Global asymptotic stability of solutions of cubic stochastic difference equations

Abstract

Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable martingale differences is proven under appropriate conditions in and#8477;1. As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift-implicit and#952;-methods with variable step sizes for ordinary stochastic differential equations driven by standard Wiener processes, is discussed.