Fixed lag smoothing of scalar diffusions. Part I. The filtering-smoothing equation
type: Item
instance of: scholarly article
title: Fixed lag smoothing of scalar diffusions. Part I. The filtering-smoothing equation
zbMATH Open document ID: 0879.93054
DOI: 10.1016/S0304-4149(96)00086-5
author: Y. Steinberg
author: R. Liptser
author: B. Z. Bobrovsky
author: Zeev Schuss
published in: Stochastic Processes and their Applications
publication date: 1997-12-10T00:00:00Z
review text: This paper is devoted to the estimation of a random signal \(x_s\) simultaneously at time \(t\) and at an earlier time \(t-\tau\), given its measurements in a nonlinear noisy channel \(y_s\), up to time \(t\). Denoting by \(Y^t\) the measurements of \(y_s\) for \(s\) in the interval \((0,t)\), the nonlinear filtering and fixed lag smoothing problem is equivalent to the calculation of the joint conditional probability distribution function \(P(x_t\leq u,x_{t-\tau} \leq v\big|Y^t)\), for \(t>\tau\) and \(u\), \(v\in \mathbb R\). The main result is a derivation of evolution equations for the normalized and unnormalized conditional joint probability density functions. The former is a stochastic integro-partial-differential equation of Kushner type and the latter is a linear stochastic partial differential equation of Zakai type. The fixed lag smoothing probability density function is obtained as a marginal of the joint probability density function.
Mathematics Subject Classification ID: 93E11
Mathematics Subject Classification ID: 60G35
zbMATH DE Number: 1095929
zbMATH Keywords: filtering
zbMATH Keywords: smoothing
zbMATH Keywords: diffusion process
reviewed by: Věra Lánská
MaRDI profile type: Publication
cites work: Q3993534
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full work available at URL: s0304-4149(96)00086-5
OpenAlex ID: W2093539345
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description: scientific article; zbMATH DE number 1095929