# On the Stability of Kalman-Bucy Diffusion Processes

arXiv: https://arxiv.org/abs/1610.04686 · HTML: https://mlsanity.com/paper/1610.04686/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Oct 2016.

## Built on by (most cited first, 2 of 2)

- [Recurrent Neural Networks and Universal Approximation of Bayesian Filters](https://mlsanity.com/paper/2211.00335/) (2022)
- [Interacting Particle Systems for Fast Linear Quadratic RL](https://mlsanity.com/paper/2406.11057/) (2024)

## Abstract (arXiv)

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.