Almost sure exponential stability of numerical solutions for stochastic delay differential equations with jumps

Abstract

This paper deals with the almost sure exponential stability of the Euler-type methods for nonlinear stochastic delay differential equations with jumps by using the discrete semimartingale convergence theorem. It is shown that the explicit Euler method reproduces the almost sure exponential stability under an additional linear growth condition. By replacing the linear growth condition with the one-sided Lipschitz condition, the backward Euler method is able to reproduce the stability property.