On the Stability of Kalman-Bucy Diffusion Processes

TL;DR

This paper reviews and refines the stability and convergence properties of the Kalman-Bucy filter, providing new bounds and inequalities for the filter and Riccati flow, crucial for continuous state-space estimation.

Contribution

It offers new exponential and comparison inequalities, along with refined bounds and eigenvalue inequalities for the Kalman-Bucy filter and Riccati equation.

Findings

Proved new bounds and inequalities for filter stability.

Established exponential convergence results.

Provided eigenvalue inequalities for Riccati flow.

Abstract

The Kalman-Bucy filter is the optimal state estimator for an Ornstein-Uhlenbeck diffusion given that the system is partially observed via a linear diffusion-type (noisy) sensor. Under Gaussian assumptions, it provides a finite-dimensional exact implementation of the optimal Bayes filter. It is generally the only such finite-dimensional exact instance of the Bayes filter for continuous state-space models. Consequently, this filter has been studied extensively in the literature since the seminal 1961 paper of Kalman and Bucy. The purpose of this work is to review, re-prove and refine existing results concerning the dynamical properties of the Kalman-Bucy filter so far as they pertain to filter stability and convergence. The associated differential matrix Riccati equation is a focal point of this study with a number of bounds, convergence, and eigenvalue inequalities rigorously proven. New results are also given in the form of exponential and comparison inequalities for both the filter and the Riccati flow.