This text is a translation of* Probabilités, processes stochastiques et applications*. It is the companion to another text by the same authors on probability and statistics, *Probabilités et introduction à la statistique: Cours et exercices corrigés*, which seems to be available only in French. The subjects of the current book are families of random variables of two kinds: those indexed by the integers to form random sequences (martingales or Markov chains, for example), or indexed by positive real numbers to form stochastic processes (such as Markov, semi-Markov or Poisson processes).

The authors begin by setting out the concepts and results that are used in the succeeding chapters. They define the functional tools needed to study random variables: generating functions, characteristic functions, the Laplace transform, and entropy. Next they review results on the sums of random variables, the main properties of the classical distributions, the various kinds of convergence and the associated limit theorems. The treatment is careful and thorough, but its intent seems to be to remind readers of what they should already know. The pace is too quick for readers who might not have seen most of the topics before.

The book’s four primary chapters are split evenly between discrete time (martingales and Markov chains) and continuous time (stochastic, Markov and semi-Markov) processes. The treatment of martingales is thorough but it moves quickly. It starts with conditional distributions and expectations and how they are computed, moves on to the basic linear model, then introduces stopping times and proves the stopping-time theorem.

The treatment of Markov chains develops their general properties and includes some applications. Most of these are pretty standard, but there are a few good examples dealing with reliability and branching processes.

The theory of stochastic processes is introduced primarily through discussion of the classic families. These include jump processes, stationary and ergodic processes, Brownian motion and other processes with independent increments, as well as point processes.

The last part of the book looks specifically at jump Markov processes and finite semi-Markov processes such as Markov renewal. The best examples here are presented in the exercises.

A number of well-chose exercises are provided for each chapter. Oddly, each set of exercises is followed immediately by complete solutions. Several of the exercises, more as the book progresses, focus on applications.

Master’s and PhD students in applied probability are the primary target audience, though the authors suggest that the book would also be of use to researchers and engineers working with stochastic processes. The prerequisites effectively include a graduate course in probability with measure theory, the Lebesgue integral and related results from real analysis.

This is a solid text, relatively fast paced with good examples and a modest number of applications. Although the exposition is mostly clear and well written, the translation from French is occasionally awkward. The style is sometimes pretty stiff and both word choice and word order are sometimes unusual enough to make the reader pause and go back to reread.

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.