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

TL;DR

This work analyzes the Milstein discretization of a multiplicative-noise SDE and its invariant measure. It proves a central limit theorem for the empirical invariant-measure statistic $\Pi_{\eta}$ and derives both normalized and self-normalized Cramér-type moderate deviations for the fluctuations $\eta^{-1/2}(\Pi_{\eta}(h) - \pi(h))$, employing a Stein-equation-based decomposition into a martingale part and a controllable remainder. The results rely on ergodicity, moment bounds, and careful tail estimates for the Milstein correction term, providing precise ranges for deviation parameters and highlighting differences from the Euler–Maruyama case. Overall, the paper advances understanding of invariant-measure estimation via high-order discretizations and quantifies rare-event behaviors in this setting, with potential implications for long-time numerical analysis of SDEs. Key contributions include (i) a CLT for $\Pi_{\eta}$, (ii) normalized and self-normalized moderate-deviation bounds with explicit rates depending on $\nabla\sigma$ and step size $\eta$, and (iii) a detailed martingale-remainder decomposition that isolates Milstein-specific effects and enables sharp probabilistic control.

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.

Theorems & Definitions (22)

• Remark 2.1
• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Remark 3.1
• Lemma 3.1
• proof
• Proposition 3.1
• proof
• Lemma 3.2