Entropic Solution of the Innovation Conjecture of T. Kailath

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 $U$.