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A measure-valued HJB perspective on Bayesian optimal adaptive control

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  • Alexander M. G. Cox
  • Sigrid Kallblad
  • Chaorui Wang

Abstract

We consider a Bayesian adaptive optimal stochastic control problem where a hidden static signal has a non-separable influence on the drift of a noisy observation. Being allowed to control the specific form of this dependence, we aim at optimising a cost functional depending on the posterior distribution of the hidden signal. Expressing the dynamics of this posterior distribution in the observation filtration, we embed our problem into a genuinely infinite-dimensional stochastic control problem featuring so-called measure-valued martingales. We address this problem by use of viscosity theory and approximation arguments. Specifically, we show equivalence to a corresponding weak formulation, characterise the optimal value of the problem in terms of the unique continuous viscosity solution of an associated HJB equation, and construct a piecewise constant and arbitrarily-close-to-optimal control to our main problem of study.

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  • Alexander M. G. Cox & Sigrid Kallblad & Chaorui Wang, 2025. "A measure-valued HJB perspective on Bayesian optimal adaptive control," Papers 2502.12957, arXiv.org.
    • Handle: RePEc:arx:papers:2502.12957
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    References listed on IDEAS

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    1. 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.
    2. Kunita, Hiroshi, 1971. "Asymptotic behavior of the nonlinear filtering errors of Markov processes," Journal of Multivariate Analysis, Elsevier, vol. 1(4), pages 365-393, December.
    3. Blount, Douglas & Kouritzin, Michael A., 2010. "On convergence determining and separating classes of functions," Stochastic Processes and their Applications, Elsevier, vol. 120(10), pages 1898-1907, September.
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