The semi-implicit Euler–Maruyama (EM) method is investigated to approximate a class of time-changed stochastic differential equations, whose drift coefficient can grow super-linearly and diffusion coefficient obeys the global Lipschitz condition. The strong convergence of the semi-implicit EM is proved and the convergence rate is discussed. When the Bernstein function of the inverse subordinator (time-change) is regularly varying at zero, we establish the mean square polynomial stability of the underlying equations. In addition, the numerical method is proved to be able to preserve such an asymptotic property. Numerical simulations are presented to demonstrate the theoretical results.

中文翻译：

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非线性时变随机微分方程的半隐式 Euler-Maruyama 方法

研究了半隐式 Euler-Maruyama (EM) 方法来逼近一类时变随机微分方程，其漂移系数可以超线性增长，扩散系数服从全局 Lipschitz 条件。证明了半隐式EM的强收敛性，讨论了收敛速度。当逆从属函数（时变）的伯恩斯坦函数在零处有规律地变化时，我们建立了基本方程的均方多项式稳定性。此外，数值方法被证明能够保持这种渐近性。给出了数值模拟来证明理论结果。