Strong rate of convergence for the Euler--Maruyama scheme of SDEs with unbounded H\"older continuous drift coefficient

TL;DR

This paper establishes the strong convergence rate of the Euler--Maruyama scheme for multi-dimensional SDEs with unbounded H"older continuous drift, using advanced stochastic analysis techniques.

Contribution

It introduces a novel approach combining the Itô--Tanaka trick and heat kernel estimates to analyze convergence with unbounded drift.

Findings

Proves strong convergence rate for Euler--Maruyama under unbounded H"older drift

Extends analysis to multi-dimensional SDEs with multiplicative noise

Employs stochastic sewing lemma with heat kernel estimates

Abstract

In this paper, we provide the strong rate of convergence for the Euler--Maruyama scheme for multi-dimensional stochastic differential equations with uniformly locally (unbounded) H\"older continuous drift and multiplicative noise. Our technique is based on It\^o--Tanaka trick (Zvonkin transformation) for unbounded drift. Moreover, in order to apply the stochastic sewing lemma, we use the heat kernel estimate for the density function of the Euler--Maruyama scheme.