Stochastic differential equations with Sobolev drifts and driven by α-stable processes
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 d avec terme de dérive dépendant du temps dans un espace de Sobolev et dirigées par un processus de Lévy α-stable symétrique avec α(1,2) et de mesure spectrale non-dégénérée. En particulier, le terme de dérive peut avoir des discontinuités de saut quand α(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 d with time-dependent Sobolev drifts, and driven by symmetric α-stable processes provided that α(1,2) and its spectral measure is non-degenerate. In particular, the drift is allowed to have jump discontinuity when α(2d d+1,2). Our proof is based on some estimates of Krylov’s type for purely discontinuous semimartingales.

DOI : 10.1214/12-AIHP476
Classification : 60H10
Mots clés : pathwise uniqueness, symmetric $\alpha $-stable process, Krylov’s estimate, fractional Sobolev space
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     title = {Stochastic differential equations with {Sobolev} drifts and driven by $\alpha $-stable processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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     publisher = {Gauthier-Villars},
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Zhang, Xicheng. Stochastic differential equations with Sobolev drifts and driven by $\alpha $-stable processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1057-1079. doi : 10.1214/12-AIHP476. http://archive.numdam.org/articles/10.1214/12-AIHP476/

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