Cram\'er-type moderate deviations for Euler-Maruyama scheme for SDE

TL;DR

This paper develops Cramér-type moderate deviation results for the Euler-Maruyama scheme applied to stochastic differential equations, providing refined probabilistic bounds and principles that improve upon recent CLT-related work.

Contribution

It introduces normalized and self-normalized Cramér-type moderate deviations for the Euler-Maruyama scheme, enhancing existing theoretical results with refined bounds and principles.

Findings

Established normalized Cramér-type moderate deviations for Euler-Maruyama scheme

Derived Berry-Esseen bounds and moderate deviation principles

Refined recent work on CLT and deviations for SDE discretizations

Abstract

In this paper, we establish normalized and self-normalized Cram\'er-type moderate deviations for Euler-Maruyama scheme for SDE. As a consequence of our results, Berry-Esseen's bounds and moderate deviation principles are also obtained. Our normalized Cram\'er-type moderate deviations refines the recent work of [Lu, J., Tan, Y., Xu, L., 2022. Central limit theorem and self-normalized Cram\'er-type moderate deviation for Euler-Maruyama scheme. Bernoulli 28(2): 937--964].