Strong existence and uniqueness for stable stochastic differential equations with distributional drift

TL;DR

This paper proves strong existence and uniqueness of solutions for a class of stochastic differential equations driven by symmetric stable Lévy processes with distributional drifts in certain Besov–Hölder spaces.

Contribution

It establishes the first strong existence and uniqueness results for SDEs with distributional drifts driven by symmetric stable Lévy processes.

Findings

Strong solutions exist and are unique for drifts in rac{1}{2}-rac{\u03b1}{2} < eta.

The results extend classical SDE theory to distributional drifts.

The paper introduces a solution concept suitable for distributional drifts.

Abstract

We consider the stochastic differential equation $$ dX_t = b(X_t) dt + dL_t,$$ where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $\alpha$-stable L\'evy processes, $\alpha \in (1, 2)$. We define the notion of solution to this equation and establish strong existence and uniqueness whenever $b$ belongs to the Besov--H\"{o}lder space $\mathcal{C}^\beta$ for $\beta >1/2-\alpha/2$.