Handbook of Brownian Motion

by Borodin, A. N.; Salminen, Paavo

• ISBN13: 9783764367053
• ISBN10: 3764367059
• Edition: 2nd
• Format: Hardcover
• Copyright: 2002-07-01
• Publisher: Birkhauser

Summary

The purpose of this book is to give an easy reference to a large number of facts and formulae associated Brownian motion. The collection contains more than 2500 numbered formulae.This book is of value as a basic reference material to researchers, graduate students, and people doing applied work with Brownian motion and diffusions. It can also be used as a source of explicit examples when teaching stochastic processes.Compared with the first edition published in 1996, this second edition has been revised and considerably expanded. More than 1000 new formulae have been added to the tables and, in particular, geometric Brownian motion is covered both in the theoretical and the formula part of the book.

Table of Contents

Preface to the first edition	ix
Preface to the second edition	xi
Notation	xiii
Part I: THEORY
Stochastic processes in general	1	(11)
Basic definitions	1	(2)
Markov processes, transition functions, resolvents, and generators	3	(2)
Feller processes, Feller-Dynkin processes, and the strong Markov property	5	(2)
Martingales	7	(5)
Linear diffusions	12	(26)
Basic facts	12	(9)
Local time	21	(4)
Passage times	25	(2)
Additive functionals and killing	27	(5)
Excessive functions	32	(3)
Ergodic results	35	(3)
Stochastic calculus	38	(13)
Stochastic integration with respect to Brownian motion	38	(4)
The Ito and Tanaka formulae	42	(2)
Stochastic differential equations - strong solutions	44	(2)
Stochastic differential equations - weak solutions	46	(1)
One-dimensional stochastic differential equations	47	(1)
The Cameron-Martin-Girsanov transformation of measure	48	(3)
Brownian motion	51	(30)
Definition and basic properties	51	(3)
Brownian local time	54	(3)
Excursions	57	(7)
Brownian bridge	64	(3)
Brownian motion with drift	67	(4)
Bessel processes	71	(5)
Geometric Brownian motion	76	(5)
Local time as a Markov process	81	(22)
Diffusion local time	81	(3)
Local time of Brownian motion	84	(6)
Local time of Brownian motion with drift	90	(4)
Local time of Bessel process	94	(5)
Summarizing tables	99	(4)
Differential systems associated to Brownian motion	103	(556)
The Feynman-Kac formula	103	(2)
Exponential stopping	105	(4)
Stopping at first exit time	109	(4)
Stopping at inverse additive functional	113	(4)
Stopping at first range time	117	(36)
Appendix 1. Briefly on some diffusions	119	(26)
Part II: TABLES OF DISTRIBUTIONS OF FUNCTIONALS OF BROWNIAN MOTION AND RELATED PROCESSES
Introduction	145	(1)
List of functionals	146	(2)
Comments and references	148	(5)
Brownian Motion	153	(97)
Exponential stopping	153	(45)
Stopping at first hitting time	198	(14)
Stopping at first exit time	212	(17)
Stopping at inverse local time	229	(13)
Stopping at first range time	242	(8)
Brownian motion with drift	250	(83)
Exponential stopping	250	(45)
Stopping at first hitting time	295	(14)
Stopping at first exit time	309	(14)
Stopping at inverse local time	323	(10)
Reflecting Brownian motion	333	(40)
Exponential stopping	333	(22)
Stopping at first hitting time	355	(9)
Stopping at inverse local time	364	(9)
Bessel process of order ν	373	(56)
Exponential stopping, ν ≥ 0	373	(25)
Stopping at first hitting time, ν > 0	398	(11)
Stopping at first exit time, ν > 0	409	(11)
Stopping at inverse local time, ν > 0	420	(9)
Bessel process of order 1/2	429	(78)
Exponential stopping	429	(34)
Stopping at first hitting time	463	(17)
Stopping at first exit time	480	(13)
Stopping at inverse local time	493	(14)
6. Bessel process of order zero	507	(15)
Stopping at first hitting time	507	(5)
Stopping at first exit time	512	(7)
Stopping at inverse local time	519	(3)
7. Ornstein-Uhlenbeck process	522	(43)
Exponential stopping	522	(20)
Stopping at first hitting time	542	(6)
Stopping at first exit time	548	(9)
Stopping at invese local time	557	(8)
8. Radial Ornstein-Uhlenbeck process	565	(41)
Exponential stopping	565	(16)
Stopping at first hitting time	581	(6)
Stopping at first exit time	587	(10)
Stopping at inverse local time	597	(9)
9. Geometric Brownian motion	606	(53)
Exponential stopping	606	(16)
Stopping at first hitting time	622	(5)
Stopping at first exit time	627	(6)
Stopping at inverse local time	633	(4)
Appendix 2. Special functions	637	(12)
Appendix 3. Inverse Laplace transforms	649	(3)
Appendix 4. Differential equations and their solutions	652	(5)
Appendix 5. Formulae for n-fold differentiation	657	(2)
Bibliography	659	(10)
Subject index	669