The Clark–Ocone formula for vector valued Wiener functionals

The classical representation of random variables as the Itô integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped with a filtration induced by a resolution of the identity on the Cameron–Martin space. The Itô integral...

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Bibliographic Details
Published inJournal of functional analysis Vol. 229; no. 1; pp. 143 - 154
Main Authors Mayer-Wolf, E., Zakai, M.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.12.2005
Subjects
Banach space valued random variables
Clark–Ocone formula
Measure preserving transformations
Rotations
Weakly adapted operators
Weakly adapted operators
Measure preserving transformations
Clark–Ocone formula
Banach space valued random variables
primary 60H07
60H25
Rotations
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Summary:The classical representation of random variables as the Itô integral of nonanticipative integrands is extended to include Banach space valued random variables on an abstract Wiener space equipped with a filtration induced by a resolution of the identity on the Cameron–Martin space. The Itô integral is replaced in this case by an extension of the divergence to random operators, and the operators involved in the representation are adapted with respect to this filtration in a suitably defined sense. A complete characterization of measure preserving transformations in Wiener space is presented as an application of this generalized Clark–Ocone formula.
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2004.11.012