A Note on the Almost Sure Exponential Stability of the Milstein Method for Stochastic Delay Differential Equations With Jumps

Abstract

In this paper we consider the Milstein method to investigate the almost sure exponential stability for nonlinear stochastic delay differential equations (SDDEs) with jump diffusion, that is, processes that change only by jumps. The class of jump-diffusion SDDEs that admits explicit solutions is rather limited. In this regard, we employ the continuous and discrete semimartingale convergence theorem. It is shown that the Milstein method reproduces the almost sure exponential stability under an additional linear growth condition on drift and the function L 1 g. In the numerical section we present the results by simulating the method on kind of equations.