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Comptes Rendus Mathématique
Volume 353, n° 1
pages 25-30 (janvier 2015)
Doi : 10.1016/j.crma.2014.10.008
Received : 22 May 2014 ;  accepted : 9 October 2014
Backward doubly stochastic differential equations with a superlinear growth generator
Équations différentielles doublement stochastiques avec un coefficient à croissance surlinéaire
 

Khaled Bahlali a, b , Rafika Gatt c , Badreddine Mansouri c
a Université de Toulon, IMATH, EA 2134, 83957 La Garde cedex, France 
b CNRS, I2M, Aix–Marseille Université, France1  
c Université de Biskra, Département de mathématiques, BP 145, Biskra, Algeria 

Abstract

We deal with backward doubly stochastic differential equations (BDSDEs) with a superlinear growth generator and a square integrable terminal datum. We introduce a new local condition on the generator, then we show that it ensures the existence and uniqueness as well as the stability of solutions. Our work goes beyond the previous results on the multidimensional BDSDEs. Although we are focused on the multidimensional case, our uniqueness result is new for one-dimensional BDSDEs, too. As an application, we establish the existence of a Sobolev solution to SPDEs with superlinear growth generator. Some illustrative examples are also presented.

The full text of this article is available in PDF format.
Résumé

Nous considérons des équations différentielles doublement stochastiques rétrogrades (EDDSR) avec un générateur de croissance surlinéaire et une donnée terminale de carré intégrable. Nous introduisons une nouvelle condition locale sur le générateur et nous montrons qu'elle assure l'existence, l'unicité et la stabilité des solutions. Même si notre intérêt porte sur le cas multidimensionnel, notre résultat est également nouveau en dimension un. Des exemples illustratifs et une application aux équations aux dérivées partielles stochastiques (EDPS) sont également donnés.

The full text of this article is available in PDF format.

 Partially supported by PHC Tassili 13MDU887 and MODTERCOM project, APEX Programme, region Provence-Alpe-Cote d'Azur.

1  (2013/2014).


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