On some Non Asymptotic Bounds for the Euler Scheme

TL;DR

This paper derives non-asymptotic bounds for Monte Carlo algorithms based on Euler discretization of diffusion processes, utilizing Gaussian concentration properties of the scheme's density to establish sharp bounds.

Contribution

It introduces a novel approach using Gaussian concentration and bounds to analyze the Euler scheme's density, providing sharp non-asymptotic bounds.

Findings

Established Gaussian concentration for the scheme's density.

Derived Gaussian upper and lower bounds for the density.

Demonstrated the sharpness of the concentration bounds.

Abstract

We obtain non asymptotic bounds for the Monte Carlo algorithm associated to the Euler discretization of some diffusion processes. The key tool is the Gaussian concentration satisfied by the density of the discretization scheme. This Gaussian concentration is derived from a Gaussian upper bound of the density of the scheme and a modification of the so-called "Herbst argument" used to prove Logarithmic Sobolev inequalities. We eventually establish a Gaussian lower bound for the density of the scheme that emphasizes the concentration is sharp.