A measure-valued HJB perspective on Bayesian optimal adaptive control
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arXiv
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| Format: | Preprint |
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2025
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| author | Cox, Alexander M. G. Källblad, Sigrid Wang, Chaorui |
| author_facet | Cox, Alexander M. G. Källblad, Sigrid Wang, Chaorui |
| contents | 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. Our setup is in sharp contrast to existing work: we include costs that depend on the full posterior distribution in a form that admits a large class of non-linear relationships. Expressing the dynamics of this posterior distribution in the observation filtration, we embed our problem into a genuinely infinite-dimensional stochastic control problem using 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. As a byproduct of our analysis, we also provide a novel stability result for a class of measure-valued SDEs which we believe is of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12957 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A measure-valued HJB perspective on Bayesian optimal adaptive control Cox, Alexander M. G. Källblad, Sigrid Wang, Chaorui Optimization and Control Probability Mathematical Finance 49L25, 60G35, 60H10, 62M20, 93E35, 93E10 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. Our setup is in sharp contrast to existing work: we include costs that depend on the full posterior distribution in a form that admits a large class of non-linear relationships. Expressing the dynamics of this posterior distribution in the observation filtration, we embed our problem into a genuinely infinite-dimensional stochastic control problem using 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. As a byproduct of our analysis, we also provide a novel stability result for a class of measure-valued SDEs which we believe is of independent interest. |
| title | A measure-valued HJB perspective on Bayesian optimal adaptive control |
| topic | Optimization and Control Probability Mathematical Finance 49L25, 60G35, 60H10, 62M20, 93E35, 93E10 |
| url | https://arxiv.org/abs/2502.12957 |