Ordinary differential equations/Probability theory
Backward doubly stochastic differential equations with a superlinear growth generator
Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 25-30.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.10.008
Bahlali, Khaled 1, 2; Gatt, Rafika 3; Mansouri, Badreddine 3

1 Université de Toulon, IMATH, EA 2134, 83957 La Garde cedex, France
2 CNRS, I2M, Aix–Marseille Université, France
3 Université de Biskra, Département de mathématiques, BP 145, Biskra, Algeria
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Bahlali, Khaled; Gatt, Rafika; Mansouri, Badreddine. Backward doubly stochastic differential equations with a superlinear growth generator. Comptes Rendus. Mathématique, Volume 353 (2015) no. 1, pp. 25-30. doi : 10.1016/j.crma.2014.10.008. http://www.numdam.org/articles/10.1016/j.crma.2014.10.008/

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