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The backward Euler-Maruyama (BEM) method is employed to approximate theinvariant measure of stochastic differential equations, where both the driftand the diffusion coefficient are allowed to grow super-linearly. The existenceand uniqueness of the invariant measure of the numerical solution generated bythe BEM method are proved and the convergence of the numerical invariantmeasure to the underlying one is shown. Simulations are provided to illustratethe theoretical results and demonstrate the application of our results in thearea of system control.

The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients