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

Abstract

The strong rate of convergence of the Euler-Maruyama scheme for nondegenerate SDEs with irregular drift coefficients is considered. In the case of $\alpha$-Hölder 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.

arXiv commentIn version 2, we have dropped the $L_1$-condition that was imposed on the drift in the $1$-dimensional case and the result now is stated for bounded measurable drift

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• The Euler-Maruyama Scheme for SDEs with Irregular Drift: Convergence Rates via Reduction to a Quadrature Problem
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• A numerical scheme for stochastic differential equations with distributional drift
• Taming singular stochastic differential equations: A numerical method
• An averaged space-time discretization of the stochastic $p$-Laplace system
• Sharp lower error bounds for strong approximation of SDEs with piecewise Lipschitz continuous drift coefficient
• Stochastic differential equations with irregular coefficients:~mind the gap!
• Error estimates of the backward Euler-Maruyama method for multi-valued stochastic differential equations
• Bicausal optimal transport for SDEs with irregular coefficients
• Existence of strong solutions for Itô's stochastic equations via approximations. Revisited