Randomised Euler-Maruyama method for SDEs with H\"older continuous drift coefficient

TL;DR

This paper analyzes the randomized Euler-Maruyama method's effectiveness for stochastic differential equations with irregular, Hölder continuous drift, demonstrating improved convergence rates over standard methods using advanced stochastic analysis techniques.

Contribution

It establishes a higher strong convergence order for the randomized EM method applied to SDEs with Hölder continuous drift, utilizing the stochastic sewing lemma for proof.

Findings

Randomized EM achieves convergence order of 1/2 + (α ∧ (β/2)) - ε.

Standard EM has lower convergence order under the same conditions.

Alternative proof provided using stochastic sewing lemma for handling time irregularity.

Abstract

In this paper, we examine the performance of randomised Euler-Maruyama (EM) method for additive time-inhomogeneous SDEs with an irregular drift. In particular, the drift is assumed to be $\alpha$-H\"older continuous in time and bounded $\beta$-H\"older continuous in space with $\alpha,\beta\in (0,1]$. The strong order of convergence of the randomised EM in $L^p$-norm is shown to be $1/2+(\alpha \wedge (\beta/2))-\epsilon$ for an arbitrary $\epsilon\in (0,1/2)$, higher than the one of standard EM, which is $\alpha \wedge (1/2+\beta/2-\epsilon)$. The proofs highly rely on the stochastic sewing lemma, where we also provide an alternative proof when handling time irregularity for a comparison.