Stochastic processes. From applications to theory. (English) Zbl 1368.60001

Del Moral, Pierre; Penev, Spiridon

Chapman & Hall/CRC Texts in Statistical Science Series. Boca Raton, FL: CRC Press (ISBN 978-1-4987-0183-9/print+ebook; 978-1-4987-0184-6/ebook). xlviii, 865 p. (2017).

This book provides an introduction to the probabilistic modelling of discrete and continuous time stochastic processes. The authors discuss a large class of models ranking from finite space valued Markov chains to jump diffusion processes, nonlinear Markov processes, self-interacting processes, mean field particle models, branching and interacting particle systems, as well as diffusions on constraint and Riemannian manifolds. The book also provides an introduction to stochastic analysis and stochastic differential calculus, including the analysis of probability measures on path spaces and the analysis of Feynman-Kac semigroups. A detailed description of some traditional and more advanced techniques based on stochastic processes is presented. The book combines rigorous statements and arguments with informal computations, illustrations, examples.
The book consists of 6 parts, divided into 30 chapters. The part titles are: “An illustrated guide”, “Stochastic simulation”, “Discrete time processes”, “Continuous time processes”, “Processes on manifolds”, “Some application areas”.
The book is almost self-contained, it contains around 500 exercises with detailed solutions. It can be useful for students, lecturers in probability theory and related topics, and practitioners interested in applications of stochastic processes.

Reviewer: Yuliya S. Mishura (Kyïv)

MSC:

60-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory
60G05	Foundations of stochastic processes
60Kxx	Special processes

Keywords:

stochastic process; Markov chain; martingale; diffusion process; jump process; nonlinear Markov process; path space measures; differential geometry; stochastic analysis; statistical physics; mean field model; gambling; ranking; control; mathematical finance