Non-asymptotic error bounds for The Multilevel Monte Carlo Euler method applied to SDEs with constant diffusion coefficient

TL;DR

This paper establishes non-asymptotic error bounds for the multilevel Monte Carlo Euler method applied to SDEs with constant diffusion, using concentration inequalities and Malliavin calculus.

Contribution

It provides the first non-asymptotic Gaussian-type concentration bounds for the multilevel Monte Carlo Euler estimator under constant diffusion.

Findings

Gaussian concentration inequality holds below a specific deviation threshold

Bounds derived for moment generating functions of scheme differences

Results are optimal in terms of variance

Abstract

In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.