Parameter Uncertainty in the Kalman-Bucy Filter

In standard treatments of stochastic filtering one first has to estimate the
values of 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. We show
how our approach may be reformulated as an optimal control problem, and proceed
to analyse the corresponding value function in some detail. In particular we
present a novel uniqueness result for the associated Hamilton-Jacobi-Bellman
equation.