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Sobolev derivatives and Itô decomposition formula for Gaussian processes using properties of their RKHSs

Sobolev derivatives and Itô decomposition formula for Gaussian processes using properties of... Malliavin calculus has been extensively developed for abstract Wiener spaces. It is of interest to develop the basic concepts of an infinite-dimensional calculus for an arbitrary Gaussian processX=(Xt), wheret $$ \in$$ T (T being a multiparameter set or, more generally, a complete separable metric space), bringing into evidence the properties of the covariance kernel (or, equivalently, the reproducing kernel Hilbert space) ofX. In this paper a definition of thekth Sobolev derivative is given and thekth chaos expansion of a functional is shown to be thekth-order divergence operator. An extension of Itô's decomposition formula is derived. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

Sobolev derivatives and Itô decomposition formula for Gaussian processes using properties of their RKHSs

Applied Mathematics and Optimization , Volume 26 (3) – Mar 10, 2005

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References (7)

Publisher
Springer Journals
Copyright
Copyright © 1992 by Springer-Verlag New York Inc.
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
DOI
10.1007/BF01371087
Publisher site
See Article on Publisher Site

Abstract

Malliavin calculus has been extensively developed for abstract Wiener spaces. It is of interest to develop the basic concepts of an infinite-dimensional calculus for an arbitrary Gaussian processX=(Xt), wheret $$ \in$$ T (T being a multiparameter set or, more generally, a complete separable metric space), bringing into evidence the properties of the covariance kernel (or, equivalently, the reproducing kernel Hilbert space) ofX. In this paper a definition of thekth Sobolev derivative is given and thekth chaos expansion of a functional is shown to be thekth-order divergence operator. An extension of Itô's decomposition formula is derived.

Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Mar 10, 2005

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