Observability and nonlinear filtering

Details Observability and nonlinear filtering Observability and nonlinear filtering

2007-08-24

arxiv.org/abs/0708.3412v2

Probab. Th. Rel. Fields 145, 35-74 (2009)

Mathematics

Probability

Scientific paper

10.1007/s00440-008-0161-y

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 in filter stability is explored. Finally, the results are partially extended to non-compact signal state spaces.

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