Essentials of Stochastic Processes

This is the third edition of a popular textbook on stochastic processes. It is intended for advanced undergraduates and beginning graduate students and aimed at an intermediate level between an undergraduate course in probability and the first graduate course that uses measure theory. The author’s intention is to make the book accessible to students having a variety of interests and a range of levels of mathematical sophistication.

The topics share much with other comparable introductory textbooks. The main items are Markov chains (in both discrete and continuous time), Poisson processes, renewal processes, and martingales. This edition replaces the previous edition’s final chapter on Brownian motion with one on mathematical finance. There is also a short appendix that is meant to provide a review of basic probability.

The author has made a number of changes from the first edition through the third edition. One of them has been to move the chapter on martingales (which students typically find difficult) toward the end of the book, and to rely on it mostly for results pertaining to mathematical finance. The changes from the second edition to the third edition are modest and amount mostly to disentangling his treatment of returns to a fixed state from the proof of the related convergence theorem.

The book has two major strengths. The first is the examples. The author strongly believes in teaching through examples, and he has a lot of them. His research interests now include stochastic processes in ecology and genetics. So he is comfortable with topics in biology and includes some biological examples. It would have been nice to see more of these. His examples have a broad reach, from gambling and games to genetics, manufacturing and computer processes. The large number of exercises also includes many of the “work out this example on your own” type.

The second major strength is an approach that begins proofs with a “why is this true section” before adding the details to complete a formal proof. Presented with this, the student can better understand what the result means and perhaps be more motivated to work through the proof.

Things that don’t work so well are the connections between sections of the book, which could be more direct but are occasionally rather obscure. In particular, the examples are sometimes not well integrated with the general exposition, so readers may be confused about what an example is intended to teach. The book also has too many typographical errors, as is unfortunately true of both earlier editions as well. Even the short paragraph added to the preface of this edition has a typo.

The author also continues to provide worked-through computational examples using TI-83 scientific calculator instructions. Not many students are likely to find this useful. If the author doesn’t want to use a more current package like Matlab or Sage, then providing something like pseudo-code would make it easier for students to do the calculations with their preferred devices and software.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.