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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:129

Article

Bandini, Elena¹  Cosso, Andrea²  Fuhrman, Marco³  Huyen Pham^4,5 

[1] Univ Milano Bicocca, Dipartimento Matemat Applicaz, Via Roberto Cozzi 55, I-20125 Milan, Italy

[2] Univ Bologna, Dipartimento Matemat, Piazza Porta S Donato 5, I-40126 Bologna, Italy

[3] Univ Milan, Dipartimento Matemat, Via Saldini 50, I-20133 Milan, Italy

[4] Univ Paris Diderot, CNRS, UMR 8001, Lab Probabilites Stat & Modelisat, Paris, France

[5] ENSAE, CREST, Palaiseau, France

关键词: Partial observation control problem;    Randomization of controls;    Dynamic programming principle;    Bellman equation;    Wasserstein space;    Viscosity solutions;   

DOI  :  10.1016/j.spa.2018.03.014

来源: Elsevier

【 摘 要 】

We study a stochastic optimal control problem for a partially observed diffusion. By using the control randomization method in Bandini et al. (2018), 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. Finally, we give an explicit solution to our HJB equation in the case of a partially observed non Gaussian linear-quadratic model. (C) 2018 Elsevier B.V. All rights reserved.

【 授权许可】

Free