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

2025/10/03 by Peng Chen, Hui Jiang, Chen, Peng +3
Economics, Econometrics and Finance · Mathematics · Business, Management and Accounting · #Stochastic processes and financial applications #Mathematical Approximation and Integration #Advanced Queuing Theory Analysis

paper · pdf · doi:10.48550/arxiv.2510.03053

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.

Citations

• Approximation of the ergodic measure of SDEs with singular drift by Euler-Maruyama scheme
• The backward Euler-Maruyama method for invariant measures of stochastic differential equations with super-linear coefficients
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• Stationary distribution of the stochastic theta method for nonlinear stochastic differential equations
• Multivariate approximations in Wasserstein distance by Stein's method and Bismut's formula
• Approximation of Invariant Measures for Regime-Switching Diffusions

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• Optimal convergence rate of modified Milstein scheme for SDEs with rough fractional diffusions
• On Milstein approximations with varying coefficients: the case of super-linear diffusion coefficients