Weak well-posedness and weak discretization error for stable-driven SDEs with Lebesgue drift

TL;DR

This paper investigates the weak well-posedness and discretization errors of stable-driven SDEs with Lebesgue drift, providing convergence rates for Euler scheme approximations under specific conditions.

Contribution

It establishes weak existence, uniqueness, and heat kernel estimates for these SDEs, and derives a convergence rate for the Euler scheme with cutoffed and randomized drifts.

Findings

Proves weak well-posedness under certain conditions.

Provides heat kernel estimates for the SDEs.

Derives explicit convergence rate for Euler scheme approximation.

Abstract

We are interested in the discretization of stable driven SDEs with additive noise for $\alpha$ $\in$ (1, 2) and Lq -- Lp drift under the Serrin type condition $\alpha$/q + d/p < $\alpha$ -- 1. We show weak existence and uniqueness as well as heat kernel estimates for the SDE and obtain a convergence rate of order (1/$\alpha$)*($\alpha$ -- 1 -- $\alpha$/q - d/p) for the difference of the densities for the Euler scheme approximation involving suitably cutoffed and time randomized drifts.