Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes

Zhang, Xicheng

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013), p. 1057-1079 / Harvested from Numdam

Résumé

Dans cet article nous prouvons l’existence et l’unicité d’équations différentielles stochastiques dans $\mathbb {R}^{d}$ avec terme de dérive dépendant du temps dans un espace de Sobolev et dirigées par un processus de Lévy $\alpha $-stable symétrique avec $\alpha \in (1,2)$ et de mesure spectrale non-dégénérée. En particulier, le terme de dérive peut avoir des discontinuités de saut quand $\alpha \in (\frac{2d}{d+1},2)$. Notre preuve est basée sur des estimations de type Krylov pour des semimartingales purement discontinues.

Abstract

In this article we prove the pathwise uniqueness for stochastic differential equations in $\mathbb {R}^{d}$ with time-dependent Sobolev drifts, and driven by symmetric $\alpha $-stable processes provided that $\alpha \in (1,2)$ and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when $\alpha \in (\frac{2d}{d+1},2)$. Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

Détail

Publié le : 2013-01-01

MR 3127913 | Zbl 1279.60074 | 1 citation dans Numdam

DOI : https://doi.org/10.1214/12-AIHP476
Classification:  60H10

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