The book is concerned with the asymptotic behaviour of stochastic differential equations including the ergodic theory, the method of averaging and the theory of stability. The author, who is one of the founders of this area in the theory of stochastic differential equations and who made fundamental contributions to this field has produced a valuable book. It should succeed in becoming the standard reference work in the field of asymptotic methods for stochastic differential equations.
The book is by no means a systematic textbook for a wide circle of readers. Instead the author has written a research level monograph on selected problems of current research interest. Some graduate training in the theory of stochastic differential equations will be necessary even for the reader who only needs to apply the presented results. The reviewer’s only criticisms are the lack of an index and the modest list of references. The book is organized into four chapters, whose contents we summarize.
General ergodic theorems, densities for transition probabilities and resolvents for Markov solutions of stochastic differential equations, and a detailed discussion of one-dimensional equations and invariant measures for stochastic equations are outlined in Ch. 1.
In Ch. II, § 1 the author investigates ordinary and stochastic equations with a small right hand side of the form \(dx=A_{\epsilon}(x,dt)\), and estimates time intervals for which x(t) differs essentially from the initial value x(0), and the asymptotic behaviour of a solution on these intervals as \(\epsilon\) \(\to 0\). In § 2 the author deals with a two-component Markov process (x(t),y(t)) in the phase space, where y(t) is a piecewise constant and finitely many changes of state take place in any finite time interval. The author considers the case when the process x(t) satisfies a diffusion equation with coefficients depending on y(t), the increase in intensity of the jumps of the process is inversely proportional to \(\epsilon\) as \(\epsilon\) \(\to 0\), and y(t) is an exponentially ergodic process for fixed \(\epsilon\). Then the limit process x(t) turns out simply to be a diffusion process with coefficients obtained from those of the pre-limit process by a certain averaging with respect to an ergodic distribution.
In § 3 the author considers systems of stochastic differential equations containing rapid variable components and he finds conditions under which the influence of those components on the remaining ones is averaged in such a way that in the limit the non-rapid components satisfy a certain stochastic equation with constant averaged coefficients.
In Ch. III, § 1 the author investigates the stability of sample paths of homogeneous Markov processes using the Lyapunov function method. § 2 of Ch. III is devoted to the stability analysis of linear equations containing differentials with respect to Wiener processes and Poisson measures with independent values. It turns out that the study of mean square stability can be reduced to the study of such stability in invariant subspaces in which the stochastic semigroup is irreducible. In § 3 sufficient conditions for stability and instability in the first approximation are found.
Ch. IV extends results of § 2 of Ch. III to linear equations with bounded coefficients in Hilbert spaces. Conditions are presented for existence and uniqueness of solutions of general equations in Hilbert spaces, conditions that amount generalizations of the “classical” conditions for the finite-dimensional case.
A special paragraph containing examples and counterexamples is devoted to clarifying the essential effects of the infinite dimensionality of the phase space and possible deviations from what has been established in the finite-dimensional case.