Exponential stability for stochastic differential equations with respect to semimartingales

Abstract

Consider a stochastic differential equation with respect to semimartingales dX(t)=AX(t) dμ(t)+G(X(t),t) dM(t) which might be regarded as a stochastic perturbed system of dX(t)=AX(t) dμ(t) . Suppose the second equation is exponentially stable almost surely. What we are interested in in this paper is to discuss the sufficient conditions under which the first equation is still exponentially stable almost surely.