Random Processes in Physics and Finance , Melvin Lax , Wei Cai , and Min Xu , Oxford U. Press, New York, 2006. $98.50 (327 pp.). ISBN 978-0-19-856776-9
The term “econophysics” was introduced just 14 years ago, but the tradition of physicists being fascinated by random processes in finance has a history much older than that. In fact, both Nicolaus Copernicus and Isaac Newton invested considerable intellectual energy in attempting to understand the economic problems of their day.
Now, centuries later, Wall Street appears to hire as many physicists as economists. (See the article “Is Economics the Next Physical Science?” by Doyne Farmer, Martin Shubik, and Eric Smith, Physics Today, Physics Today 0031-9228 58 9 2005 37 https://doi.org/10.1063/1.2117821 September 2005, page 37 .) One wonders why. Is it because physicists are trained to solve the problems that are so difficult that knowing where to begin is unclear? Whatever the reason, just as physics departments in the US now prepare students for possible careers at the interface between physics and biology, they may want to emulate several excellent examples of departments at European institutions that prepare physics students for possible careers at the interface between physics and economics.
One challenge in adequately preparing physics students to contribute to economics has been the absence of a comprehensive and rigorous monograph that presents the physical laws governing random processes, which, in turn, provide the starting point for understanding economic fluctuations. Random Processes in Physics and Finance, by the late Melvin Lax and completed by coauthors Wei Cai and Min Xu, rises admirably to the challenge. One could certainly find no finer teacher than Lax to pen an introduction to the random-process techniques used in finance today. Lax was a Distinguished Professor of Physics at the City College of New York. I first came to admire his pedagogical gifts more than 30 years ago when he, with Robert Brout, Freeman Dyson, Mark Kac, and Leo Kadanoff, taught a statistical physics summer-school institute at Brandeis University in Massachusetts.
Chapter 1 is a 43-page summary of probability, the theory of random events. On page 1 Lax makes the humbling statement that there are three approaches to the very definition of probability. In chapter 2 he introduces the concept of random processes, a sequence of random events extended over a period of time, and emphasizes the simplifying ideas that underlie stationary, Gaussian, and Markovian processes. The following is an example of the liveliness of his prose: “Markovian processes are the analog of students who can remember only the last thing they have been told.”
Chapter 4 covers the concept of the noise spectrum and how it is measured. It is one of the nicest treatments I have seen on the subject; the development of the Wiener–Khinchine theorem relating the spectrum to the autocorrelation function is especially impressive. The Langevin treatment of the Fokker–Planck process, covered in chapter 10, differs from the stochastic differential equation approach, which incorporates Ito’s calculus lemma, a mathematical tool used in financial quantitative analysis. Other departures from traditional mathematical finance texts make Lax’s book a refreshing and much clearer read. For example, delta functions are used to avoid the need for abstract Lebesgue measures and Stieltjes integrals.
Throughout the book Lax emphasizes experimental measurement, an approach sadly missing from some traditional texts on mathematical finance. The numerous examples from experimental physics, such as noise in homogeneous semiconductors and the random walk of light in turbid media, allow the heavily abstract formalisms to come alive.
Examples from finance occupy only the last 30 or so pages, chapters 16 and 17, which are about 10% of the book. Lax covers some topics that are standard in mathematical finance texts, such as the Black–Scholes differential equation, as well as subjects that are new to me, for example, the Slepian functions. Not discussed, presumably for lack of space, is the rather large set of empirical facts that has emerged in recent years and has challenged the degree to which the assumptions that underlie the tractable mathematical models are valid. That the well-known “fat tail phenomena” are not simple perturbations on a Gaussian distribution calls into question the validity of many current economic models. But that omission is not so serious because the empirical facts are already discussed in other recent books on econophysics. Examples include the second edition of Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management (Cambridge U. Press, 2003) by Jean-Philippe Bouchaud and Marc Potters and An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge U. Press, 2000), which Rosario Mantegna and I wrote.
Sadly, Lax passed away in 2002 at age 80, and his almost completed manuscript was brought to its present form as a labor of love by his tremendously capable and dedicated former students, Cai and Xu. One hopes that this text will inspire other physics students to continue Lax’s legacy and contribute to this growing, diverse field.
