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

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 $L^q([0,T];{\mathcal{C}}_b^{\alpha }({\mathbb{R}}^d))$ with $\alpha \in (0,1)$ and $q\in (2\mathrm{/}(1+\alpha ),\mathrm{\infty }]$. For every $p\ge 2$, by using stochastic sewing and/or the Itô-Tanaka trick, we obtain the $L^p$-convergence rates: $(1+\alpha )\mathrm{/}2$ for $q\in [2,\mathrm{\infty }]$ and $(1-1\mathrm{/}q)$ for $q\in (2\mathrm{/}(1+\alpha ),2)$. Moreover, we prove that the unique strong solution can be constructed via the Picard iteration.