论文信息 - Probability densities for conditional statistics in the cubic sensor problem

Probability densities for conditional statistics in the cubic sensor problem

AbstractThis 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.

Daniel Ocone

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