Stability in the small moment sense of the backward Euler–Maruyama method for stochastic differential equations with super-linear coefficients

Abstract

For stochastic differential equations (SDEs) with super-linear drift and diffusion coefficients, the backward Euler–Maruyama (BEM) method is considered to reproduce the stability of the underlying SDEs. The pth moment exponential stability for some small p∈(0,1) and the almost sure exponential stability of the BEM method are proved. The results in this paper partially extend those in Higham, Mao and Yuan (2007). Numerical simulations are provided to illustrate the theoretical results.