Uniform observability of hidden Markov models and filter stability for unstable signals
Ramon van Handel
Source: Ann. Appl. Probab. Volume 19, Number 3 (2009), 1172-1199.
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.
Primary Subjects: 93E11
Secondary Subjects: 60J25, 62M20, 93B07, 93E15
Keywords: Nonlinear filtering; prediction; asymptotic stability; observability; hidden Markov models; uniform approximation; merging of probability measures
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Permanent link to this document: http://projecteuclid.org/euclid.aoap/1245071023
Digital Object Identifier: doi:10.1214/08-AAP576
Zentralblatt MATH identifier:
1165.93034
Mathematical Reviews number (MathSciNet):
MR2537203
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