On the performance of the Euler-Maruyama scheme for SDEs with discontinuous drift coefficient

TL;DR

This paper demonstrates that the Euler-Maruyama scheme achieves an $L_p$-error rate of at least 1/2 for scalar SDEs with discontinuous drift and Lipschitz diffusion, matching the rate for Lipschitz coefficients.

Contribution

It proves that the Euler-Maruyama scheme attains a 1/2 error rate for SDEs with discontinuous drift, extending known results to less regular coefficients.

Findings

Euler-Maruyama achieves 1/2 $L_p$-error rate for discontinuous drift SDEs.

Error rate matches that of Lipschitz coefficient SDEs.

Results hold for all $p \\in [1,\\infty)$.

Abstract

Recently a lot of effort has been invested to analyze the $L_p$-error of the Euler-Maruyama scheme in the case of stochastic differential equations (SDEs) with a drift coefficient that may have discontinuities in space. For scalar SDEs with a piecewise Lipschitz drift coefficient and a Lipschitz diffusion coefficient that is non-zero at the discontinuity points of the drift coefficient so far only an $L_p$-error rate of at least $1/(2p)-$ has been proven. In the present paper we show that under the latter conditions on the coefficients of the SDE the Euler-Maruyama scheme in fact achieves an $L_p$-error rate of at least $1/2$ for all $p\in [1,\infty)$ as in the case of SDEs with Lipschitz coefficients.