Strong Rate of Convergence for the Euler-Maruyama Approximation of Stochastic Differential Equations with Irregular Coefficients

TL;DR

This paper establishes the strong convergence rate of the Euler-Maruyama method for multi-dimensional stochastic differential equations with irregular coefficients, including discontinuous drifts and H"older continuous diffusions.

Contribution

It provides the first explicit strong convergence rate for Euler-Maruyama applied to SDEs with irregular, discontinuous drift coefficients and H"older continuous diffusion coefficients.

Findings

Established strong convergence rate for Euler-Maruyama with irregular coefficients

Proved convergence under one-sided Lipschitz and H"older conditions

Applicable to multi-dimensional SDEs with discontinuous drifts

Abstract

We consider the Euler-Maruyama approximation for multi-dimensional stochastic differential equations with irregular coefficients. We provide the rate of strong convergence where the possibly discontinuous drift coefficient satisfies a one-sided Lipschitz condition and the diffusion coefficient is H\"older continuous.