Strong convergence of the Euler scheme for singular kinetic SDEs driven by $\alpha$-stable processes

TL;DR

This paper proves the strong convergence of an Euler scheme for singular kinetic SDEs driven by α-stable processes, establishing a specific convergence rate under certain regularity conditions.

Contribution

It extends the analysis of Euler schemes to second-order kinetic SDEs with α-stable noise and anisotropic Hölder continuous drifts, providing explicit convergence rates.

Findings

Convergence rate of the Euler scheme is established.

Results generalize previous first-order SDE findings.

Applicable for α in (1,2) with specific drift regularity.

Abstract

We study the strong approximation of the solutions to singular stochastic kinetic equations (also referred to as second-order SDEs) driven by $\alpha$-stable processes, using an Euler-type scheme inspired by [11]. For these equations, the stability index $\alpha$ lies in the range $(1,2)$, and the drift term exhibits anisotropic $\beta$-H\"older continuity with $\beta >1 - \frac{\alpha}{2}$. We establish a convergence rate of $(\frac{1}{2} + \frac{\beta}{\alpha(1+\alpha)} \wedge \frac{1}{2})$, which aligns with the results in [4] concerning first-order SDEs.