Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE

Idris Kharroubi (1) , Huyen Pham (2, 3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 CREST - Centre de Recherche en Économie et Statistique
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

We aim to provide a Feynman-Kac type representation for Hamilton-Jacobi-Bellman equation, in terms of Forward Backward Stochastic Differential Equation (FBSDE) with a simulatable forward process. For this purpose, we introduce a class of BSDE where the jumps component of the solution is subject to a partial nonpositive constraint. Existence and approximation of a unique minimal solution is proved by a penalization method under mild assumptions. We then show how minimal solution to this BSDE class provides a new probabilistic representation for nonlinear integro-partial differential equations (IPDEs) of Hamilton-Jacobi-Bellman (HJB) type, when considering a regime switching forward SDE in a Markovian framework. Moreover, we state a dual formula of this BSDE minimal solution involving equivalent change of probability measures. This gives in particular an original representation for value functions of stochastic control problems including controlled diffusion coefficient.

Mots clés: semiconcave approximation; nonlinear Integral PDE; viscosity solutions; inf-convolution; regime-switching jump-diffusion; Hamilton-Jacobi-Bellman equation; constrained BSDE; BSDE with jumps

Domaines: Probabilités [math.PR]

Idris Kharroubi, Huyen Pham. Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE. Annals of Probability, 2015, 43 (4), pp.1823-1865. ⟨hal-00761057v2⟩