Entropic solution of the innovation conjecture of T. Kailath

Abstract

On a general filtered probability space $(\Omega ,\mathcal{F},({\mathcal{F}}_t,t\in [0,1]),P)$, for a given signal $U_t=B_t+{\int }_0^t{\dot{u}}_{s}ds$, where $(B_t,t\in [0,1\left]\right)$ is a Brownian motion and $\dot{u}$ is adapted and in $L^2(dt\times dP)$, we prove that the filtration of $U$, denoted $({\mathcal{U}}_t,t\in [0,1\left]\right)$, is equal to the filtration of its innovation process $Z$, which is defined as $Z_t=U_t-{\int }_0^t{E}_P[{\dot{u}}_s\mid {\mathcal{U}}_s]ds$, $t\in [0,1]$, if and only if $$H\left(Z\right(\nu )\mid \mu )=\frac{1}{2}{E}_{\nu }\left[{\int }_0^1\right|E_P[{\dot{u}}_s\mid {\mathcal{U}}_s]{|}^{2}ds]$$ where $d\nu =exp(-{\int }_0^1{E}_P[{\dot{u}}_s\mid {\mathcal{U}}_s]d{Z}_s-\frac{1}{2}{\int }_0^1|E_P[{\dot{u}}_s\mid {\mathcal{U}}_s]{|}^{2}ds)dP$ in the case in which the density has expectation $1$; otherwise, we give a localized version of the same strength with a sequence of stopping times of the filtration of $U$.