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

J. Berestycki (1) , L. Döring , L. Mytnik , L. Zambotti (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We study a class of self-similar jump type SDEs driven by Holder 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 in some cases 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

Domaines: Probabilités [math.PR]

J. Berestycki, L. Döring, L. Mytnik, L. Zambotti. Hitting properties and non-uniqueness for SDEs driven by stable processes. Stochastic Processes and their Applications, 2015, 125 (3), pp.918-940. ⟨10.1016/j.spa.2014.10.012⟩. ⟨hal-01164571⟩