Handbook of Brownian Motion - Facts and Formulae

ISBN-13: 9783764367053 / Angielski / Twarda / 2002 / 685 str.

A. N. Borodin; P. Salminen; Andrei N. Borodin

Opis

There are two parts in this book. The first part is devoted mainly to the proper- ties of linear diffusions in general and Brownian motion in particular. The second part consists of tables of distributions of functionals of Brownian motion and re- lated 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.

Spis treści

I: Theory.- I. Stochastic processes in general.- II. Linear diffusions.- III. Stochastic calculus.- IV. Brownian motion.- V. Local time as a Markov process.- VI. Differential systems associated to Brownian motion.- Appendix 1. Briefly on some diffusions.- II: TABLES OF DISTRIBUTIONS OF FUNCTIONALS OF BROWNIAN MOTION AND RELATED PROCESSES.- 1. Brownian motion.- 2. Brownian motion with drift.- 3. Reflecting Brownian motion.- 4. Bessel process of order ?.- 5. Bessel process of order 1/2.- 6. Bessel process of order zero.- 7. Ornstein-Uhlenbeck process.- 8. Radial Ornstein-Uhlenbeck process.- 9. Geometric Brownian motion.- Appendix 2. Special functions.- Appendix 3. Inverse Laplace transforms.- Appendix 4. Differential equations and their solutions.- Appendix 5. Formulae for n-fold differentiation.

Blurb

The purpose of this book is to give an easy reference to a large number of facts and formulae associated with Brownian motion. The book consists of two parts. The first one - theory part - is devoted to properties of linear diffusions in general and Brownian motion in particular. Results are given mainly without proofs. The second one - formula part - is a table of distributions of functionals of Brownian motion and related processes. 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.

“This is an extremely useful handbook (...) It is without any previous example in its own subject matter: the very first of its kind. The primary aim of this handbook is to serve as an easily accessible reference for an incredibly large number of facts concerning the distribution of various functionals of Brownian motion and some related stochastic processes.”

Mathematical Reviews (review of 1st edition)

"In every respect, this most reliable handbook of Brownian motion and its friends is a volume to cherish. I can highly recommend it to researchers and users of probability alike. The authors are to be congratulated on their great job in bringing all of these facts and fomulas together."

Journal of the American Statistical Association (review of 1st edition)