Applebaum, D. (2009) Universal Malliavin calculus in Fock and Levy-Ito spaces. Communications on Stochastic Analysis, 3. pp. 119-41. ISSN 0973-9599
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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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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. |
Dates: |
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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: | 16 Nov 2015 11:48 |
Status: | Published |
Refereed: | Yes |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:9804 |