Approximation of invariant measures of a class of backward Euler-Maruyama scheme for stochastic functional differential equations

Shi, Banban and Wang, Ya and Mao, Xuerong and Wu, Fuke (2024) Approximation of invariant measures of a class of backward Euler-Maruyama scheme for stochastic functional differential equations. Journal of Differential Equations, 389. pp. 415-456. ISSN 0022-0396 (https://doi.org/10.1016/j.jde.2024.01.025)

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Abstract

This paper is concerned with approximations of invariant probability measures for stochastic functional differential equations (SFDEs) using a backward Euler-Maruyama (BEM) scheme under one-sided Lipschitz condition on the drift coefficient. Firstly, the strong convergence of the numerical "segment sequence" from the BEM scheme on finite time interval [0, T] is established. In addition, it is also demonstrated that the numerical segment sequence from the BEM scheme is a Markov process, and the corresponding discrete-time semigroup generated by this BEM scheme admits a unique numerical invariant probability measure. Finally, it is revealed that the numerical invariant probability measure converges to the underlying one in a Wasserstein distance.

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