A tamed Euler scheme for SDEs with non-locally integrable drift coefficient

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 $L^p$. Our proof strategy involves controlling the inverse moments of the distance of scheme and the true solution to the singularity set. We additionally show that our setting applies to the case of two scalar valued particles with singular interaction kernel. To the best of the authors’ knowledge, this is the first work to prove strong convergence of an Euler-type scheme in the case of non-locally integrable drift.