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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Abstract

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.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1479373254

Digital Object Identifier
doi:10.1214/15-AIHP704

Mathematical Reviews number (MathSciNet)
MR3573301

Zentralblatt MATH identifier
1355.60076

Subjects
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]

Keywords
Stable driven SDEs Weak Hörmander condition Parametrix Martingale problem

Citation

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. https://projecteuclid.org/euclid.aihp/1479373254


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