On the regularisation of the noise for the Euler-Maruyama scheme with irregular drift

TL;DR

This paper demonstrates that the Euler-Maruyama scheme for SDEs with irregular drift coefficients converges at a rate close to 1/2 by leveraging the noise's regularising effect, extending previous results to broader classes of coefficients.

Contribution

The paper improves the convergence rate analysis of Euler-Maruyama for irregular drifts, showing near 1/2 rate for all positive Hölder exponents and extending to Dini continuous and bounded measurable coefficients.

Findings

Convergence rate is arbitrarily close to 1/2 for all α>0.

Extension of results to Dini continuous coefficients.

Applicable to bounded measurable coefficients in one dimension.

Abstract

The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $\alpha$-H\"older drift in the recent literature the rate $\alpha/2$ was proved in many related situations. By exploiting the regularising effect of the noise more efficiently, we show that the rate is in fact arbitrarily close to $1/2$ for all $\alpha>0$. The result extends to Dini continuous coefficients, while in $d=1$ also to all bounded measurable coefficients.