Multiresolution Hilbert approach to multidimensional Gauss-Markov processes
type: Item
instance of: scholarly article
title: Multiresolution Hilbert approach to multidimensional Gauss-Markov processes
zbMATH Open document ID: 1259.60092
DOI: 10.1155/2011/247329
author: Thibaud Taillefumier
author: Jonathan Touboul
published in: International Journal of Stochastic Analysis
publication date: 2012-05-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.0533
review text: Summary: The study of multidimensional stochastic processes involves complex computations in intricate functional spaces. In particular, the diffusion processes, which include the practically important Gauss-Markov processes, are ordinarily defined through the theory of stochastic integration. Here, inspired by the Lévy-Ciesielski construction of the Wiener process, we propose an alternative representation of multidimensional Gauss-Markov processes as expansions on well-chosen Schauder bases, with independent random coefficients of normal law with zero mean and unit variance. We thereby offer a natural multiresolution description of the Gauss-Markov processes as limits of finite-dimensional partial sums of the expansion, that are strongly almost-surely convergent. Moreover, such finite-dimensional random processes constitute an optimal approximation of the process, in the sense of minimizing the associated Dirichlet energy under interpolating constraints. This approach allows for a simpler treatment of problems in many applied and theoretical fields, and we provide a short overview of applications we are currently developing.
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60H35
zbMATH DE Number: 6031631
zbMATH Keywords: optimal approximation of a process
zbMATH Keywords: Lévy-Ciesielski construction
zbMATH Keywords: Wiener process
zbMATH Keywords: Gauss-Markov processes
Wikidata QID: Q58688897
MaRDI profile type: Publication
OpenAlex ID: W2028385554
arXiv ID: 1009.0533
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description: scientific article; zbMATH DE number 6031631