Probability and Its Applications

Series Editors Thomas Liggett Charles Newman Loren Pitt

Andrei N. Borodin Paavo Salminen

Handbook of Brownian Motion -Facts and Formulae

Second Edition

Springer Basel AG

Authors' addresses:

Andrei N. Borodin Steklov Mathematical Institute St. Petersburg Division Fontanka 27 191011 St. Petersburg Russia

Paavo Salminen Äbo Akademi University Mathematical Institute Fänriksgatan 3 20500 Turku Finland

2000 Mathematics Subject Classification 60J65, 60J60, 60H05, 60H10, 60J55

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA

Deutsche Bibliothek Cataloging-in-Publication Data

Borodin, Andrej N.: Handbook of Brownian motion : facts and formulae / Andrei N. Borodin ; Paavo Salminen. - 2. ed. - Basel; Boston ; Berlin : Birkhäuser, 2002

(Probability and its applications) ISBN 978-3-0348-9462-3 ISBN 978-3-0348-8163-0 (eBook) DOI 10.1007/978-3-0348-8163-0

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.

© 2002 Springer Basel AG Originally published by Birkhäuser Verlag in 2002 Softcover reprint of the hardcover 2nd edition 2002 Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced from chlorine-free pulp. TCF oo ISBN 978-3-0348-9462-3

9 8 7 6 5 4 3 2 1 www.birkhauser-science.com

CONTENTS

Preface to the first edition

Preface to the second edition

Notation

Part I: THEORY

Chapter I. Stochastic processes in general

1. Basic definitions 2. Markov processes, transition functions, resolvents,

and generators 3. Feller processes, Feller-Dynkin processes,

and the strong Markov property 4. Martingales

Chapter II. Linear diffusions

1. Basic facts 2. Local time 3. Passage times 4. Additive functionals and killing 5. Excessive functions 6. Ergodic results

Chapter III. Stochastic calculus

ix

xi

xiii

1

1

3

5 7

12

12 21 25 27 32 35

38

1. Stochastic integration with respect to Brownian motion 38 2. The Ito and Tanaka formulae 42 3. Stochastic differential equations - strong solutions 44 4. Stochastic differential equations - weak solutions 46 5. One-dimensional stochastic differential equations 47 6. The Cameron-Martin-Girsanov transformation of measure 48

Chapter IV. Brownian motion

1. Definition and basic properties 2. Brownian local time 3. Excursions 4. Brownian bridge 5. Brownian motion with drift 6. Bessel processes 7. Geometric Brownian motion

51

51 54 57 64 67 71 76

vi CONTENTS

Chapter V. Local time as a Markov process

1. Diffusion local time 2. Local time of Brownian motion 3. Local time of Brownian motion with drift 4. Local time of Bessel process 5. Summarizing tables

Chapter VI. Differential systems associated to Brownian motion

1. The Feynman-Kac formula 2. Exponential stopping 3. Stopping at first exit time 4. Stopping at inverse additive functional 5. Stopping at first range time

Appendix 1. Briefly on some diffusions

Part II: TABLES OF DISTRIBUTIONS OF FUNCTIONALS OF BROWNIAN MOTION AND RELATED PROCESSES

Introduction

1. List of functionals 2. Comments and references

1. Brownian motion

1. Exponential stopping 2. Stopping at first hitting time 3. Stopping at first exit time 4. Stopping at inverse local time 5. Stopping at first range time

2. Brownian motion with drift

1. Exponential stopping 2. Stopping at first hitting time 3. Stopping at first exit time 4. Stopping at inverse local time

81

81 84 90 94 99

103

103 105 109 113 117

119

145

146 148

153

153 198 212 229 242

250

250 295 309 323

3. Reflecting Brownian motion

1. Exponential stopping 2. Stopping at first hitting time 4. Stopping at inverse local time

4. Bessel process of order ZJ

1. Exponential stopping, ZJ 2 a 2. Stopping at first hitting time, ZJ > a 3. Stopping at first exit time, ZJ > a 4. Stopping at inverse local time, ZJ > a

5. Bessel process of order 1/2

1. Exponential stopping 2. Stopping at first hitting time 3. Stopping at first exit time 4. Stopping at inverse local time

6. Bessel process of order zero

2. Stopping at first hitting time 3. Stopping at first exit time 4. Stopping at inverse local time

7. Ornstein-Uhlenbeck process

1. Exponential stopping 2. Stopping at first hitting time 3. Stopping at first exit time 4. Stopping at inverse local time

8. Radial Ornstein-Uhlenbeck process

1. Exponential stopping 2. Stopping at first hitting time 3. Stopping at first exit time 4. Stopping at inverse local time

9. Geometric Brownian motion

1. Exponential stopping 2. Stopping at first hitting time 3. Stopping at first exit time 4. Stopping at inverse local time

Appendix 2. Special functions

Appendix 3. Inverse Laplace transforms

333

333 355 364

373

373 398 409 420

429

429 463 480 493

507

507 512 519

522

522 542 548 557

565

565 581 587 597

606

606 622 627 633

637

649

vii

viii CONTENTS

Appendix 4. Differential equations and their solutions 652

Appendix 5. Formulae for n-fold differentiation 657

Bibliography 659

Subject index 669

PREFACE TO THE FIRST EDITION

There are two parts in this book. The first part is devoted mainly to the proper­ties of linear diffusions in general and Brownian motion in particular. The second part consists of tables of distributions of functionals of Brownian motion and re­lated processes.

The primary aim of this book is to give an easy reference to a large number of facts and formulae associated to Brownian motion. We have tried to do this in a "handbook-style". By this we mean that results are given without proofs but are equipped with a reference where a proof or a derivation can be found. It is our belief and experience that such a material would be very much welcome by students and people working with applications of diffusions and Brownian motion. In discussions with many of our colleagues we have found that they share this point of view.

Our original plan included more things than we were able to realize. It turned out very soon when trying to put the plan into practice that the material would be too wide to be published under one cover. Excursion theory, which most of the recent results concerning linear Brownian motion and diffusions can be classified as, is only touched upon slightly here, not to mention Brownian motion in several dimensions which enters only through the discussion of Bessel processes. On the other hand, much attention is given to the theory of local time. The Markov property of local time, the so called Ray-Knight theorems, are discussed in a chapter of their own, and one can find a large number of distributions associated with local times in the tables. The concept of local time has proved to be of importance in many applications of diffusions; for instance, in economics, mathematical finance, and optimal control. We hope that the material presented in this book will be useful in these and other applications.

In the tables there are more than 1550 numbered formulae associated directly or via some transformations to Brownian motion. The systematic structure of the tables is explained in Introduction on p. 1201 . Some of the formulae are certainly obvious, tautological, and perhaps too complicated to be of any real use. In spite of this a decision was made to keep the structure and not to expand the tables with more "exotic" formulae. The main reason for this decision was that nothing like these tables has been published before, and so we think that one should start from the "bottom". We also believe that the structure of the tables makes them easy to

IOn p. 145 in this edition.

x

use, especially for people who do not deal with Brownian motion or diffusions on a daily basis. It is, further, our hope that the presentation has some pedagogical merits when used in teaching and by students.

As mentioned above results in this book are usually given without proofs. There are many superb research monographs and text books where diffusions and Brownian motion play the central role, and where proofs of the theorems in our book are given. In particular, we have been using books by Dynkin (1965), Blu­menthal and Getoor (1968), McKean (1969), Ito and McKean (1974, 2nd ed.), Gikhman and Skorokhod (1974, 1975, 1979), Williams (1979), Ikeda and Watan­abe (1981), Knight (1981), Chung and Williams (1983), Doob (1984), Rogers and Williams (1987), Revuz and Yor (1991), and Karatzas and Shreve (1991, 2nd ed.). Many of these, e.g., Doob (1984), Rogers and Williams (1987), Revuz and Yor (1991), Karatzas and Shreve (1991, 2nd ed.), give, besides the proofs, also histor­ical comments and notes concerning the origin and development of the theorems. Therefore, whenever possible, a reference is given to a monograph. We want to stress heavily (the obvious thing) that the presentation in Part I is not everywhere meant to be mathematically complete. To achieve this one must in several places study the different sources given in the text and via them, perhaps, also the original research papers.

Paul McGill, Jim Pitman and Marc Yor read parts of the material ofthe first and latter versions. We thank them for their comments, criticism and encouragement. We are also grateful to Goran Hognas and Juha Vuolle-Apiala for their comments to a more final version.

During the course of this work we have received financial support from the Academy of Finland, the Research Institute of the Abo Akademi Foundation, the Foundation of the University of Turku, and the Russian Foundation of Fundamental Research, grant No. 93-01-01454. This support is gratefully acknowledged. It is also our pleasure to thank the mathematical departments at the Steklov Institute, St. Petersburg's branch, the University of Turku, and the Abo Akademi University for their hospitality.

Our special thanks goes to P.E. Baskakova for helping us with the complicated typing work.

Finally, we want to thank the staff of Birkhauser for their patience when this project was dragging in time, and for their excellent work when preparing the real thing. It is also our pleasure to thank the copy-editor whose careful work made the text more readable.

St. Petersburg, Russia, and Abo (Turku), Finland, March 1996

A.N. Borodin P. Salminen

PREFACE TO THE SECOND EDITION

Compared with the first edition published in 1996, this second edition is revised and considerably expanded. The number of new formulae in the tables is more than 1000 and some of the old formulae in the tables and especially in Appendix 1 have found a new equivalent form.

It is our feeling that the first edition has been used, in some extent, for problems arising in financial mathematics. To meet better requirements from this field we have included in the theory part a new subsection on some recent results on geo­metric Brownian motion and in the formula part there is also a new section with ca. 200 formulae devoted to geometric Brownian motion.

Section V on Ray~Knight theorems has undergone much revision. Our aim has been to make it more readable and also to present more detailed explanations. We have also included summarizing tables for ease of reference.

There is a new subsection in Section VI on the Feynman~Kac formula concerning functionals stopped at the first range time. This is the random time when the maximum minus minimum of a process reaches a given level. We think that the first range time is an important stopping time with many applications in different fields. A number of formulae associated to the first range time of Brownian motion are displayed in the formula part of the book.

New processes are added to Appendix 1 "Briefly on some diffusions". In particu­lar, we have studied radial Ornstein~Uhlenbeck processes and their squares. These processes are stationary with interesting equilibrium probability distributions; like gamma-, Rayleigh-, Maxwell- and X2-distributions. Radial Ornstein~Uhlenbeck processes appear also in the formula part with ca. 160 new formulae.

We have also expanded the introduction to the tables with new references and some explicit examples. We hope that these examples make it easier to use and understand the results in Section VI. There are also three new appendices and the old appendices have been expanded. The first new appendix is on formulae for inverse Laplace transforms, the next one on second order ordinary differential equations and their fundamental solutions, and the third one on formulae for n-fold differentiation of the product and composition of two functions. The appendix on differential equations is needed when applying the Feynman~Kac formula, and the appendix on differentiation is useful when computing, e.g., moments of a functional of a random process by differentiating a Laplace transform.

This second edition has about 200 pages more than the first edition. However, the added material covers more than that. To keep the book a reasonable size we have not included some formulae from the first edition to the second one. From

xii PREFACE TO THE SECOND EDITION

our point of view the excluded formulae are simple consequences of others and can be constructed easily, if needed.

We apologize sincerely for the misprints in the first edition and hope that the positive experiences with the book have weighed more than the negative ones. We have worked hard to correct all the misprints and other errors. However, due to the nature of the book and the fast modern word processing technology it would be too optimistic to claim that this second edition does not have any defects. We would therefore appreciate feedback from our readers also on this edition. Our e-mail addresses are printed on the information page of this book.

We express our warm thanks to Marc Yor for encouragement, many concrete proposals for improvements, pointing out misprints, and giving new references. We are also grateful to Jim Pitman for comments and interest in this work. Andrei Sudakov helped us with the secrets of the AMS-1E;Xpackage and we thank him for his help.

Financial support from the Research Institute of the Abo Akademi Foundation, the Magnus Ehrnrooth's Foundation (Finland), Vaisala Foundation (Finland), and the Russian Foundation of Fundamental Research, grant No. 99-01-0112 are grate­fully acknowledged.

St. Petersburg, Russia, and Abo (Turku), Finland, October 2001

A.N. Borodin P. Salminen

NOTATION

Chapters are referred to with Roman numerals. Inside the chapters we use running Arabic numbering for the items. For example, V.6 is item 6 in the fifth chapter, Al.1 is item 1 in the first appendix which is at the end of Part 1, and A2.2 is item 2 in the second appendix at the end of Part 2. When referring to an item in the same chapter the Roman numeral is omitted and we write, e.g., No.6.

The symbol := means "is defined to be equal to" ,

'" means "is identical in law to" .

(0, F, P) - probability space.

w - element in 0.

t f-+ w(t) - continuous function in the canonical framework, see 1.12.

(0, F, {Ft }, P) - filtered probability space, see 1.3.

{Ft : t 2 a}, {Ft } - filtration, see 1.3.

{FtC: t 2 a}, {Ft} - canonical filtration, see I.12.

{F~: t 2 a}, {Fn - natural filtration, see 1.4.

B(E) - Borel a-algebra of subsets of E.

C(E) - continuous functions from E to R

Cb (E) - continuous and bounded functions from E to R

Co (E) - continuous and bounded functions from E to ffi. vanishing at infinity.

Exp(a) - exponential distribution with parameter a.

T '" Exp(a) - random variable T is exponentially distributed with parameter a.

I - interval on R

l, r - left and right, respectively, endpoint of I.

int I - interior of I.

C1(I) - continuously differentiable functions from I to R

xiv NOTATION

C2(I) - two times continuously differentiable functions from I to R

C;(I) - bounded two times continuously differentiable functions from I to R

Bs(I) - bounded Borel-measurable functions from I to R

Leb - Lebesgue measure.

E{x} - Dirac measure at x.

bO(t) - Dirac b-function.

p x - probability measure associated with a process started at x.

Ex - expectation operator associated with a process started at x.

Ot - shift operator, see 1.13.

llA (w) - indicator function.

[) Ex(~(w);1](w) E dy) := [)yEx(~(w);1](w) < y)dy.

Q - infinitesimal generator, see 1.8, 11.7.

m - speed measure, see 11.4.

k - killing measure, see 11.4.

s - scale function, see 11.4.

p(t; x, y) - transition density with respect to the speed measure, see 11.4.

GOI(x, y), G(x, y) := Go(x, y) - Green function, see 11.11.

'ljJ0I - fundamental increasing solution, see 11.10.

'Po: - fundamental decreasing solution, see 11.10.

j+ - right derivative of j with respect to the scale function.

j- - left derivative of j with respect to the scale function.

£(t, x) - local time with respect to the Lebesgue measure.

g(t,z) = inf{s: £(s,z) > t} - inverse local time.

L(t,x) -local time with respect to the speed measure, see 11.13.

a V b:= max{a, b}.

a 1\ b:= min{a, b}.

Mt := sup{ Xs: 0::; S ::; t} - maximum of the process X.

Mt := inf{Xs: 0::; s ::; t} - minimum of the process X.

fI (t) := inf {s < t: X 8 = Mt } - first location of the maximum before time t.

H(t) := inf{ s < t: Xs = Md - first location of the minimum before time t.

Hz := inf{ t: X t = z} - first hitting time of z.

Ha,b := inf{t: Xs rt. (a, b)} - first exit time from (a, b) .

.c::/ - inverse Laplace transform with respect to 'Y.

t

f(t) * g(t) := f f(s)g(t - s) ds - convolution of the functions f and g. o

sh, ch, th, cth - hyperbolic functions, see A2.1.

r - Gamma function, see A2.2.

Jv , Yv - Bessel functions, see A2.3.

Iv, Kv - modified Bessel functions, see A2.4.

Ai - Airy function, see A2.5.

Hen - Hermite polynomials, see A2.6.

Dv - parabolic cylinder functions, see A2.7.

Erf, Erfc, Erfi, Erfid - error functions, see A2.8.

M(a, b, .), U(a, b, .) - Kummer functions, see A2.9.

Mn,m, Wn,m - Whittaker functions, see A2.9.

For further notations associated to special functions, see A2.

xv