastic Differential Equations as Markov

Processes ................................................................ 220

8.4 A Few Examples of Stochastic Differential Equations................ 225

8.4.1 The Ornstein–Uhlenbeck Process.............................. 225

8.4.2 Geometric Brownian Motion ................................... 226

8.4.3 Bessel Processes ................................................ 227

9 Local Times................................................................... 235

9.1 Tanaka’s Formula and the Definition of Local Times ................. 235

9.2 Continuity of Local Times and the Generalized Itô Formula ......... 239

9.3 Approximations of Local Times........................................ 247

9.4 The Local Time of Linear Brownian Motion .......................... 249

9.5 The Kallianpur–Robbins Law .......................................... 254

Contents xiii

Erratum ........................................................................... E1

A1 The Monotone Class Lemma ............................................... 261

A2 Discrete Martingales......................................................... 263

References......................................................................... 267

Index ............................................................................... 271

Chapter 1

Gaussian Variables and Gaussian Processes

Gaussian random processes play an important role both in theoretical probability

and in various applied models. We start by recalling basic facts about Gaussian ran￾dom variables and Gaussian vectors. We then discuss Gaussian spaces and Gaussian

processes, and we establish the fundamental properties concerning independence

and conditioning in the Gaussian setting. We finally introduce the notion of a

Gaussian white noise, which will be used to give a simple construction of Brownian

motion in the next chapter.

1.1 Gaussian Random Variables

Throughout this chapter, we deal with random variables defined on a probability

space .˝; F; P/. For some of the existence statements that follow, this probability

space should be chosen in an appropriate way. For every real p 1, Lp.˝; F; P/,

or simply Lp if there is no ambiguity, denotes the space of all real random variables

X such that jXj

p is integrable, with the usual convention that two random variables

that are a.s. equal are identified. The space Lp is equipped with the usual norm.

A real random variable X is said to be a standard Gaussian (or normal) variable

if its law has density

pX.x/ D 1

p2 exp.x2

2 /

with respect to Lebesgue measure on R. The complex Laplace transform of X is

then given by

EŒezX D ez2=2; 8z 2 C:

© Springer International Publishing Switzerland 2016

J.-F. Le Gall, Brownian Motion, Martingales, and Stochastic Calculus,

Graduate Texts in Mathematics 274, DOI 10.1007/978-3-319-31089-3_1

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