Eventual uniform asymptotic stability for stochastic differential equation based on semimartingale with spatial parameter (Q756269)

The author considers the stochastic differential equation based on a semimartingale with spatial parameter \[ (1)\quad \phi_ t=x_ 0+\int^{t}_{t_ 0}F(\phi_ s,ds)\text{ on } t\geq t_ 0, \] and the perturbed system \[ (2)\quad \psi_ t=x_ 0+\int^{t}_{t_ 0}F(\psi \alpha_ s,ds)+\int^{t}_{t_ 0}G(\psi_ s,ds)\text{ on } t\geq t_ 0. \] Sufficient conditions are given under which the eventual uniform asymptotic stability of (1) is shared by (2). The paper is concluded with a discussion of some corollaries holding for classical Itô equations and stochastic differential equations with respect to semimartingales.