Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs

K. Bahlali; E. Essaky; M. Hassani

doi:10.1016/j.crma.2010.03.006

Probability Theory

Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs
[EDSR multidimensionnelles à croissance surlinéaire : Application aux systèmes d'EDP dégénérées]

Présenté par : Paul Malliavin

K. Bahlali ¹ ; E. Essaky ² ; M. Hassani ²

¹ IMATH, UFR Sciences, UTV, B.P. 132, 83957 La Garde cedex, France
² Université Cadi Ayyad, Laboratoire de Statistique des Processus, Marrakech, Maroc

Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 677-682.

Abstract (VO)
Résumé

We establish the existence and uniqueness as well as the stability of p-integrable solutions to multidimensional backward stochastic differential equations (BSDEs) with super-linear growth coefficient and a p-integrable terminal condition $(p > 1)$ . The generator could neither be locally monotone in the variable y nor locally Lipschitz in the variable z. As application, we establish the existence and uniqueness of weak (Sobolev) solutions to the associated systems of semilinear parabolic PDEs. The uniform ellipticity of the diffusion matrix is not required. Our result covers, for instance, certain systems of PDEs with logarithmic nonlinearities which arise in physics.

Nous établissons l'existence, l'unicité et la stability des solutions fortes pour des équations différentielles stochastiques rétrogrades (EDSR) avec une condition terminale p-integrable $(p > 1)$ et un coefficient admettant des croissances surlinéaires en les deux variables y et z. De plus, ce dernier peut être ni locallement monotone en y ni localement Lipschitz en z. Nous montrons également l'existence et l'unicité des solutions faibles pour les systèmes d'EDP associés. L'uniforme ellipticité n'est pas requise pour la matrice de diffusion.

Reçu le : 2009-08-23
Accepté le : 2010-02-20
Publié le : 2010-05-10

DOI : 10.1016/j.crma.2010.03.006

Affiliations des auteurs :

K. Bahlali ¹ ; E. Essaky ² ; M. Hassani ²

¹ IMATH, UFR Sciences, UTV, B.P. 132, 83957 La Garde cedex, France
² Université Cadi Ayyad, Laboratoire de Statistique des Processus, Marrakech, Maroc

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K. Bahlali; E. Essaky; M. Hassani. Multidimensional BSDEs with super-linear growth coefficient: Application to degenerate systems of semilinear PDEs. Comptes Rendus. Mathématique, Volume 348 (2010) no. 11-12, pp. 677-682. doi : 10.1016/j.crma.2010.03.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.006/

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