A course in applied stochastic processes.

*(English)*Zbl 1137.60001
Texts and Readings in Mathematics 40. New Delhi: Hindustan Book Agency (ISBN 978-81-85931-69-2/hbk). xi, 214 p. (2006).

The book grew out of lectures given by the authors at the Indian Statistical Institute for students at Master’s level in Statistics. It consists of five chapters, starting with Chapter 0 on Probability Tools and Techniques. Topics introduced in this chapter range from the definition of probability, random variables and expectation to discrete martingales, Markov chains and continuous time processes. Chapter 1 is concerned with branching processes and, in particular Galton-Watson type branching processes are discussed and questions of the expected size of a generation, extinction probabilities and the asymptotic behaviour of of the process are studied. Several examples, such as family trees and customer queues, as well as generalisations are presented. The following chapter provides an introduction to genetics, that is first the main relevant ideas from biology are recalled, then some basic mathematical results in this area are presented, which are related to the variations in gene frequency in a population, e.g., the Hardy-Weinberg laws. The chapter provides the basis for the material in Chapter 3 on Markov models in genetics. The models discussed include the Wright-Fisher model, several versions of Moran’s model, e.g., for haploid or diploid populations, and Kimura’s model. At the end of the chapter an outline of the idea of a diffusion approximation of the Wright-Fisher model is given. In the final chapter the authors consider models in epidemics. The basic stochastic models are now continuous time Markov chains and questions of interest are how the epidemic changes with respect to the numbers of infected and susceptibles or what the duration and total size of the epidemic is. A number of threshold theorems are employed to study these questions. Each chapter contains exercises and references and suggestions for further reading.

The book provides a good introduction to the topic and can be used as material for lectures or as a basis for projects. It is well-written, although some misprints could have been more carefully corrected before publishing.

The book provides a good introduction to the topic and can be used as material for lectures or as a basis for projects. It is well-written, although some misprints could have been more carefully corrected before publishing.

Reviewer: Evelyn Buckwar (Berlin)

##### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |

60J85 | Applications of branching processes |

60J20 | Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

92D10 | Genetics and epigenetics |

92D30 | Epidemiology |