This is a continuation of the first author's earlier paper  jointly with Pang and Deng, in which the authors established some sufficient conditions under which the EulerMaruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in  is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will then show that the backward EM method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs. © 2010 Elsevier B.V. All rights reserved.
Mao, X., Shen, Y., & Gray, A. (2011). Almost sure exponential stability of backward EulerMaruyama discretizations for hybrid stochastic differential equations. Journal of Computational and Applied Mathematics, 235(5), 1213–1226. https://doi.org/10.1016/j.cam.2010.08.006