A Tamed Euler Scheme for SDEs with Non-Locally Integrable Drift Coefficient

TL;DR

This paper introduces a tamed Euler scheme for stochastic differential equations with non-locally integrable drift, proving strong convergence at rate 1/2, which is novel for such challenging cases.

Contribution

The paper develops the first proven strong convergence Euler-type scheme for SDEs with non-locally integrable drift coefficients.

Findings

Converges in L^p at rate 1/2

Applicable to particles with singular interaction kernels

First proof of strong convergence in this setting

Abstract

In this article we show that for SDEs with a drift coefficient that is non-locally integrable, one may define a tamed Euler scheme that converges in $L^p$ at rate $1/2$ to the true solution. The taming is required in this case since one cannot expect the regular Euler scheme to have finite moments in $L^p$. We additionally show that our setting applies to the case of two scalar valued particles with singular interaction kernel. To the best of the author's knowledge, this is the first work we are aware of to prove strong convergence of an Euler-type scheme in the case of non-locally integrable drift.