Convergence in total variation of the Euler-Maruyama scheme applied to diffusion processes with measurable drift coefficient and additive noise

TL;DR

This paper establishes the convergence rates of the Euler-Maruyama scheme for diffusion processes with measurable drift and additive noise, showing improved convergence under certain divergence conditions, supported by numerical validation.

Contribution

It proves weak convergence in total variation for the Euler-Maruyama scheme with minimal regularity assumptions on the drift, including divergence conditions that enhance convergence order.

Findings

Weak convergence order 1/2 in total variation for measurable drift.

Improved convergence order 1 under divergence conditions on the drift.

Numerical experiments confirm theoretical convergence rates.

Abstract

We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension $d$ with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order $1/2$ in total variation distance. When the drift has a spatial divergence in the sense of distributions with $\rho$-th power integrable with respect to the Lebesgue measure in space uniformly in time for some $\rho \ge d$, the order of convergence at the terminal time improves to $1$ up to some logarithmic factor. In dimension $d=1$, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.