Invariant measures and a stability theorem for locally Lipschitz stochastic delay equations
Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1121-1146.

Nous considérons une équation 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.

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.

DOI : 10.1214/10-AIHP396
Classification : 34K50, 60G48
Mots-clés : Delay equation, invariant measure, Lévy process, semimartingale, Skorohod space, stability, tightness, variation-of-constants formula 
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     title = {Invariant measures and a stability theorem for locally {Lipschitz} stochastic delay equations},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {1121--1146},
     publisher = {Gauthier-Villars},
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Stojkovic, I.; van Gaans, O. Invariant measures and a stability theorem for locally Lipschitz stochastic delay equations. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1121-1146. doi : 10.1214/10-AIHP396. http://archive.numdam.org/articles/10.1214/10-AIHP396/

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