Abstract
We consider a stochastic control problem with unknowns, inspired by neuroscience. In this model, a neuron is characterized by its membrane potential and emits discharges randomly, with a rate that depends on its potential and an unknown parameter. Only the instants of the neuron's discharges are observed. The control acts on the membrane potential as a drift term. The question is how to choose it optimally in order to best estimate the unknown parameter. We'll show how to reformulate the problem using Girsanov's theorem and how to establish a dynamic programming principle from a Bayesian point of view. We'll see that in some cases, the value function can be obtained as the unique viscosity solution of a finite-dimensional Hamilton-Jacobi-Bellman equation.
Joint work with Nicolas Baradel.
