Table of Contents
International Journal of Stochastic Analysis
Volume 2011, Article ID 247329, 89 pages
http://dx.doi.org/10.1155/2011/247329
Research Article

Multiresolution Hilbert Approach to Multidimensional Gauss-Markov Processes

1Lewis-Sigler Institute, Princeton University, Carl Icahn Laboratory, Princeton, NJ 08544, USA
2Laboratory of Mathematical Physics, The Rockefeller University, New York, NY 10065, USA
3Mathematical Neuroscience Laboratory, Collège de France, CIRB, 11 Place Marcelin Berthelot, CNRS UMR 7241 and INSERM U 1050, Université Pierre et Marie Curie ED, 158 and Memolife PSL, 75005 Paris, France
4INRIA BANG Laboratory, Paris, France

Received 28 April 2011; Accepted 6 October 2011

Academic Editor: Agnès Sulem

Copyright © 2011 Thibaud Taillefumier and Jonathan Touboul. 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 study of the 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.