no. 9
Optimal control/Calculus of variations
Dynamic programming for mean-field type control
[Programmation dynamique pour les problèmes de contrôle à champs moyen]

Présenté par : Alain Bensoussan

Laurière, Mathieu1 ; Pironneau, Olivier1
1 LJLL, Université Pierre-et-Marie-Curie (Paris-6), 4, place Jussieu, 75005 Paris, France
Comptes Rendus. Mathématique, Tome 352 (2014) no. 9, pp. 707-713.

Pour les problèmes de contrôle stochastique à champs moyen, la programmation dynamique ne s'applique pas sans adaptation ; mais si l'on reformule le problème avec l'équation de Fokker–Planck, on peut le faire en utilisant une fonctionnelle valeur {τ,ρτ(⋅)}→V(τ,ρτ) comme dans l'analyse des problèmes de jeux à champs moyen par P.-L. Lions (2007) [10]. Les résultats sont appliqués à un problème d'optimisation de portefeuille et à un problème de risque systémique.

For mean-field type control problems, stochastic dynamic programming requires adaptation. We propose to reformulate the problem as a distributed control problem by assuming that the PDF ρ of the stochastic process exists. Then we show that Bellman's principle applies to the dynamic programming value function V(τ,ρτ), where the dependency on ρτ is functional as in P.-L. Lions' analysis of mean-field games (2007) [10]. We derive HJB equations and apply them to two examples, a portfolio optimization and a systemic risk model.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.07.008
Laurière, Mathieu 1 ; Pironneau, Olivier 1

1 LJLL, Université Pierre-et-Marie-Curie (Paris-6), 4, place Jussieu, 75005 Paris, France 
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Laurière, Mathieu; Pironneau, Olivier. Dynamic programming for mean-field type control. Comptes Rendus. Mathématique, Tome 352 (2014) no. 9, pp. 707-713. doi : 10.1016/j.crma.2014.07.008. http://www.numdam.org/articles/10.1016/j.crma.2014.07.008/

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