This paper applies an explicit numerical method, i.e. the truncated Euler–Maruyama (EM) scheme, to investigate numerical approximations for nonlinear and non-autonomous stochastic differential equations (SDEs). The explicit method reproduces asymptotic stability of the equations. Under a weak condition and with locally Lipschitz continuous coefficients, strong convergence of the truncated EM scheme for the SDEs is proved. Combing the Hölder continuous condition for temporal variables, we obtain the convergence rate that is related to the Hölder continuity. Moreover, employing the discrete nonnegative semimartingale convergence theorem, we reformulate the explicit EM scheme to approximate asymptotic stability in an infinite horizon. Numerical simulations are provided to illustrate the validity of our theoretical results.

本文采用显式数值方法（即截断的Euler-Maruyama（EM）方案）来研究非线性和非自治随机微分方程（SDE）的数值逼近。显式方法再现了方程的渐近稳定性。在弱条件下，在局部具有Lipschitz连续系数的情况下，证明了SDE的截短EM方案的强收敛性。结合时间变量的Hölder连续条件，我们得到与Hölder连续性有关的收敛速度。此外，使用离散非负半mart收敛定理，我们重新构造了显式EM方案，以逼近无限远的渐近稳定性。提供数值模拟来说明我们理论结果的有效性。