$L^p$-sup Convergence of the Euler-Maruyama Scheme for SDEs with Distributional Besov Drift

TL;DR

This paper establishes convergence rates for the Euler-Maruyama scheme applied to one-dimensional SDEs with distributional Besov space drifts, extending previous results to broader function spaces and providing explicit $L^1$-$ ext{sup}$ norm rates.

Contribution

It extends convergence results of the Euler-Maruyama scheme to SDEs with drifts in negative order Besov spaces, using the Yamada-Watanabe technique and deriving explicit $L^1$-$ ext{sup}$ norm rates.

Findings

Proves $L^p$ convergence rates for all $p \\geq 2$.

Derives explicit $L^1$-$\\sup$ norm convergence rate.

Extends analysis to SDEs with distributional Besov drift.

Abstract

In this paper we extend existing results on the numerical approximation of one-dimensional SDEs with drift in a negative order Besov space and driven by Brownian motion. Using the Yamada-Watanabe approximation technique, we prove rates in $L^p$, for all $p\geq 2$, applying a Gronwall-type lemma previously used in the literature for SDEs with H\"older continuous coefficients. Additionally, we obtain an explicit convergence rate in the $L^1$-$\sup$ norm.