On kailath's innovations conjecture hold

With z. a signal process, w. a Brownian motion, and y<inf>t</inf> = ∫<inf>0</inf>^t z<inf>s</inf>ds + w<inf>t</inf>, a noisy observation, the innovations problem is to determine whether y. is adapted to the innovations process v., which is also a Brownian motion, and is defined using the estimate <tex>$\hat {z}_t = E\{z_{t}\vert y_{s}, 0 \leq s \leq t\}$</tex> by <tex>$y_{t} = ∫_{0}^{t} \hat {z}_{s}ds + v_{t}$</tex>. The closely related σ-algebras problem in stochastic DEs is to determine, for a given causal drift α, when a solution of dξ = α(t, ξ)dt + dw is a causal functional of w. Previous results on these problems are reviewed and extended. In particular, we broach and answer positively the physically important case of the innovations problem in which the signal satisfies a stochastic de with drift depending in part on the noisy observations. This case is important because it models a system observed through noise and controlled by feedback of these noisy observations. The last part of the paper shows that the innovations problem has a positive resolution if and only if on some probability space there is a Brownian motion W and a causal solution ξ of dξ = α(t, ξ)dt + dW, where α expresses the estimator <tex>$\hat z;$</tex> that is, α is a causal functional such that <tex>$\hat z_{t} = α(t, y)$</tex>.