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

TL;DR

This paper analyzes the convergence rates of the Euler-Maruyama method for SDEs with low-regularity drift, establishing new rates under minimal regularity assumptions using advanced stochastic techniques.

Contribution

It provides the first detailed $L^p$-convergence rates for Euler-Maruyama applied to SDEs with irregular drift coefficients in Lebesgue-Hölder spaces.

Findings

Convergence rate of (1+α)/2 for q ≥ 2.

Convergence rate of (1 - 1/q) for q in (2/(1+α), 2).

Strong solutions can be constructed via Picard iteration.

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