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Universal Malliavin calculus in Fock and Levy-Ito spaces

Applebaum, D. (2009) Universal Malliavin calculus in Fock and Levy-Ito spaces. Communications on Stochastic Analysis, 3. pp. 119-41. ISSN 0973-9599

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Abstract

We review and extend Lindsay's work on abstract gradient and divergence operators in Fock space over a general complex Hilbert space. Precise 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 Levy-It^o space in terms of multiple Wiener-Levy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform now identies Levy-It^o space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general Levy process. In particular we establish maximal domains of these operators and obtain the It^o-Skorohod isometry on its maximal domain.

Item Type: Article
Copyright, Publisher and Additional Information: This is an author produced version of a paper subsequently published in Communications on Stochastic Processes.
Keywords: Fock space, exponential vector, universal annihilation and creation operators, number operator, Lindsay-Malliavin transform, Levy process, multiple Wiener-Levy integrals, It^o representation theorem, chaos decomposition, duality transform, stochastic (Doleans- Dade) exponential, gradient, divergence, Malliavin derivative, It^o-Skorohod isometry.
Institution: The University of Sheffield
Academic Units: The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield)
Depositing User: Miss Anthea Tucker
Date Deposited: 30 Sep 2009 11:26
Last Modified: 08 Feb 2013 16:59
Status: Published
Refereed: Yes
URI: http://eprints.whiterose.ac.uk/id/eprint/9804

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