Observability and nonlinear filtering

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 proc...

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Bibliographic Details
Published in:Probability theory and related fields 2009-09, Vol.145 (1-2), p.35-74
Main Author: van Handel, Ramon
Format: Article
Language:English
Subjects:
Economics
Exact sciences and technology
Finance
General topics
Inference from stochastic processes
time series analysis
Insurance
Linear algebra
Management
Markov analysis
Markov chains
Markov processes
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Noise
Nonlinear filters
Nonlinear programming
Observability (systems)
Operations Research/Decision Theory
Probability
Probability and statistics
Probability Theory and Stochastic Processes
Quantitative Finance
Sciences and techniques of general use
Signal processing
Special processes (renewal theory, markov renewal processes, semi-markov processes, statistical mechanics type models, applications)
Stability
Statistics
Statistics for Business
Studies
Theoretical
White noise
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Summary: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.
ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-008-0161-y