The Annals of Applied Probability

Uniform observability of hidden Markov models and filter stability for unstable signals

Ramon van Handel

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Abstract

A hidden Markov model is called observable if distinct initial laws give rise to distinct laws of the observation process. Observability implies stability of the nonlinear filter when the signal process is tight, but this need not be the case when the signal process is unstable. This paper introduces a stronger notion of uniform observability which guarantees stability of the nonlinear filter in the absence of stability assumptions on the signal. By developing certain uniform approximation properties of convolution operators, we subsequently demonstrate that the uniform observability condition is satisfied for various classes of filtering models with white-noise type observations. This includes the case of observable linear Gaussian filtering models, so that standard results on stability of the Kalman–Bucy filter are obtained as a special case.

Article information

Source
Ann. Appl. Probab. Volume 19, Number 3 (2009), 1172-1199.

Dates
First available in Project Euclid: 15 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1245071023

Digital Object Identifier
doi:10.1214/08-AAP576

Mathematical Reviews number (MathSciNet)
MR2537203

Zentralblatt MATH identifier
1165.93034

Subjects
Primary: 93E11: Filtering [See also 60G35]
Secondary: 60J25: Continuous-time Markov processes on general state spaces 62M20: Prediction [See also 60G25]; filtering [See also 60G35, 93E10, 93E11] 93B07: Observability 93E15: Stochastic stability

Keywords
Nonlinear filtering prediction asymptotic stability observability hidden Markov models uniform approximation merging of probability measures

Citation

van Handel, Ramon. Uniform observability of hidden Markov models and filter stability for unstable signals. Ann. Appl. Probab. 19 (2009), no. 3, 1172--1199. doi:10.1214/08-AAP576. https://projecteuclid.org/euclid.aoap/1245071023


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