Explicit approximation of the invariant measure for SDDEs with the nonlinear diffusion term

Abstract

To our knowledge, the 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 the nonlinear diffusion term and establish the measure approximation theory. Precisely, we construct a function-valued explicit truncated Euler-Maruyama segment process (TEMSP) and prove that it admits a unique ergodic numerical invariant measure. We also prove that the numerical invariant measure converges to the underlying one of SDDE in the Fortet-Mourier distance. Finally, we give an example and numerical simulations to support our theory.