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To our knowledge, the existing measure approximation theory requires thediffusion term of the stochastic delay differential equations (SDDEs) to beglobally Lipschitz continuous. Our work is to develop a new explicit numericalmethod for SDDEs with the nonlinear diffusion term and establish the measureapproximation theory. Precisely, we construct a function-valued explicittruncated Euler-Maruyama segment process (TEMSP) and prove that it admits aunique ergodic numerical invariant measure. We also prove that the numericalinvariant measure converges to the underlying one of SDDE in the Fortet-Mourierdistance. Finally, we give an example and numerical simulations to support ourtheory.

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