Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Invariant measures and a stability theorem for locally Lipschitz stochastic delay equations

I. Stojkovic and O. van Gaans

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Abstract

We consider a stochastic delay differential equation with exponentially stable drift and diffusion driven by a general Lévy process. The diffusion coefficient is assumed to be locally Lipschitz and bounded. Under a mild condition on the large jumps of the Lévy process, we show existence of an invariant measure. Main tools in our proof are a variation-of-constants formula and a stability theorem in our context, which are of independent interest.

Résumé

Nous considérons une equation différentielle stochastique retardée à dérive exponentiellement stable et conduite par un processus de Lévy général. Le coefficient de la diffusion est seulement supposé satisfaire une condition lipschitzienne locale et être borné. En supposant une condition additionnelle faible sur les grands sauts du processus de Lévy, nous démontrons l’existence d’une mesure invariante. Les principaux ingrédients de la preuve sont une formule pour les variations des constantes et un théorème de stabilité par rapport aux perturbations des conditions initiales, qui sont d’un intérêt indépendant.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 4 (2011), 1121-1146.

Dates
First available in Project Euclid: 6 October 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1317906504

Digital Object Identifier
doi:10.1214/10-AIHP396

Mathematical Reviews number (MathSciNet)
MR2884227

Zentralblatt MATH identifier
1278.34093

Subjects
Primary: 34K50: Stochastic functional-differential equations [See also , 60Hxx] 60G48: Generalizations of martingales

Keywords
Delay equation Invariant measure Lévy process Semimartingale Skorohod space Stability Tightness Variation-of-constants formula

Citation

Stojkovic, I.; van Gaans, O. Invariant measures and a stability theorem for locally Lipschitz stochastic delay equations. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 4, 1121--1146. doi:10.1214/10-AIHP396. https://projecteuclid.org/euclid.aihp/1317906504


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