Stationary distribution of the Milstein scheme for stochastic differential delay equations with first-order convergence.

• We are very grateful to the editor and the reviewer for evaluating our manuscript and for the comments. In accordance with the comments, we have carefully and thoroughly revised and rewritten our manuscript. To our best knowledge, this work is the first paper to consider the stationary distribution of the Milstein scheme for stochastic differential delay equations. We reveal that the distribution of numerical segment process converges exponentially to the underlying one in the Wasserstein metric. Then the first-order convergence of numerical stationary distribution to exact stationary distribution is presented. Finally, we have demonstrated the reliability of the theoretical results through abundant numerical experiments. This paper focuses on the stationary distribution of the Milstein scheme for stochastic differential delay equations. The numerical segment process is constructed, which is proved to be a time homogeneous Markov process. We show that this numerical segment process admits a unique numerical stationary distribution. Then we reveal that the distribution of numerical segment process converges exponentially to the underlying one in the Wasserstein metric. Moreover, the first-order convergence of numerical stationary distribution to exact stationary distribution is presented. Finally, abundant numerical experiments confirm the reliability of theoretical findings. [ABSTRACT FROM AUTHOR]