Paper: arXiv 2502.12957

Authors: Alexander M. G. Cox, Sigrid Källblad, 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. 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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents a highly sophisticated theoretical framework using infinite-dimensional stochastic control, measure-valued martingales, and viscosity solutions to address Bayesian optimal adaptive control. While it provides strong mathematical rigor and novel theoretical contributions, it is purely theoretical and completely lacks empirical backtesting or real-world data application.

Research Flowchart

  flowchart TD
    A[Research Goal: Bayesian Adaptive Control with Non-Linear Posterior Costs] --> B[Inputs: Hidden Signal, Noisy Observation Drift, Control Parameters]
    B --> C[Methodology: Embed Dynamics into Infinite-Dim SDEs via Measure-Valued Martingales]
    C --> D[Analysis: Establish Weak Formulation & Viscosity Solution Framework]
    D --> E[Computation: Approximation Arguments & Piecewise Constant Control Construction]
    E --> F[Outcomes: Unique Viscosity Solution to HJB Equation, Arbitrarily-Close-to-Optimal Control, & SDE Stability Result]