On diffusion approximations for filtering
type: Item
instance of: scholarly article
title: On diffusion approximations for filtering
zbMATH Open document ID: 0732.60054
DOI: 10.1016/0304-4149(91)90092-Q
author: R. Liptser
author: Wolfgang J. Runggaldier
published in: Stochastic Processes and their Applications
publication date: 1991-01-01T00:00:00Z
review text: Following the earlier paper by \textit{H. J. Kushner} and \textit{W. J. Runggaldier} [IEEE Trans. Autom. Control AC-32, 123-133 (1987; Zbl 0617.93060)], the authors consider a family of processes \((X^{\epsilon},Y^{\epsilon})\) where \(X^{\epsilon}=(X^{\epsilon}_ t)\) is unobservable, while \(Y^{\epsilon}=(Y^{\epsilon}_ t)\) is observable. The family is given by a model that is nonlinear in the observations, has coefficients that may be rapidly oscillating, and additive disturbances that may be wide- band and non-Gaussian. Using results of diffusion approximation for semimartingales, they show the convergence in distribution (for \(\to 0)\) of \((X^{\epsilon},Y^{\epsilon})\) to a process (X,Y) that satisfies a linear-Gaussian model. Applying the Kalman-Bucy filter for (X,Y) to \((X^{\epsilon},Y^{\epsilon})\), they obtain a linear filter estimate for \((X^{\epsilon}_ t)\), given the observations \(\{Y^{\epsilon}_ s,\quad 0\leq s\leq t\}.\) Such filter estimate is shown to possess the property of asymptotic (for \(\epsilon \to 0)\) optimality of its variance. The results are also applied to show the effects that a limiter in the observation equation may have on the signal-to-noise ratio and thus on the filter variance.
Mathematics Subject Classification ID: 60G35
Mathematics Subject Classification ID: 93E11
zbMATH DE Number: 4211186
zbMATH Keywords: diffusion approximation
zbMATH Keywords: additive wide bandwidth noise
zbMATH Keywords: diffusion approximation for semimartingales
reviewed by: Robert J. Elliott
MaRDI profile type: Publication
cites work: Q5560061
cites work: Q5667328
cites work: Q3344923
cites work: Filtering and control for wide bandwidth noise driven systems
cites work: Q3995465
cites work: Semimartingales and measure preserving flows
cites work: Q4069566
full work available at URL: 0304-4149(91)90092-q
OpenAlex ID: W2069597515
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description: scientific article; zbMATH DE number 4211186