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

2011/11/18 by Julien Berestycki, Berestycki, Julien, Leif Doering +5 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60J80 (Primary) 60G18 (Secondary) #Economic theories and models #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1111.4388

openalex publication_date 2011/11/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study a class of self-similar jump type SDEs driven by Hölder-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.

Citations

• A Jump Type SDE Approach to Positive Self-Similar Markov Processes

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