Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

A parametrix approach for some degenerate stable driven SDEs

Lorick Huang and Stéphane Menozzi

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We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak Hörmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process.


Pour une équation différentielle stochastique dégénérée dirigée par un processus stable et dont les coefficients vérifient une condition de Hörmander faible, nous établissons sous de faibles hypothèses de régularité l’unicité au problème de martingale sous des contraintes de dimensions. Par ailleurs, lorsque le bruit est scalaire et tempéré, nous obtenons des bornes de densité reflétant le caractère multi-échelle du processus.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 4 (2016), 1925-1975.

Received: 17 February 2014
Revised: 27 July 2015
Accepted: 29 July 2015
First available in Project Euclid: 17 November 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G52: Stable processes
Secondary: 35K65: Degenerate parabolic equations 35R09: Integro-partial differential equations [See also 45Kxx]

Stable driven SDEs Weak Hörmander condition Parametrix Martingale problem


Huang, Lorick; Menozzi, Stéphane. A parametrix approach for some degenerate stable driven SDEs. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 4, 1925--1975. doi:10.1214/15-AIHP704.

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