Parameter Uncertainty in the Kalman-Bucy Filter

TL;DR

This paper addresses the issue of parameter uncertainty in the continuous-time Kalman-Bucy filter by reformulating it as an optimal control problem and analyzing the resulting Hamilton-Jacobi-Bellman equation.

Contribution

It introduces a novel approach to incorporate parameter uncertainty into the Kalman-Bucy filter via optimal control reformulation and provides a new uniqueness result for the HJB equation.

Findings

Reformulation of parameter uncertainty as an optimal control problem

Analysis of the value function in the reformulated problem

A new uniqueness theorem for the associated Hamilton-Jacobi-Bellman equation

Abstract

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.