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.

翻译：据我们所知，现有的测度逼近理论要求随机延迟微分方程（SDDEs）的扩散项满足全局Lipschitz连续条件。本文旨在针对具有非线性扩散项的SDDEs提出一种新的显式数值方法，并建立测度逼近理论。具体而言，我们构造了一个函数值显式截断Euler-Maruyama分段过程（TEMSP），并证明其具有唯一遍历的数值不变测度。进一步证明，该数值不变测度在Fortet-Mourier距离下收敛于SDDE的原始不变测度。最后，通过算例与数值模拟验证了理论结果的有效性。