Explicit approximation of the invariant measure for stochastic delay differential equations with the nonlinear diffusion term

Li, Xiaoyue and Mao, Xuerong and Song, Guoting (2023) Explicit approximation of the invariant measure for stochastic delay differential equations with the nonlinear diffusion term. Journal of Theoretical Probability. ISSN 0894-9840 (https://doi.org/10.1007/s10959-023-01290-5)

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Abstract

To our knowledge, existing measure approximation theory requires the diffusion term of the stochastic delay differential equations (SDDEs) to be globally Lipschitz continuous. Our work is to develop a new explicit numerical method for SDDEs with nonlinear diffusion term and establish the measure approximation theory. Precisely, we construct a function-valued explicit truncated Euler–Maruyama segment process and prove that it admits a unique ergodic numerical invariant measure. We also prove that the numerical invariant measure converges to the underlying invariant measure of the SDDE in the Fortet–Mourier distance. Finally, we give an example and numerical simulations to support our theory.

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