Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem |
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Authors: | Elena Bandini Andrea Cosso Marco Fuhrman Huyên Pham |
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Institution: | 1. Dipartimento di Matematica Applicazioni, Università degli studi di Milano-Bicocca, Via Roberto Cozzi, 55, 20125 Milano, Italy;2. Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5, 40126 Bologna, Italy;3. Dipartimento di Matematica, Università degli studi di Milano, via Saldini 50, 20133 Milano, Italy;4. Laboratoire de Probabilités, Statistique et Modélisation, CNRS, UMR 8001, Université Paris Diderot, France;5. CREST-ENSAE, France |
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Abstract: | 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. |
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Keywords: | 93E20 60G35 60H30 Partial observation control problem Randomization of controls Dynamic programming principle Bellman equation Wasserstein space Viscosity solutions |
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