Diffusions, Markov Processes, and Martingales: Volume 1, Foundations (Cambridge Mathematical Library)

Rating:
Summary: Excellent Treatment of Theory of Diffusion, Martingales, Ito
Review: Although not an easy read, this book contains a wealth of information on diffusion, martingales and Ito calculus. Reading difficulty is comparable to Karatzas/Shreve. Mastery of topics included enables the reader to get understanding of most of the current research papers in this field.

Rating:
Summary: A Great Book
Review: This book and its companion volume are a well organized and relatively easy-to-read introduction to a wide variety of ideas in stochastic processes. It is not only a great reference (I always keep it on my desk) but it also has a solid expositional style that fully motivates concepts as they are introduced. The Ito Calculus volume goes deeper than a number of other books on topic including information on integration wrt to a general semimartingale instead of just BM and even an introduction to stochastic calculus on manifolds. My only complaints about the book are that it is separated into two volumes which can be kind of a pain and that its coverage of the SDE/PDE relationship is weak. I would recommend reading Karatzas&Shreeve in addition to this book to fill in some the SDE/PDE details and to get another point of view on the somewhat difficult topic of stochastic analysis

Rating:
Summary: A Beautiful Survey of Markov Processes
Review: This book, the first in a two volume set, is a wonderful survey of some of the most important results in modern mathematics. The books begin with Brownian motion, review results from measure theory, and proceed all the way up to the general theory of Markov processes. As a researcher in econometrics and finance, I have found these books incredibly useful.

Several things really set these books apart. First, the authors do a great job motivating the subject matter, giving the reader a sense of why technical topics are important. Although mathematical purists may quibble with this, it gives readers with backgrounds outside of pure mathematics a really useful perspective, and makes the progression of topics flow smoothly throughout the two volumes. Second, these books actually manage to provide motivation and intuition without sacrificing rigor, which is truly an amazing accomplishment. Finally, the price is outstanding--I would challenge anyone to find a text in this area that covers half as much ground for less than twice the price of R&W's books!

While on a similar technical level to Karatzas and Shreve, these books offer much more breadth and intuition at the cost of a few technical details and little treatment of PDEs (this is really my only complaint). Both are indispensible references, but Rogers and Williams is one of the finest mathematical texts I have encountered.

Rating:
Summary: A Great Book
Review: This is a great book. It is not difficult to read. The style is very informal and at times actually humourous. It does not follow the definition-lemma-proof way of doing things at the expense of leaving simple definitions out, but these can be easily found somewhere else. The book contains an enormous amount of information, and the authors are clearly men of great knowledge and depth. The book is very nicely produced (from a 1st edition) by Cambridge U Press. Very clearly printed, and at a low price for the volume. I highly recommend both volumes to anyone who works in stochastic processes, or mathematical finance (assuming one wants to learn things, rather than just talk about them).