  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.
