Observability and nonlinear filtering

Abstract

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.

References in corpus (2)

• The stability of conditional Markov processes and Markov chains in random environments
• A simple proof of Kaijser's unique ergodicity result for hidden Markov $\alpha$-chains

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• An explicit Floquet-type representation of Riccati aperiodic exponential semigroups
• Likelihood Scouting Via Map Inversion For A Posterior-Sampled Particle Filter
• Phase Transitions in Nonlinear Filtering
• Variance Decay Property for Filter Stability
• Stability of Non-linear Filter for Deterministic Dynamics