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Observability and nonlinear filtering

van Handel, Ramon (2009) Observability and nonlinear filtering. Probability Theory and Related Fields, 145 (1-2). pp. 35-74. ISSN 0178-8051.

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This paper develops a connection between the asymptotic stability of nonlinear filters and a notion of observability. We consider a general class of hidden Markov models in continuous time with compact signal state space, and call such a model observable if no two initial measures of the signal process give rise to the same law of the observation process. We demonstrate that observability implies stability of the filter, i.e., the filtered estimates become insensitive to the initial measure at large times. For the special case where the signal is a finite-state Markov process and the observations are of the white noise type, a complete (necessary and sufficient) characterization of filter stability is obtained in terms of a slightly weaker detectability condition. In addition to observability, the role of controllability is explored. Finally, the results are partially extended to non-compact signal state spaces.

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Additional Information:© Springer-Verlag 2008. Received: 27 August 2007 / Revised: 26 May 2008 / Published online: 17 June 2008. The author is indebted to Pavel Chigansky for countless useful discussions on the topic of this article. Mathematics Subject Classification (2000) 93E11 · 60J25 · 62M20 · 93B05 · 93B07 · 93E15.
Subject Keywords:Nonlinear filtering; Asymptotic stability; Observability; Detectability; Controllability; Hidden Markov models
Issue or Number:1-2
Record Number:CaltechAUTHORS:20090908-152746744
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:15679
Deposited By: Tony Diaz
Deposited On:30 Sep 2009 20:54
Last Modified:03 Oct 2019 01:01

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