Some stabilities of stochastic differential equations with delay in the G-framework and Euler–Maruyama method

Abstract

This paper discusses some stabilities of stochastic differential equations with delay in the G-framework (G-SDDEs, in short) and Euler-Maruyama method. We construct a weaker condition instead of using the Lyapunov functional method to obtain the p-th moment exponential stability of the G-SDDE. We prove that the Euler-Maruyama method can reproduce the p-th moment exponential stability of the G-SDDE under some step size restrictions. We also prove that the stability of the discrete Euler-Maruyama method is equivalent to that of the continuous Euler-Maruyama method. Furthermore, we introduce some sufficient conditions under which the p-th moment exponential stability of the G-SDDE can imply its quasi sure exponential stability. Finally, we give two numerical examples to confirm the theoretical results.