Entropic Solution of the Innovation Conjecture of T. Kailath

TL;DR

This paper establishes a condition involving entropy and conditional expectations under which the filtration of a signal equals that of its innovation process, extending classical results in stochastic filtering.

Contribution

It provides a new entropic characterization of the equivalence of filtrations for a filtered probability space, including a localized version with stopping times.

Findings

Filtration of the signal equals that of the innovation process under a specific entropy condition.

Derived a formula relating entropy of the innovation process to the conditional expectation of the signal's derivative.

Extended the result to a localized setting with stopping times for broader applicability.

Abstract

On a general filtered probability space, for a given signal $U_t=B_t+\int_0^t\dot{u}_sds$, we prove that the filtration of $U$ is equal to the filtration of its innovation process $Z$ if and only if $$ H(Z(\nu)|\mu)=\half E_\nu[\int_0^1|E_P[\dot{u}_s|\calU_s]|^2ds] $$ where $d\nu=\exp(-\int_0^1 E_P[\dot{u}_s|\calU_s]dZ_s-\half \int_0^1|E_P[\dot{u}_s|\calU_s]|^2 ds)dP$ in case the density has expectation one, otherwies we give a localized version of the same strength with a sequence of stopping times of the filtration of $U$.