Central limit theorem and Cramér-type moderate deviations for Milstein scheme

In this paper, we investigate the Milstein numerical scheme with step size $\eta$ for a stochastic differential equation driven by multiplicative Brownian motion. Under some appropriate coefficient conditions, the continuous-time system and its discrete Milstein scheme approximation each possess unique invariant measures, which we denote by $\pi$ and ${\pi }_{\eta }$ respectively. We first establish a central limit theorem for the empirical measure ${\mathrm{\Pi }}_{\eta }$, a statistical consistent estimator of ${\pi }_{\eta }$. Subsequently, we derive both normalized and self-normalized Cramér-type moderate deviations.