Discrete time nonlinear filters with informative observations are stable

TL;DR

This paper demonstrates that in discrete-time nonlinear filtering, highly informative observations can ensure the filter's stability regardless of the underlying signal's properties, especially when the observation function is invertible.

Contribution

It establishes that informative observations guarantee filter stability without requiring strong ergodic conditions on the signal, extending previous results.

Findings

Filter stability is achieved with informative observations regardless of signal properties.

Invertible observation functions enable stability in total variation under mild conditions.

Weak stability can be strengthened to total variation stability with additional continuity assumptions.

Abstract

The nonlinear filter associated with the discrete time signal-observation model $(X_k,Y_k)$ is known to forget its initial condition as $k\to\infty$ regardless of the observation structure when the signal possesses sufficiently strong ergodic properties. Conversely, it stands to reason that if the observations are sufficiently informative, then the nonlinear filter should forget its initial condition regardless of any properties of the signal. We show that for observations of additive type $Y_k=h(X_k)+\xi_k$ with invertible observation function $h$ (under mild regularity assumptions on $h$ and on the distribution of the noise $\xi_k$), the filter is indeed stable in a weak sense without any assumptions at all on the signal process. If the signal satisfies a uniform continuity assumption, weak stability can be strengthened to stability in total variation.