Parameter uncertainty in the Kalman--Bucy filter

In standard treatments of stochastic filtering one first has to estimate the parameters of the model. Simply running the filter without considering the reliability of this estimate does not take into account this additional source of statistical uncertainty. We propose an approach to address this problem when working with the continuous time Kalman--Bucy filter, by making evaluations via a nonlinear expectation. We show how our approach may be reformulated as an optimal control problem, and proceed to analyze the corresponding value function. In particular we present a novel uniqueness result for the associated Hamilton--Jacobi--Bellman equation