Randomised Euler-Maruyama Method for SDEs with H\"older Continuous Drift Coefficient Driven by $\alpha$-stable L\'evy Process

TL;DR

This paper analyzes the randomized Euler-Maruyama method's convergence for SDEs with irregular drift driven by symmetric alpha-stable Lévy processes, extending previous results to a broader class of stochastic differential equations.

Contribution

It establishes the strong convergence order of the randomized EM method for SDEs with H"older continuous drift driven by alpha-stable Lévy processes, extending prior Gaussian noise results.

Findings

Convergence order exceeds standard EM for irregular drifts.

Validation through numerical experiments confirms theoretical results.

Extension of convergence analysis to alpha-stable Lévy driven SDEs.

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 $\alpha$-table process, $\alpha\in (1,2)$. In particular, the drift is assumed to be $\beta$-H\"older continuous in time and bounded $\eta$-H\"older continuous in space with $\beta,\eta\in (0,1]$. The strong order of convergence of the randomised EM in $L^p$-norm is shown to be $1/2+(\beta \wedge (\eta/\alpha)\wedge(1/2))-\varepsilon$ for an arbitrary $\varepsilon\in (0,1/2)$, higher than the one of standard EM, which cannot exceed $\beta$. The result for the case of $\alpha \in (1,2)$ extends the almost optimal order of convergence of randomised EM obtained in (arXiv:2501.15527) for SDEs driven by Gaussian noise ($\alpha=2$), and coincides with the performance of EM method in simulating time-homogenous SDEs driven by $\alpha$-stable process considered in (arXiv:2208.10052). Various experiments are presented to validate the theoretical performance.