This book is an excellent reference for those interested in how probability theory can be applied to concrete problems arising in engineering, biology, management science and operation research.
Roughly speaking, a stochastic process is a function, defined on a set T, whose values are random variables. A typical example of stochastic process is a random walk, where a particle moves choosing its direction according to a given probability. Other common examples are the arrivals of customers to a bank, according to some probability law: the function assigning to t the number of customers present at time t is then a stochastic process. In this example, the customers may form one or more queues, and the service time usually follows a probabilistic distribution. This kind of situation is the object of queuing theory, which is extensively studied in chapter 6. The most common probability distributions encountered in practice (such as Poisson, uniform, Bernoulli) are explained in the first chapter, together with some basic concepts of probability theory, such as expectations, moment-generating functions, conditional probabilities and so on.
The third chapter, the longest, is devoted to the study of Markov processes, both discrete and continuous. A Markov process is a particular kind of stochastic process in which the evolution of the system after a given time is independent of how the system reached that state. This is in fact a realistic assumption in many situations. Markov chains are used, for example, to predict the evolution of infectious diseases through a given population.
Each chapter is followed by a long list of problems, so the reader has the opportunity to check systematically his/her understanding of the text; the solutions of the even-numbered exercises are given at the end of the book. Several tables and figures enrich the book.
Turning to more technical remarks, the book is certainly very well written, the proofs are clear and the examples are often illuminating. However, this is not a theoretical book on stochastic processes and it requires only a basic knowledge of calculus; the proofs of some general abstract results are sometimes omitted (for example: the Perron-Frobenius theorem, that justifies the existence of the limiting probabilities of Markov processes in § 3.2.3). Note also that there is no use of abstract measure theory or Lebesgue integration; this book should be largely accessible to everyone with a mathematical background including a course in calculus.
This book will be valuable to graduate students or professionals whose first objective is to use probabilistic methods in modelling and simulation. The large number of problems makes it suitable also for self-study.
Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at email@example.com.
Preface.- Review of probability theory.- Stochastic processes.- Markov chains.- Diffusion processes.- Poisson processes.- Queueing theory.- Appendix A. Statistical tables.- Appendix B. Answers to even-numbered exercises.- References.- Index