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,现有测量近似理论要求全球Lipschitz(SDDEs)延迟差分方程式(SDDEs)的传播术语持续。我们的工作是为非线性扩散术语的SDDEs开发一个新的明确的数字方法,并确立测量近近似理论。确切地说,我们构建了一个功能估值的明显短脱的 Euler-Maruyama 段进程(TEMSP ), 并证明它承认了独特的异变性数值测量。 我们还证明了数字变量测量方法与Fortet- Mourier 距离的SDDE基本方法相融合。 最后,我们举了一个实例和数字模拟来支持我们的理论。</s>