Central limit theorem for the multilevel Monte Carlo Euler method

TL;DR

This paper establishes a central limit theorem for the multilevel Monte Carlo Euler method, providing theoretical insights into its asymptotic behavior and optimal parameter choices, enhancing understanding of its efficiency.

Contribution

It proves a central limit theorem for the multilevel Monte Carlo Euler method, including a stable law convergence and explicit variance characterization.

Findings

Central limit theorem established for the method

Explicit description of optimal parameters

Characterization of limiting variance

Abstract

This paper focuses on studying the multilevel Monte Carlo method recently introduced by Giles [Oper. Res. 56 (2008) 607-617] which is significantly more efficient than the classical Monte Carlo one. Our aim is to prove a central limit theorem of Lindeberg-Feller type for the multilevel Monte Carlo method associated with the Euler discretization scheme. To do so, we prove first a stable law convergence theorem, in the spirit of Jacod and Protter [Ann. Probab. 26 (1998) 267-307], for the Euler scheme error on two consecutive levels of the algorithm. This leads to an accurate description of the optimal choice of parameters and to an explicit characterization of the limiting variance in the central limit theorem of the algorithm. A complexity of the multilevel Monte Carlo algorithm is carried out.