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Journal of Applied Mathematics and Stochastic Analysis
Volume 14, Issue 3, Pages 215-226
http://dx.doi.org/10.1155/S104895330100017X

Optimal linear filtering of general multidimensional Gaussian processes and its application to Laplace transforms of quadratic functionals

1Institute for Information Transmission Problems, Bolshoi Karetnii per. 19, Moscow 101447, Russia
2Université J. Fourier, Laboratoire de Modélisation et Calcul, BP 53, Grenoble Cedex 9 38041, France

Received 1 July 2000; Revised 1 March 2001

Copyright © 2001 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The optimal filtering problem for multidimensional continuous possibly non-Markovian, Gaussian processes, observed through a linear channel driven by a Brownian motion, is revisited. Explicit Volterra type filtering equations involving the covariance function of the filtered process are derived both for the conditional mean and for the covariance of the filtering error. The solution of the filtering problem is applied to obtain a Cameron-Martin type formula for Laplace transforms of a quadratic functional of the process. Particular cases for which the results can be further elaborated are investigated.