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) no. 4, pp. 1057-1079.

Dans cet article nous prouvons l’existence et l’unicité d’équations différentielles stochastiques dans ${ℝ}^d$ 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.

In this article we prove the pathwise uniqueness for stochastic differential equations in ${ℝ}^d$ 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.

Classification : 60H10
Mots clés : pathwise uniqueness, symmetric $\alpha $-stable process, Krylov’s estimate, fractional Sobolev space

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