On Kailath's Innovations Conjecture Hold

Estimation of signals from past observations of them corrupted by noise is a classical problem of filtering theory. The following is a standard mathematical idealization of this problem: The signal z. is a measurable stochastic process with Ezt the noise w. is a Brownian motion, and the observations consist of the process (1) Define it = E{zty,, 0 ^ s ^ 2}, the expected value of zt given the past of the observations up to t. It can be shown 1 that if yV zds °o a.s., then there is a measurable version of z. with Jo* £2,ds « a.s. The innovations process for this setup is defined to be 981 and it is a basic result of Frost 2 and also of Kailath 3 that, under weak conditions, v. is itself a Wiener process with respect to the observations. Thus, (1) is equivalent to the integral equation (2) which reduces the general case (1) to that in which z. is adapted to y., a special property useful in questions of absolute continuity in filtering and detection. Since i. is of necessity adapted to y., Eq. (2) purports to define y. in terms of v.; the innovations problem, first posed by Frost, 2 is precisely to determine whether it really does. Frost asked: Do the innovations contain all the information in the observations? £By (2) they do not contain more.]] In the language of probability this is to ask whether the c-algebras t h a t the processes generate are the same up to null sets; i.e., is ^o 4 « ^ = *{>>*, « ^ t 4 3lo (mod P ) ?