The Euler-Maruyama method for SDEs with low-regularity drift

Abstract

We study the strong $L^p$-convergence rates of the Euler-Maruyama method for stochastic differential equations driven by Brownian motion with low-regularity drift coefficients. Specifically, the drift is assumed to be in the Lebesgue-Hölder spaces $L^q([0,T];{\mathcal{C}}_b^{\hat{I}}\pm ({\mathbb{R}}^d))$ with $\hat{I}\pm \in (0,1)$ and $q\in (2/(1+\hat{I}\pm ),\infty ]$. For every $p\ge 2$, by using stochastic sewing and/or the Itô-Tanaka trick, we obtain the $L^p$-convergence rates: $(1+\hat{I}\pm )/2$ for $q\in [2,\infty ]$ and $(1-1/q)$ for $q\in (2/(1+\hat{I}\pm ),2)$. Moreover, we prove that the unique strong solution can be constructed via the Picard iteration.