On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients

TL;DR

This paper investigates the convergence rates of the Euler-Maruyama method for one-dimensional stochastic differential equations with irregular coefficients, establishing strong convergence rates even with discontinuous or H"older continuous drifts.

Contribution

It provides new strong convergence rate results for Euler-Maruyama applied to SDEs with irregular coefficients, including discontinuous drifts.

Findings

Strong rate of 1/2 for equations with discontinuous drift

Explicit rates for H"older continuous drift with nonconstant diffusion

Applicable to a broad class of irregular coefficient SDEs

Abstract

We study the strong rates of the Euler-Maruyama approximation for one dimensional stochastic differential equations whose drift coefficient may be neither continuous nor one-sided Lipschitz and diffusion coefficient is H\"older continuous. Especially, we show that the strong rate of the Euler-Maruyama approximation is 1/2 for a large class of equations whose drift is not continuous. We also provide the strong rate for equations whose drift is H\"older continuous and diffusion is nonconstant