Applications of Mathematics. 23. Berlin: Springer-Verlag. xxxv, 632 p. (1992).
The first half of this monograph on numerical methods for approximating the solutions of stochastic differential equations (SDEs) is devoted to furnishing the necessary preliminary background. After spending the first two chapters reviewing basic ideas from probability and stochastic processes, the authors present in Chapters 3 and 4 the pertinent results from the theories of the Itô integral and the Stratonovich integral and their respective associated SDEs. In Chapter 5 a thorough treatment of stochastic Taylor series is given which forms the theoretical backbone for most of the numerical methods discussed later in the text. In Chapters 6 and 7 the authors pause to emphasize the practical importance of SDEs by presenting several significant applications of SDEs. A brief summary of the basic theoretical ideas and the major methods associated with numerical solution of deterministic ordinary differential equations is given in Chapter 8.
Chapter 9 begins addressing the primary objective of the book by extending the deterministic ideas of Chapter 8 to the stochastic case and by developing the stochastic Euler method. Then methods yielding approximations that converge strongly to the solution of the SDE are developed in Chapter 10 (Taylor methods), Chapter 11 (Runge-Kutta and multistep methods), and Chapter 12 (Implicit methods useful for stiff differential equations). In Chapter 13 the authors pause again to present applications where the numerical methods of Chapters 10, 11, 12 are quite useful. After that, methods yielding approximations that converge weakly to the solution of the SDE are developed in Chapter 14 (Taylor methods), Chapter 15 (Runge-Kutta, extrapolation, implicit and predictor-corrector methods), and Chapter 16 (Variance reduction methods). The book concludes with a brief Chapter 17 containing applications of the weakly converging methods.
The book is designed to be more accessible by allowing readers, who so desire, to omit unessential theoretical discussion. Occasional exercises (with solutions of the non-computer exercises in the back of the book) are included for readers who would profit from an interactive approach. Many of the stochastic numerical methods presented have been developed just within the last decade. This monograph provides an extensive introduction to a very new and rapidly growing area of mathematics and as such is a welcome and valuable addition to the literature.