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

TL;DR

This paper introduces a stronger form of observability for hidden Markov models that ensures filter stability even when the underlying signal process is unstable, extending classical results to broader classes of models.

Contribution

The paper defines uniform observability and proves it guarantees filter stability without requiring signal stability, covering models like linear Gaussian filters.

Findings

Uniform observability implies filter stability for unstable signals.

The condition is verified for various models, including linear Gaussian filters.

Standard Kalman--Bucy filter stability results are recovered as a special case.

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.