Stability of Stochastic Delay Differential Equation with a Small Parameter.

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution xε(·) of the underlying delay equation for ε > 0 small enough. Perturbed Liapunov function method is used in the analysis. [ABSTRACT FROM AUTHOR]