This paper is concerned with long-time strong approximations of SDEs with non-globally Lipschitz coefficients.Under certain non-globally Lipschitz conditions, a long-time version of fundamental strong convergence theorem is established for general one-step time discretization schemes. With the aid of the fundamental strong convergence theorem, we prove the expected strong convergence rate over infinite time for two types of schemes such as the backward Euler method and the projected Euler method in non-globally Lipschitz settings. Numerical examples are finally reported to confirm our findings.

翻译：本文研究具有非全局Lipschitz系数的随机微分方程的长时间强逼近问题。在特定非全局Lipschitz条件下，我们为一般单步时间离散化格式建立了长时间版本的基本强收敛定理。借助该基本强收敛定理，我们证明了在非全局Lipschitz设定下两类数值格式（如后向欧拉法和投影欧拉法）在无限时间区间上的期望强收敛速率。最后通过数值算例验证了理论结果。