Almost Sure Exponential Stability of Numerical Solutions for Stochastic Pantograph Differential Equations with Poisson Jumps

Abstract

The stability analysis of the numerical solutions of stochastic models has gained great interest, but there is not much research about the stability of stochastic pantograph differential equations. This paper deals with the almost sure exponential stability of numerical solutions for stochastic pantograph differential equations interspersed with the Poisson jumps by using the discrete semimartingale convergence theorem. It is shown that the Euler–Maruyama method can reproduce the almost sure exponential stability under the linear growth condition. It is also shown that the backward Euler method can reproduce the almost sure exponential stability of the exact solution under the polynomial growth condition and the one-sided Lipschitz condition. Additionally, numerical examples are performed to validate our theoretical result.

Keywords

• stochastic pantograph differential equation with jumps
• Poisson process
• Euler–Maruyama method
• backward Euler–Maruyama method almost sure exponential stability
• Lipschitz condition
• polynomial growth condition