Parameter uncertainty in stochastic filtering

In standard treatments of stochastic filtering one first requires the various parameters of the model. Simply running a filter with estimated parameters without considering the reliability of this estimate does not take into account this additional source of statistical uncertainty. We propose a novel approach to address this problem by making evaluations via a nonlinear expectation. We show how our approach may be reformulated as a pathwise stochastic optimal control problem, where the optimisation is performed for each fixed realisation of the observation process, and proceed to analyse the corresponding value function. We focus in particular on two finite-dimensional continuous-time filters, namely the Kalman-Bucy and Wonham filters. In each case, we present novel comparison results for the viscosity solution of the associated Hamilton-Jacobi-Bellman equation. To enable this analysis we consider pathwise optimal control problems in generality. In particular we investigate the degeneracy induced by directly controlling the coefficient of the noise term, and we propose a simple procedure to resolve this degeneracy whilst retaining dynamic programming. As we will see, such control problems may be rephrased as the optimal control of rough differential equations, thus demonstrating an original application of rough path theory to statistics.