Polynomial and exponential stability of $θ$-EM approximations to a class of stochastic differential equations

Both the mean square polynomial stability and exponential stability of $θ$ Euler-Maruyama approximation solutions of stochastic differential equations will be investigated for each $0\leθ\le 1$ by using an auxiliary function $F$ (see the following definition (2.3)). Sufficient conditions are obtained to ensure the polynomial and exponential stability of the numerical approximations. The results in Liu et al [12] will be improved and generalized to more general cases. Several examples and non stability results are presented to support our conclusions.