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 {X t} be a Brownian motion observed by a cubic sensor corrupted by white noise, and let\(\hat \phi \) denote the unnormalized conditional estimate of φ(X i ). If φ1,...,φ n 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.
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Research supported in part by NSF Grant No. MCS-8301880 and by the Institute for Mathematics and It Applications, Minneapolis, Minnesota.
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Ocone, D. Probability densities for conditional statistics in the cubic sensor problem. Math. Control Signal Systems 1, 183–202 (1988). https://doi.org/10.1007/BF02551408
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DOI: https://doi.org/10.1007/BF02551408