Stability of Non-linear Filter for Deterministic Dynamics

TL;DR

This paper demonstrates the stability of nonlinear filters for deterministic systems under broad conditions, extending previous stochastic-focused results and linking filter behavior to system dynamics.

Contribution

It proves filter stability for deterministic dynamics with general assumptions on state space and observations, unlike prior stochastic-based results.

Findings

Nonlinear filters are stable for deterministic systems under certain conditions.

The asymptotic structure of the filtering distribution relates to the system's dynamical properties.

Examples provided illustrate systems satisfying the stability assumptions.

Abstract

This papers shows that nonlinear filter in the case of deterministic dynamics is stable with respect to the initial conditions under the conditions that observations are sufficiently rich, both in the context of continuous and discrete time filters. Earlier works on the stability of the nonlinear filters are in the context of stochastic dynamics and assume conditions like compact state space or time independent observation model, whereas we prove filter stability for deterministic dynamics with more general assumptions on the state space and observation process. We give several examples of systems that satisfy these assumptions. We also show that the asymptotic structure of the filtering distribution is related to the dynamical properties of the signal.