Exponential mean-square stability of numerical solutions for stochastic delay integro-differential equations with Poisson jump

Abstract

Abstract In this paper, we investigate the exponential mean-square stability for both the solution of n-dimensional stochastic delay integro-differential equations (SDIDEs) with Poisson jump, as well for the split-step θ-Milstein (SSTM) scheme implemented of the proposed model. First, by virtue of Lyapunov function and continuous semi-martingale convergence theorem, we prove that the considered model has the property of exponential mean-square stability. Moreover, it is shown that the SSTM scheme can inherit the exponential mean-square stability by using the delayed difference inequality established in the paper. Eventually, three numerical examples are provided to show the effectiveness of the theoretical results.

Keywords

• Split-step θ-Milstein scheme
• Exponential mean-square stability
• Stochastic delay integro-differential equations
• Poisson jump
• Lyapunov function