On the exchange of intersection and supremum of sigma-fields in filtering theory

TL;DR

This paper constructs a specific Markov process demonstrating that the nonlinear filtering process can lack unique ergodicity, challenging a previous assumption related to sigma-field operations in filtering theory.

Contribution

It provides a counterexample showing the non-commutativity of intersection and supremum of sigma-fields in nonlinear filtering, refuting a longstanding conjecture.

Findings

Counterexample with trivial tail sigma-field and nondegenerate observations

Non-uniqueness of ergodic measures in the constructed filter

Refutation of a conjecture based on an erroneous prior proof

Abstract

We construct a stationary Markov process with trivial tail sigma-field and a nondegenerate observation process such that the corresponding nonlinear filtering process is not uniquely ergodic. This settles in the negative a conjecture of the author in the ergodic theory of nonlinear filters arising from an erroneous proof in the classic paper of H. Kunita (1971), wherein an exchange of intersection and supremum of sigma-fields is taken for granted.