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.

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