Optimal control under unknown intensity with Bayesian learning

Abstract

We investigate an optimal control problem motivated by neuroscience, where the dynamics is driven by a Poisson process with a controlled stochastic intensity and an unknown parameter. Given a prior distribution for the unknown parameter, we describe its evolution using Bayes' rule. We reformulate the optimization problem by applying Girsanov's theorem and establish a dynamic programming principle. Finally, we characterize the value function as the unique viscosity solution to a finite-dimensional Hamilton-Jacobi-Bellman equation, which can be solved numerically.