Numerical approximations of stochastic delay differential equations with delayed impulses

Abstract

This paper focuses on approximating stochastic delay differential equations with delayed impulses using Euler-Maruyama-type approximations. One key difference from previous literature is that the impulsive perturbations considered in this paper are past-dependent. Additionally, both the time delays in the stochastic delay differential equations and in the impulsive functions are functions of time. We establish the mean square convergence of the Euler-Maruyama approximations under a local Lipschitz condition and a linear growth condition. Furthermore, we determine the order of convergence under a global Lipschitz condition and provide an illustrative example.