Weak Error on the densities for the Euler scheme of stable additive SDEs with Besov drift

TL;DR

This paper investigates the weak error convergence rates of Euler-Maruyama discretizations for stable additive SDEs with distributional Besov drift, providing new insights into density approximation accuracy.

Contribution

It establishes weak error convergence rates for Euler schemes applied to SDEs driven by stable processes with Besov space drifts, extending existing theory to more irregular coefficients.

Findings

Derived convergence rates depending on parameters

Extended weak error analysis to stable processes with Besov drifts

Provided conditions for weak well-posedness

Abstract

We are interested in the Euler-Maruyama dicretization of the formal SDE, $dX_t=b(t,X_t)dt+dZ_t$, where $Z$ is a symmetric isotropic d dimensional stable process of index $\alpha\in (1,2)$, and $b$ is distributional. It belongs to a mix Lebesgue-Besov space. The associated parameters satisfy some constraints which guarantee weak-well posedness. Defining an appropriate Euler scheme, we obtain a convergence rate for the weak error on the densities. The rate depends on the parameters.