Weak error on the densities for the Euler scheme of stable additive SDEs with H{\"o}lder drift

TL;DR

This paper investigates the weak error convergence of the Euler scheme applied to stable additive SDEs with Hölder continuous drift, showing a rate depending on the stability index and drift regularity.

Contribution

It introduces a randomized Euler scheme for stable SDEs and establishes the convergence rate of the weak error on densities.

Findings

Weak error on densities converges at rate (α + β - 1)/α.

Randomized Euler scheme effectively approximates densities of stable SDEs.

Convergence rate depends on stability index and drift regularity.

Abstract

We are interested in the Euler-Maruyama dicretization of the SDE dXt =b(t,Xt)dt+ dZt, X0 =x$\in$Rd, where Zt is a symmetric isotropic d-dimensional $\alpha$-stable process, $\alpha$ $\in$ (1, 2] and the drift b $\in$ L$\infty$ ([0,T],C$\beta$(Rd,Rd)), $\beta$ $\in$ (0,1), is bounded and H{\"o}lder regular in space. Using an Euler scheme with a randomization of the time variable, we show that, denoting $\gamma$\,:= $\alpha$ + $\beta$ -- 1, the weak error on densities related to this discretization converges at the rate $\gamma$/$\alpha$.