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

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 $\beta$-Hölder 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.

arXiv comment28 pages