For stochastic stochastic differential equations with Markovian switching, whose drift and diffusion coefficients are allowed to contain superlinear terms, the backward Euler-Maruyama (BEM) method is proposed to approximate the invariant measure. The existence and uniqueness of the invariant measure of the numerical solution generated by the BEM method is proved. Then the convergence of the numerical invariant measure to its underlying counterpart is shown. The results obtained in this work release the requirement of the global Lipschitz condition on the diffusion coefficient in [X. Li et al. SIAM J. Numer. Anal. 56(3)(2018), pp. 1435-1455]. Numerical simulations are provided to demonstrate those theoretical results.

翻译：针对漂移项和扩散项允许包含超线性项的马尔可夫切换型随机微分方程，本文提出采用向后欧拉-丸山（BEM）方法逼近其不变测度。首先证明了BEM方法生成的数值解的不变测度的存在唯一性，进而证明了数值不变测度对其原系统不变测度的收敛性。本文所得结果放宽了[X. Li et al. SIAM J. Numer. Anal. 56(3)(2018), pp. 1435-1455]中对扩散系数需满足全局Lipschitz条件的要求。数值模拟验证了相关理论结果。