Book review

Random processes with independent increments are the primary and most important stochastic processes. Still today they constitute a huge area of mathematical and physical research including very many applications of different types. The author, a leading authority in the field of stochastic processes, here presents an introduction into the foundations of the theory together with selected special topics. This book is, in the proper sense, not a textbook on Poisson processes, Wiener processes and related classes of stochastic processes although Poisson processes give one framework for the development of the whole theory. Rather, it is a modern representation of the general characteristics of basic random processes written with full mathematical rigour. It is well translated from the Russian original of 1986.

The text starts with a short zeroth chapter on the fundamentals of probability spaces and preliminary information about random processes. Then follow five chapters of nearly equal weight. The first chapter is devoted to sums of independent random variables tantamount to simple random walks and including the renewal scheme. The main contents of the next chapter are general processes with independent increments viewed as random measures with independent values; here random measures with alternating signs and stochastic integrals as well as random linear functionals and generalized functions are thoroughly discussed. In the following two chapters further general properties of processes are considered including the decomposition of processes, homogeneous processes and composed Poisson processes. Finally, multiplicative processes are the topic of the fifth chapter; these processes are a generalization of the proceses with independent increments and a cornerstone of the modern noncommutative probability theory basic for quantum stochastics. In particular, multiplicative processes in Abelian groups and stochastic semigroups of linear operators in multidimensional spaces are discussed.

There are an appendix on local properties of sample functions, a short list of notations, bibliographical notes, a list of references including books and important original papers (with full titles), and a really too short subject index.

The text is written in a lively style. The book is well printed and well produced. It can render good services as a textbook on a postgraduate level and as a valuable source book. A full background of analysis and probability theory is necessary to read it. This volume represents an important contribution to the vast literature on stochastic processes. It can be warmly recommended to all mathematicians, physicists and experts of applied sciences engaged in the theory of stochastic processes. For libraries it is a must.