{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,30]],"date-time":"2026-09-30T05:07:34Z","timestamp":1790744854222,"version":"4.1.0"},"reference-count":46,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DE-120102873"],"award-info":[{"award-number":["DE-120102873"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.1137\/16m1102707","type":"journal-article","created":{"date-parts":[[2017,12,19]],"date-time":"2017-12-19T13:16:29Z","timestamp":1513689389000},"page":"4015-4047","source":"Crossref","is-referenced-by-count":37,"title":["On the Stability of Kalman--Bucy Diffusion Processes"],"prefix":"10.1137","volume":"55","author":[{"given":"Adrian N.","family":"Bishop","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Pierre","family":"Del Moral","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,12,19]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"H. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank,\n                      Matrix Riccati Equations in Control and Systems Theory\n                      , Birkha\u0308user, Basel, 2003.","DOI":"10.1007\/978-3-0348-8081-7"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/0016-0032(71)90016-0"},{"key":"atypb3","doi-asserted-by":"crossref","unstructured":"S. Bittanti, A. J. Laub, and J. C. Willems,\n                      The Riccati Equation\n                      , Springer-Verlag, Berlin, 1991.","DOI":"10.1007\/978-3-642-58223-3"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0000(67)80025-4"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1016\/S0022-0000(72)80026-6"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1972.1099862"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/0313043"},{"key":"atypb8","unstructured":"R. S. Bucy and P. D. Joseph,\n                      Filtering for Stochastic Processes with Applications to Guidance\n                      , Interscience, New York, 1968."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/0503065"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1981.1102812"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1995.1169"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1109\/9.148372"},{"key":"atypb13","doi-asserted-by":"crossref","unstructured":"P. Del Moral,\n                      Feynman-Kac Formulae\n                      , Springer-Verlag, New York, 2004.","DOI":"10.1007\/978-1-4684-9393-1"},{"key":"atypb14","doi-asserted-by":"crossref","unstructured":"P. Del Moral,\n                      Mean Field Simulation for Monte Carlo Integration\n                      , Chapman & Hall\/CRC Monographs on Statistics & Applied Probability 126, CRC Press, Boca Raton, FL, 2013.","DOI":"10.1201\/b14924"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"P. Del Moral, A. Kurtzmann, and J. Tugaut,\n                      On the Stability and the Uniform Propagation of Chaos of Extended Ensemble Kalman-Bucy Filters\n                      , preprint,arXiv:1606.08256, 2016.","DOI":"10.1137\/16M1087497"},{"key":"atypb16","unstructured":"P. Del Moral, A. Kurtzmann, and J. Tugaut,\n                      On the Stability and the Exponential Concentration of Extended Kalman-Bucy Filters\n                      , preprint,arXiv:1606.08251, 2016."},{"key":"atypb17","doi-asserted-by":"crossref","unstructured":"P. Del Moral and J. Tugaut,\n                      On the Stability and the Uniform Propagation of Chaos Properties of Ensemble Kalman-Bucy Filters\n                      , preprint,arXiv:1605.09329, 2016.","DOI":"10.1137\/16M1087497"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1109\/9.940944"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s10236-003-0036-9"},{"key":"atypb20","unstructured":"M. Fiedler,\n                      Special Matrices and Their Applications in Numerical Mathematics\n                      , 2nd ed., Dover, Mineola, NY, 2008."},{"key":"atypb21","first-page":"531","author":"Gevers M.","year":"1986","journal-title":"Amsterdam"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1972.1099902"},{"key":"atypb23","first-page":"102","volume":"5","author":"Kalman R. E.","year":"1960","journal-title":"Bol. Soc. Mat. Mex."},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1115\/1.3662552"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1972.1099861"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1115\/1.3662604"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1115\/1.3658902"},{"key":"atypb28","doi-asserted-by":"crossref","unstructured":"D. Khoshnevisan,\n                      Analysis of Stochastic Partial Differential Equations\n                      , American Mathematical Society, Providence, RI, 2014.","DOI":"10.1090\/cbms\/119"},{"key":"atypb29","unstructured":"V. Kres\u030cimir,\n                      Damped Oscillations of Linear Systems: A Mathematical Introduction\n                      , Lecture Notes in Mathematics 2023, Springer-Verlag, Berlin, 2011."},{"key":"atypb30","first-page":"42","volume":"9","author":"Kucera V.","year":"1973","journal-title":"Kybernetika"},{"key":"atypb31","unstructured":"F. le Gland, V. Monbet, and V.D. Tran,\n                      Large Sample Asymptotics for the Ensemble Kalman Filter\n                      , Research report RR-7014, INRIA, 2009, inria-00409060."},{"key":"atypb32","unstructured":"A. J. Majda and X. T. Tong,\n                      Robustness and Accuracy of Finite Ensemble Kalman Filters in Large Dimensions\n                      , preprint,arXiv:1606.09321, 2016."},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1137\/S00361445024180"},{"key":"atypb34","doi-asserted-by":"crossref","unstructured":"R. T. O'Brien,\n                      On stabilizing solutions to the Riccati differential equation\n                      , in Proceedings of the 30th Southeastern Symposium on System Theory, 1998, pp. 80-84.","DOI":"10.1109\/SSST.1998.660023"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/S0363012993256617"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.1986.1104355"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1214\/14-AAP1061"},{"key":"atypb38","unstructured":"W. T. 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