Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem

Elena Bandini (1) , Andrea Cosso (2) , Marco Fuhrman (2) , Huyên Pham (3, 4)

1 Dipartimento di Economia e Finanza
2 POLIMI - Politecnico di Milano [Milan]
3 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
4 CREST - Centre de Recherche en Économie et Statistique

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in [4], we prove a corresponding randomized dynamic programming principle (DPP) for the value function, which is obtained from a flow property of an associated filter process. This DPP is the key step towards our main result: a characterization of the value function of the partial observation control problem as the unique viscosity solution to the corresponding dynamic programming Hamilton-Jacobi-Bellman (HJB) equation. The latter is formulated as a new, fully non linear partial differential equation on the Wasserstein space of probability measures. An important feature of our approach is that it does not require any non-degeneracy condition on the diffusion coefficient, and no condition is imposed to guarantee existence of a density for the filter process solution to the controlled Zakai equation, as usually done for the separated problem. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear quadratic model.

Mots clés: partial observation control problem; randomization of controls; dynamic programming principle; Bellman equation; Wasserstein space; viscosity solutions AMS 2010 subject classification:

Domaines: Probabilités [math.PR]; Optimisation et contrôle [math.OC]

Elena Bandini, Andrea Cosso, Marco Fuhrman, Huyên Pham. Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem . 2016. ⟨hal-01362059⟩