Hitting properties and non-uniqueness for SDE driven by stable processes

TL;DR

This paper investigates the hitting properties and non-uniqueness phenomena in self-similar jump SDEs driven by stable processes, providing conditions for extinction and pathwise uniqueness under specific constraints.

Contribution

It introduces a necessary and sufficient condition for finite-time extinction and explores pathwise uniqueness in a restricted sense for certain parameter regimes.

Findings

Condition for almost sure extinction derived

Pathwise uniqueness holds among solutions spending negligible time at zero

Power transformation is crucial for analysis

Abstract

We study a class of self-similar jump type SDEs driven by H\"older-continuous drift and noise coefficients. Using the Lamperti transformation for positive self-similar Markov processes we obtain a necessary and sufficient condition for almost sure extinction in finite time. We then show that for certain parameters pathwise uniqueness holds in a restricted sense, namely among solutions spending a Lebesgue-negligible amount of time at 0. A direct power transformation plays a key role.