Polynomial stability for perturbed stochastic differential equations with respect to semimartingales

The aim of this paper is to investigate the almost surely polynomial stability of the stochastic differential equation with respect to semimartingales d[phi]t = F([phi]t, t) d[mu]t + G([phi]t) dMt + f([phi]t, t) d[mu]t + g([phi]t) dMt under the condition that its unperturbed equation d[psi]t = F([psi]t, t) d[psi]t + G([psi]t, t) dMt is polynomially stable almost surely. Several useful corollaries are obtained in dealing with the classical Itô equations. The results are also extended to the more general stochastic differential equation based on semimartingales with spatial parameters.