Well-posedness of density dependent SDE driven by $\alpha$-stable process with H\"{o}lder drifts

TL;DR

This paper establishes the well-posedness of density-dependent SDEs driven by $ ext{alpha}$-stable processes, using Euler approximation and Schauder estimates, under minimal regularity assumptions on the drift.

Contribution

It proves weak and strong well-posedness of such SDEs with new techniques, requiring only continuity in density for existence and H"older regularity for strong uniqueness.

Findings

Existence of solutions via Euler approximation.

Weak uniqueness under Lipschitz condition in density.

Strong uniqueness with H"older continuity in space.

Abstract

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by $\alpha$-stable processes with $\alpha \in(1,2)$. The existence part is based on Euler's approximation as \cite{HRZ20}, while, the uniqueness is based on the Schauder estimates in Besov spaces for nonlocal Fokker-Planck equations. For the existence, we only assume the drift being continuous in the density variable. For the weak uniqueness, the drift is assumed to be Lipschitz in the density variable, while for the strong uniqueness, we also need to assume the drift being $\beta_0$-order H\"older continuous in the spatial variable, where $\beta_0\in(1-\alpha/2,1)$.