On pathwise uniqueness for stochastic differential equations driven by stable L\'evy processes

TL;DR

This paper investigates pathwise uniqueness for one-dimensional stochastic differential equations driven by stable Lévy processes, revealing different conditions for uniqueness depending on the stability index and symmetry of the process.

Contribution

It provides new results on pathwise uniqueness for SDEs driven by stable Lévy processes, including weaker conditions under certain stability indices and symmetry assumptions.

Findings

Pathwise uniqueness holds for $ ext{alpha} ext{ in } (1,2)$ under specific regularity conditions.

For $ ext{alpha} ext{ in } (0,1)$, an equivalent SDE satisfies pathwise uniqueness under weaker assumptions.

Results depend on the stability index and symmetry of the Lévy process.

Abstract

We study a one-dimensional stochastic differential equation driven by a stable L\'evy process of order $\alpha$ with drift and diffusion coefficients $b,\sigma$. When $\alpha\in (1,2)$, we investigate pathwise uniqueness for this equation. When $\alpha\in (0,1)$, we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $\alpha\in (0,1)$ or $\alpha \in (1,2)$ and on whether the driving stable process is symmetric or not. Our assumptions involve the regularity and monotonicity of $b$ and $\sigma$.