Comptes Rendus
no. 1
Optimal control/Probability theory
Existence of an optimal control for a system driven by a degenerate coupled forward–backward stochastic differential equations
[Existence d'un contrôle optimal pour des équations différentielles stochastiques progressives–rétrogrades couplées dégénérées]

Présenté par : the Editorial Board

Khaled Bahlali1 ; Omar Kebiri2 ; Ahmed Mtiraoui1, 3
1 Université de Toulon, IMATH, EA 2134, 83957 La Garde cedex, France
2 Université de Tlemcen, Laboratoire de probabilités et statistiques, Algérie
3 Faculté des sciences de Sfax, Département de mathématiques, BP 1171, 3000 Sfax, Tunisia
Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 84-89.

Nous établissons l'existence d'un contrôle optimal, pour un système modélisé par une équation différentielle stochastique progressive–rétrograde (EDSPR) couplée, dont le coefficient de diffusion peut dégénérer (i.e. est non nécessairement uniformément elliptique). Par une double régularisation, nous construisons une suite de contrôles optimaux markoviens. Nous passons ensuite à la limite pour établir l'existence d'un contrôle optimal markovien. L'hypothèse de convexité de Filippov est utilisée pour montrer que le contrôle optimal ainsi construit est strict. Le résultat étend en un sens ceux obtenus dans [2,4] aux systèmes d'EDS–EDSR couplés.

We establish the existence of an optimal control for a system driven by a coupled forward–backward stochastic differential equation (FBDSE) whose diffusion coefficient may degenerate (i.e. are not necessary uniformly elliptic). The cost functional is defined as the initial value of the backward component of the solution. We construct a sequence of approximating controlled systems, for which we show the existence of a sequence of feedback optimal controls. By passing to the limit, we get the existence of a feedback optimal control. Filippov's convexity condition is used to ensure that the optimal control is strict. The present result extends those obtained in [2,4] to controlled systems of coupled SDE–BSDE.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.11.012
Khaled Bahlali 1 ; Omar Kebiri 2 ; Ahmed Mtiraoui 1, 3

1 Université de Toulon, IMATH, EA 2134, 83957 La Garde cedex, France
2 Université de Tlemcen, Laboratoire de probabilités et statistiques, Algérie
3 Faculté des sciences de Sfax, Département de mathématiques, BP 1171, 3000 Sfax, Tunisia 
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Khaled Bahlali; Omar Kebiri; Ahmed Mtiraoui. Existence of an optimal control for a system driven by a degenerate coupled forward–backward stochastic differential equations. Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 84-89. doi : 10.1016/j.crma.2016.11.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.11.012/

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☆ This work is partially supported by PHC Tassili 13MDU887.

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