Almost Sure Convergence of Stochastic Differential Equations of Jump-Diffusion Type

Abstract

We consider a class of stochastic differential equations of jump-diffusion type

$$d{{x}_{t}} = f\left( {t,{{x}_{t}}} \right)dt + G\left( {t,{{x}_{t}}} \right)d{{W}_{t}} + \int_{U} {h\left( {t - ,u,{{x}_{{t - }}}} \right)} N\left( {dt,du} \right),$$

where f, G, and h are càdlàg adapted processes; W _t is an m-dimensional standard Brownian motion and N(dt, du) is a Poisson random counting measure with deterministic characteristic measure ⋋(du) on the measurable space U The pathwise unique strong solution and the stability problem for a sequence of stochastic systems with small perturbations in almost sure convergence are discussed by means of a stochastic version of Gronwall’s inequality.