New formulas concerning Laplace transforms of quadratic forms for general Gaussian sequences
type: Item
instance of: scholarly article
title: New formulas concerning Laplace transforms of quadratic forms for general Gaussian sequences
zbMATH Open document ID: 1022.60037
DOI: 10.1155/S1048953302000266
author: M. Viot
author: Marina Kleptsyna
author: Alain Le Breton
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 2003-06-23T00:00:00Z
review text: Summary: Various methods to derive new formulas for the Laplace transforms of some quadratic forms of Gaussian sequences are discussed. In the general setting, an approach based on the resolution of an appropriate auxiliary filtering problem is developed; it leads to a formula in terms of the solutions of Volterra-type recursions describing characteristics of the corresponding optimal filter. In the case of Gauss-Markov sequences, where the previous equations reduce to ordinary forward recursive equations, an alternative approach prices another formula; it involves the solution of a backward recursive equation. Comparing the different formulas for the Laplace transforms, various relationships between the corresponding entries are identified. In particular, relationships between the solutions of matched forward and backward Riccati equations are thus proved probabilistically; they are proved again directly. In various specific cases, a further analysis of the concerned equations lead to completely explicit formulas for the Laplace transform.
Mathematics Subject Classification ID: 60G15
Mathematics Subject Classification ID: 60G44
Mathematics Subject Classification ID: 62M20
zbMATH DE Number: 1933141
zbMATH Keywords: Gaussian sequences
zbMATH Keywords: quadratic forms
zbMATH Keywords: Laplace transform
zbMATH Keywords: martingale
zbMATH Keywords: optimal filtering
zbMATH Keywords: filtering error
MaRDI profile type: Publication
full work available at URL: s1048953302000266
OpenAlex ID: W2038443638
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description: scientific article; zbMATH DE number 1933141