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

Abstract

In this paper, we investigate the Milstein numerical scheme with step size $\hat{I}\cdot$ 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 $\ddot{I}€$ and $\ddot{I}{€}_{\hat{I}}\cdot$ respectively. We first establish a central limit theorem for the empirical measure ${\hat{I}}_{\hat{I}}\cdot$, a statistical consistent estimator of $\ddot{I}{€}_{\hat{I}}\cdot$. Subsequently, we derive both normalized and self-normalized Cramér-type moderate deviations.