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Book review

Author(s) Sobczyk, K.
Publisher Kluwer Academic Publishers
Year of publication 1991
Reviewed by Isaac Elishakoff

Differential equations play a central role in applications of mathematics to natural and engineering sciences. This monograph deals with stochastic differential equations, i.e.differential equations for stochastic processes. The basic problem is to find the probabilistic characteristics of the solution of such equations.

The book consists of six chapters. First chapter gives the short resume of the stochastic processes, including a brief account of probability and random variables, basic concepts of stochastic processes, including Gaussian processes, stationary processes, Markov processes, processes with independent increments, joint stochastic processes, martingales, white noise and processes with values in Hilbert space.

The second chapter deals with stochastic calculus; namely it first discusses the mean-square calculus covered in almost every classical textbook on random vibrations; then author gives an account of Ho's stochastic integral-topics which usually were outside the scope of "engineering" books. The third chapter is devoted to Ho's stochastic differential equations, whereas the fourth chapter gives an exposition of linear systems with random excitations, nonlinear systems, and stochastic systems. Chapter five deals with numerical methods of solutions of stochastic differential equations. These five chapters in 337 pages constitute the heart of the book.

The next 60 pages include some applications: random vibrations of road vehicles, response of structures to turbulent fields, response of structures to earthquake excitations, response of structures to sea waves, and stochastic stability of structures.

For the next edition of the book one would desire exposition of methods dealing with the structural reliability which should be the final product of structural analysis due to random disturbances. One would also want to see the coverage of finite element methods to solve the Fockker-Planck equations, as well as discussion of stochastic finite element method. Also the eminent questions on the interrelation of the probabilistic methods of analysis with its alternatives - fuzzy set based analysis as well as set valued analysis, like convex modelling - are posed when discussing the uncertainties in loads, initial conditions, boundary conditions or parametric uncertainties. Engineers are looking for answers to these fundamental equations.

To sum up, this is an excellent book dedicated to integration of mathematical and engineering aspects of stochastic differential equations. The preface of Prof. M. Hazewinkel is indeed very appropriate one. He mentions: "There are several good books on stochastic differential equations. But a book that combines a thorough, self contained treatment of the topic with actual real life applications, a book that also really tells the reader what all this beautiful theory is good for, and, moreover, discusses how to deal with these things numerically for simulations purposes, is rare. This is such a book..." It is a pleasure to recommend, therefore, the book to engineering researchers and students working in the field of stochastic differential equations.