Probability densities for conditional statistics in the cubic sensor problem

Abstract

This paper applies the techniques of Malliavin’s stochastic calculus of variations to Zakai’s equation for the one-dimensional cubic sensor problem in order to study the existence of densities of conditional statistics. Let {Xt} be a Brownian motion observed by a cubic sensor corrupted by white noise, and let\(\hat \phi \) denote the unnormalized conditional estimate of φ(Xi). If φ1,...,φn are linearly independent, and if\(\hat \Phi = (\hat \phi _1 ,...,\hat \phi _n )\), it is shown that the probability distribution of\(\hat \Phi \) admits a density with respect to Lebesgue measure for anyn. This implies that, at any fixed time, the unnormalized conditional density cannot be characterized by a finite set of sufficient statistics.