Department of Applied Mathematics University of Leeds, Leeds ls2 9jt, UK

UK Nonlinear News Book Review

An Introduction to Continuous-Time Stochastic Processes.
Theory, Models, and Applications to Finance, Biology, and Medicine

V. Capasso and D. Bakstein

This book provides a mathematical overview of the theory of con tinuous-time stochastic processes, with emphasis on stochastic differential equations (SDEs). Applications in finance and population modelling are also briefly reviewed.

The authors divide their monograph into three sections. Part I provides a rigorous introduction to probability and stochastic processes. The fundamentals of probability are introduced from Lebesgue measure theory, assum ing that the reader is familiar with the motivations and notions of probability. Markov, Poisson, Weiner and Lévy processes are carefully defined before the fundamentals of Ito calculus are reviewed. Part I concludes with an overview of SDEs, discussing issues of existence, uniqueness and stability. The second part of the book focuses on the authors' interests in financial and biological applications of SDEs. The Black-Scholes model of an arbitrage-free financial markets is explained from first principles, and possible extensions (such as non-constant interest rates) are discussed. To introduce biological applications of SDEs, the authors describe simple models of birth-and-death processes, and stochastic epidemic models. Part III of the monograph comprises stand-alone appendices on measure and integration, metric spaces, and stability properties of ordinary differential equations.

The primary audience for this book will be mathematicians (both pure and applied) active in other areas who require an introduction to stochastic theory. Scientists already working in the applications of SDEs will also benefit from this mathematically rigorous reference text. The core of the text on Ito calculus was developed from course material, and would be suitable supplementary reading for graduate or advanced undergraduate students of stochastic theory (who already have a sound grasp of discrete-time stochastic processes, and appropriate mathematical training in calculus and analysis).

The style of the text, in particular the first part of the monograph, is concise and rigorous. More lengthy proofs are omitted, with comprehensive references given for further study. Each chapter concludes with a set of exercises inviting readers to prove supplementary results and review particular aspects of the theory. To keep the introductory chapters to moderate length, the authors have chosen not to include numerical exercises and illustrations in the introductory chapters.

In summary, I have found this to be a useful reference text, and would recommend it to those wishing to delve in to the mathematical theory of stochastic processes.

Reviewed by Paul D. Baxter, University of Leeds.

UK Nonlinear News would like to thank Birkhäuser for providing a copy of this volume for review.