The strong convergence of numerical methods for stochastic differential equations (SDEs) for $t\in[0,\infty)$ is proved. The result is applicable to any one-step numerical methods with Markov property that have the finite time strong convergence and the uniformly bounded moment. In addition, the convergence of the numerical stationary distribution to the underlying one can be derived from this result. To demonstrate the application of this result, the strong convergence in the infinite horizon of the backward Euler-Maruyama method in the $L^p$ sense for some small $p\in (0,1)$ is proved for SDEs with super-linear coefficients, which is also a a standalone new result. Numerical simulations are provided to illustrate the theoretical results.

翻译：针对随机微分方程（SDEs）在$t\in[0,\infty)$区间上的数值方法，证明了其强收敛性。该结果适用于任何具有马尔可夫性、有限时间强收敛性及一致有界矩的单步数值方法。此外，由此结果可推导出数值平稳分布向真实平稳分布的收敛性。为展示该结果的应用，针对具有超线性系数的随机微分方程，证明了后向欧拉-马鲁亚马方法在$L^p$意义下（$p\in(0,1)$较小）的无限时域强收敛性，这本身也是一个独立的新结果。数值仿真验证了理论结果。