Kurt Jacobs: Stochastic Processes for Physicists

Abstract

Kurt Jacobs’ book on stochastic processes is intended for a one-semester course for physics graduate students. The high point of the book is the last chapter, where Jacobs accomplishes a very useful task: he demystifies the abstract language of modern probability theory to make it accessible for physicists. I particularly enjoyed his translation of several passages from recent work in control theory—as one might expect, there is often less there than meets the eye. Jacobs is to be commended for these passages. Unfortunately, the rest of the book is less satisfactory. Those looking for a brief but comprehensive introduction to the subjects we all learned from the classics by N. van Kampen (curiously, not referenced) or C. Gardiner will be disappointed. There is no mention of the binomial distribution, the central limit theorem is mentioned in passing without stating the result completely, the phases “random walk” and “Markov process” never appear, etc. Despite the title, the book is mostly about the Ito calculus for stochastic differential equations. The exposition is uneven and sometimes seems too elementary: at one point Jacobs feels obliged to explain at length what a partial differential equation is. Then, when the going gets rough he resorts to stating “it turns out”; that is, most of the hard results are given without proof. For my taste, too many things “turn out” in this book, even things whose proof is not all that hard, like the Wiener-Khintchine theorem. Almost all of the results in Chap. 6 on numerical methods “turn out”. Chapter 6 is a cookbook with little insights offered. The applications are mostly outside of physics. For example, in the chapter on Gaussian noise there is a brief treatment of Brownian motion, followed by a detailed and careful exposition of the Black-Scholes equation for option pricing. To be fair, before the financial crisis of 2007–2008 many physics students did want to know this stuff. Now, this topic seems less relevant. The editing is careless—in a book from Cambridge University Press, at that. Some of the errors are minor and occasionally amusing. Still, is it too much to insist that the famous