Almost sure and moment exponential stability of Euler-Maruyama method for non-autonomous stochastic differential equations

For autonomous stochastic differential equations (SDEs), it has been shown that the Euler-Maruyama (EM) method produces almost sure exponential stability and moment exponential stability of SDEs under some conditions which include the linear growth condition. In this work, we extend these results to non-autonomous SDEs, and prove that the exact solution is almost sure exponential stability and moment exponential stability under the local linear growth condition which is weaker than the linear growth condition. It is shown that the EM method maintains corresponding stability. A numerical example is presented to illustrate the effectiveness of this result.