Weak error for SDEs with additive stable noise and singular drift: choose the test function in the same space as the drift!

TL;DR

This paper studies weak error estimates for SDEs with additive stable noise and singular drifts, highlighting the importance of selecting test functions with matching regularity to the drift to improve convergence rates.

Contribution

It introduces a framework where choosing test functions with the same regularity as the drift enhances weak error analysis for SDEs with singular drifts.

Findings

Improved convergence rates for densities with Lebesgue or H{"o}lder drifts.

Preservation of convergence rates for singular generalized test functions.

Emphasizes the importance of matching test function regularity to the drift.

Abstract

We emphasize that for a stochastic differential equation with isotropic stable additive noise and non Lipschitz drift, when considering an appropriate discretization scheme and the associated weak error, it is somehow natural to consider a test function having the same spatial regularity as the drift involved. We will in particular focus on drifts belonging to Lebsegue, H{\"o}lder or Besov spaces with negative regularity index in their spatial variable. Choosing such a test function allows to improve the convergence rate previously obtained on the densities (for Lebesgue or H{\"o}lder drifts) or preserve the rate for possibly singular generalized test functions (for Besov spaces with negative regularity).