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.

中文翻译：

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泊松跳变随机时滞积分-微分方程数值解的指数均方稳定性

在本文中，我们研究了具有Poisson跳变的n维随机延迟积分微分方程（SDIDEs）的解，以及为实现该方法的分步θ-Milstein（SSTM）方案的指数均方稳定性。建议的模型。首先，利用李雅普诺夫函数和连续半semi收敛定理，证明了所考虑的模型具有指数均方稳定性。此外，证明了通过使用本文中建立的延迟差分不等式，SSTM方案可以继承指数均方稳定性。最终，提供了三个数值示例来说明理论结果的有效性。