Entropic Solution of the Innovation Conjecture of T. Kailath

Abstract

On a general filtered probability space, for a given signal $U_t=B_t+{\int }_0^t{\dot{u}}_{s}ds$, we prove that the filtration of $U$ is equal to the filtration of its innovation process $Z$ if and only if $$ H(Z(ν)|μ)=\half E_ν[\int_0^1|E_P[\dot{u}_s|\calU_s]|^2ds] $$ where $dν=\exp(-\int_0^1 E_P[\dot{u}_s|\calU_s]dZ_s-\half \int_0^1|E_P[\dot{u}_s|\calU_s]|^2 ds)dP$ 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 $U$.

Some typos have been corrected

References in corpus (1)

• Analysis on Wiener Space and Applications

Cited by in corpus (1)

• Some applications of the Tsirelson spectral measures to noise filtering problems