Randomised Euler-Maruyama Method for SDEs with Hölder Continuous Drift Coefficient Driven by $\hat{I}\pm$-stable Lévy Process

Abstract

In this paper, we examine the performance of randomised Euler-Maruyama (EM) method for additive time-inhomogeneous SDEs with an irregular drift driven by symmetric $\hat{I}\pm$-table process, $\hat{I}\pm \in (1,2)$. In particular, the drift is assumed to be ${\hat{I}}^2$-Hölder continuous in time and bounded $\hat{I}\cdot$-Hölder continuous in space with ${\hat{I}}^2,\hat{I}\cdot \in (0,1]$. The strong order of convergence of the randomised EM in $L^p$-norm is shown to be $1/2+({\hat{I}}^2\wedge (\hat{I}\cdot /\hat{I}\pm )\wedge (1/2))-\epsilon$ for an arbitrary $\epsilon \in (0,1/2)$, higher than the one of standard EM, which cannot exceed ${\hat{I}}^2$. The result for the case of $\hat{I}\pm \in (1,2)$ extends the almost optimal order of convergence of randomised EM obtained in (arXiv:2501.15527) for SDEs driven by Gaussian noise ($\hat{I}\pm =2$), and coincides with the performance of EM method in simulating time-homogenous SDEs driven by $\hat{I}\pm$-stable process considered in (arXiv:2208.10052). Various experiments are presented to validate the theoretical performance.