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.
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https://hal.inria.fr/hal-01669244
Contributor : Pierre Del Moral <>
Submitted on : Wednesday, December 20, 2017 - 4:32:52 PM
Last modification on : Thursday, January 11, 2018 - 5:22:02 PM

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Adrian Bishop, Pierre Del Moral. On the Stability of Kalman--Bucy Diffusion Processes. SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2017, 55 (6), pp.4015 - 4047. ⟨10.1137/16M1102707⟩. ⟨hal-01669244⟩

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