Stability of Stochastic Differential Equations

In Chap. 1 we studied problems of stability under random perturbations of the parameters. We noted there that no significant results can be expected unless the random perturbations possess sufficiently favorable mixing properties. Fortunately, in practical applications one may often assume that the “noise” has a “short memory interval.” The natural limiting case of such noise is of course white noise. Thus it is very important to study the stability of solutions of Ito equations since this is equivalent to the study of stability of systems perturbed by white noise. Generalization of well known results on stability and instability for the deterministic ODE in terms of the Lyapunov functions are proven for SDE. Conditions for stability and instability of moments are also proven.