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

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  • Bandini, Elena
  • Cosso, Andrea
  • Fuhrman, Marco
  • Pham, Huyên

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.

Suggested Citation

  • Bandini, Elena & Cosso, Andrea & Fuhrman, Marco & Pham, Huyên, 2019. "Randomized filtering and Bellman equation in Wasserstein space for partial observation control problem," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 674-711.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:2:p:674-711
    DOI: 10.1016/j.spa.2018.03.014
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    References listed on IDEAS

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    1. Bandini, Elena & Fuhrman, Marco, 2017. "Constrained BSDEs representation of the value function in optimal control of pure jump Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 127(5), pages 1441-1474.
    2. Cosso, Andrea & Fuhrman, Marco & Pham, Huyên, 2016. "Long time asymptotics for fully nonlinear Bellman equations: A backward SDE approach," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 1932-1973.
    3. Choukroun, Sébastien & Cosso, Andrea & Pham, Huyên, 2015. "Reflected BSDEs with nonpositive jumps, and controller-and-stopper games," Stochastic Processes and their Applications, Elsevier, vol. 125(2), pages 597-633.
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    Citations

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    Cited by:

    1. Martini, Mattia, 2023. "Kolmogorov equations on spaces of measures associated to nonlinear filtering processes," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 385-423.
    2. Calvia, Alessandro & Ferrari, Giorgio, 2021. "Nonlinear Filtering of Partially Observed Systems Arising in Singular Stochastic Optimal Control," Center for Mathematical Economics Working Papers 651, Center for Mathematical Economics, Bielefeld University.
    3. Alexander M. G. Cox & Sigrid Kallblad & Martin Larsson & Sara Svaluto-Ferro, 2021. "Controlled Measure-Valued Martingales: a Viscosity Solution Approach," Papers 2109.00064, arXiv.org, revised Aug 2023.
    4. Fuhrman, Marco & Morlais, Marie-Amélie, 2020. "Optimal switching problems with an infinite set of modes: An approach by randomization and constrained backward SDEs," Stochastic Processes and their Applications, Elsevier, vol. 130(5), pages 3120-3153.
    5. Peter Bank & Yan Dolinsky, 2023. "Optimal investment with a noisy signal of future stock prices," Papers 2302.10485, arXiv.org, revised Dec 2023.
    6. Bandini, Elena & Calvia, Alessandro & Colaneri, Katia, 2022. "Stochastic filtering of a pure jump process with predictable jumps and path-dependent local characteristics," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 396-435.

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