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arXiv · 2510.03053

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

Peng Chen·Hui Jiang·Jing Wang
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Abstract

In this paper, we investigate the Milstein numerical scheme with step size $η$ 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 $π$ and $π_η$ respectively. We first establish a central limit theorem for the empirical measure $Π_η$, a statistical consistent estimator of $π_η$. Subsequently, we derive both normalized and self-normalized Cramér-type moderate deviations.

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Peng Chen, Hui Jiang, Jing Wang. 2025-10-03. Central limit theorem and Cramér-type moderate deviations for Milstein scheme. https://arxiv.org/abs/2510.03053

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