Central Limit Theorem for Irregular Discretization Scheme of Multilevel Monte Carlo Method

TL;DR

This paper establishes a central limit theorem for the error distribution in an irregular discretization scheme of stochastic differential equations, extending to multilevel Monte Carlo methods, with implications for understanding error behavior in numerical SDE solutions.

Contribution

It introduces a CLT for irregular discretization errors in SDE solutions and applies it to multilevel Monte Carlo schemes, advancing theoretical understanding of error distributions.

Findings

Proves a CLT with rate √n for irregular discretization errors.

Derives a CLT of Lindeberg-Feller type for multilevel Monte Carlo schemes.

Provides theoretical foundation for error analysis in irregular SDE discretizations.

Abstract

In this paper, we study the asymptotic error distribution for a two-level irregular discretization scheme of the solution to the stochastic differential equations (SDE for short) driven by a continuous semimartingale and obtain a central limit theorem for the error processes with the rate $\sqrt{n}$. As an application, in the spirit of the result of Ben Alaya and Kebaier, we get a central limit theorem of the Linderberg-Feller type for the irregular discretization scheme of the multilevel Monte Carlo method.