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Stochastic Processes

Andrei N. Borodin
Publication Date: 
Number of Pages: 
Probability and Its Applications
[Reviewed by
Richard Durrett
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The back cover says

This book provides a rigorous yet accessible introduction to the theory of stochastic processes. A significant part of the book is devoted to the classic theory of stochastic processes. In turn, it also presents proofs of well-known results, sometimes together with new approaches. Moreover, the book explores topics not previously covered elsewhere, such as distributions of functionals of diffusions stopped at different times, the Brownian local times, diffusions with jumps, and an invariance principle for random walks and local times.

The preface says that “the aim of the book is to give a rigorous and at the same time accessible presentation of the theory of stochastic processes.” I do not think that he has achieved his aim. The first thing that should be clarified about this book is that it is not the Stochastic Processes course that comes between elementary and measure-theoretic probability. It is one that comes after at least a semester or preferably a whole year of measure theoretic probability.

The main reasons for buying this book are the chapters on the distribution of functionals of Brownian motion, and on diffusions with jumps (III and VI respectively). I can only recommend these for people who are already experts on the subject. If you are not then Revuz and Yor’s encyclopedic book on Continuous Martingales and Brownian Motion is a better choice for a complete treatment, while LeGall’s Brownian Motion, Martingales, and Stochastic Calculus is much more comprehensible introduction. 

Borodin’s most famous publication is his Handbook of Brownian Motion — Facts and Formulas written with P. Salminen, which has been cited 2054 times. If you can’t find the formula in this book it does not exist. A number of his other highly cited works concern local time and convergence to it, including two papers published in Probability Theory and Related Fields in 1986, so you can expect that material covered in Chapter V to be authoritative. I am sure that experts will find new insights here, but I am afraid that even very well trained students will find it difficult.

Richard Durrett taught at UCLA and Cornell before he came to Duke in 2010. He is a member of the National Academy of Science, who for the last thirty years has used probability to study problems that arise from ecology, genetics, and cancer modeling.

See the table of contents in the publisher's webpage.