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Universal Malliavin Calculus in Fock and Levy-It^o Spaces

Summary: Universal Malliavin Calculus in Fock and
L´evy-It^o Spaces
David Applebaum,
Probability and Statistics Department,
University of Sheffield,
Hicks Building, Hounsfield Road,
Sheffield, England, S3 7RH
e-mail: D.Applebaum@sheffield.ac.uk
We review and extend Lindsay's work on abstract gradient and diver-
gence operators in Fock space over a general complex Hilbert space. Pre-
cise expressions for the domains are given, the L2
-equivalence of norms is
proved and an abstract version of the It^o-Skorohod isometry is established.
We then outline a new proof of It^o's chaos expansion of complex L´evy-It^o
space in terms of multiple Wiener-L´evy integrals based on Brownian mo-
tion and a compensated Poisson random measure. The duality transform
now identifies L´evy-It^o space as a Fock space. We can then easily obtain
key properties of the gradient and divergence of a general L´evy process.
In particular we establish maximal domains of these operators and obtain


Source: Applebaum, David - Department of Probability and Statistics, University of Sheffield


Collections: Mathematics