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

Davood Ahmadian,Omid Farkhondeh Rouz

doi:10.1186/s13660-020-02452-3

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.

Highlights

1 Introduction In special cases, stochastic delay differential equations (SDDEs) and stochastic delay integro-differential equations (SDIDEs) are a type of stochastic differential equations (SDEs), which has been discussed in a variety of sciences such as the mathematical model [1], economy [2], infectious diseases [3], and population dynamics [4]
For SDIDEs, most of the related existing literature focused on the linear models
As for SDIDEs, Ding et al [17] studied the stability of the semi-implicit Euler method for linear SDIDEs

Summary

Introduction

Stochastic delay differential equations (SDDEs) and stochastic delay integro-differential equations (SDIDEs) are a type of stochastic differential equations (SDEs), which has been discussed in a variety of sciences such as the mathematical model [1], economy [2], infectious diseases [3], and population dynamics [4]. Li and Gan investigated the exponential mean-square stability of theta method for nonlinear SDIDEs by the technique with the Barbalat lemma in the literature [20]. Mo et al [24] discuss the exponential mean-square stability of the θ -method for neutral stochastic delay differential equations with jumps. Li and Zhu [28] investigated the pth moment exponential stability and almost surely exponential stability of stochastic delay differential equations with Poisson jump. In order to fill this gap, we introduce the SSTM scheme for n-dimensional SDIDEs with Poisson jump by some numerical integration technique and perform a stability analysis of the proposed scheme.

Preliminary results

The SSTM scheme with Poisson jump

Conclusion