Exponential stability of stochastic differential equations. (English) Zbl 0806.60044

Pure and Applied Mathematics, Marcel Dekker. 182. New York: Marcel Dekker, Inc.. xii, 307 p. $ 125.00/hbk (1994).
This monograph is the first systematic treatment of stability properties of stochastic differential equations driven by nonlinear integrators. Such equations have been looked at e.g. by H. Kunita [Stochastic flows and stochastic differential equations (1990; Zbl 0743.60052)] and by R. A. Carmona and D. Nualart [Nonlinear stochastic integrators, equations and flows (1990; Zbl 0772.60041)].
The first three (out of eight) chapters are introductions to nonlinear integrators and stochastic (functional) differential equations based on nonlinear integrators. In the remaining chapters the author provides sufficient criteria for almost sure and moment exponential stability of stochastic differential (delay) equations. One key tool is an adaptation of Lyapunov’s second method. The monograph also contains many examples. The reader is assumed to have some background in probability and measure theory (like Brownian motion, measurability etc). Martingales, stochastic integrals w.r.t. nonlinear integrators and their properties, generalized ItĂ´’s formula etc. are provided without proof (and without precise reference).
The book will certainly be a valuable reference for anyone interested in the qualitative theory of stochastic differential equations.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
93E15 Stochastic stability in control theory