  Both the mean square polynomial stability and exponential stability of
$\theta$ Euler-Maruyama approximation solutions of stochastic differential
equations will be investigated for each $0\le\theta\le 1$ 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.
