The stability of conditional Markov processes and Markov chains in random environments

TL;DR

This paper proves the weak ergodicity and stability of conditional Markov signals in random environments, extending results to continuous time and addressing a longstanding gap in nonlinear filter theory.

Contribution

It establishes the weak ergodicity of conditional Markov processes in random environments, providing new insights into filter stability and resolving a key proof gap in prior work.

Findings

Conditional signals are weakly ergodic under certain conditions

Results extend to continuous time models

Addresses a long-standing gap in nonlinear filter stability proof

Abstract

We consider a discrete time hidden Markov model where the signal is a stationary Markov chain. When conditioned on the observations, the signal is a Markov chain in a random environment under the conditional measure. It is shown that this conditional signal is weakly ergodic when the signal is ergodic and the observations are nondegenerate. This permits a delicate exchange of the intersection and supremum of $\sigma$-fields, which is key for the stability of the nonlinear filter and partially resolves a long-standing gap in the proof of a result of Kunita [J. Multivariate Anal. 1 (1971) 365--393]. A similar result is obtained also in the continuous time setting. The proofs are based on an ergodic theorem for Markov chains in random environments in a general state space.