On the Stability of Kalman-Bucy Diffusion Processes

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.

Updated with corrections and some (minor) additions

References in corpus (2)

• On the Stability and the Exponential Concentration of Extended Kalman-Bucy filters
• Performance of Ensemble Kalman filters in large dimensions

Cited by in corpus (15)

• A perturbation analysis of stochastic matrix Riccati diffusions
• On the stability of matrix-valued Riccati diffusions
• On the Mathematical Theory of Ensemble (Linear-Gaussian) Kalman-Bucy Filtering
• Perturbations and projections of Kalman-Bucy semigroups
• Analysis of a localised nonlinear Ensemble Kalman Bucy Filter with complete and accurate observations
• On One-Dimensional Riccati Diffusions
• Stability Properties of Systems of Linear Stochastic Differential Equations with Random Coefficients
• An explicit Floquet-type representation of Riccati aperiodic exponential semigroups
• Stability of Non-linear Filter for Deterministic Dynamics
• Quantum harmonic oscillators and Feynman-Kac path integrals for linear diffusive particles
• Backward Nonlinear Smoothing Diffusions
• Rigorous Analysis for Efficient Statistically Accurate Algorithms for Solving Fokker-Planck Equations in Large Dimensions
• Asymptotic Properties of Linear Filter for Noise Free Dynamical System
• Multilevel Estimation of Normalization Constants Using the Ensemble Kalman-Bucy Filter
• A second order analysis of McKean-Vlasov semigroups