Randomised Euler-Maruyama method for SDEs with Hölder continuous drift coefficient

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 $\hat{I}\pm$-Hölder continuous in time and bounded ${\hat{I}}^2$-Hölder continuous in space with $\hat{I}\pm ,{\hat{I}}^2\in (0,1]$. The strong order of convergence of the randomised EM in $L^p$-norm is shown to be $1/2+(\hat{I}\pm \wedge ({\hat{I}}^2/2))-\hat{I}\mu$ for an arbitrary $\hat{I}\mu \in (0,1/2)$, higher than the one of standard EM, which is $\hat{I}\pm \wedge (1/2+{\hat{I}}^2/2-\hat{I}\mu )$. The proofs highly rely on the stochastic sewing lemma, where we also provide an alternative proof when handling time irregularity for a comparison.