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arXiv · 0808.2593

Universal Malliavin Calculus in Fock and Lévy-Itô Spaces

David Applebaum
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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 $L^2$-equivalence of norms is proved and an abstract version of the Itô-Skorohod isometry is established. We then outline a new proof of Itô's chaos expansion of complex Lévy-Itô space in terms of multiple Wiener-Lévy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform now identifies Lévy-Itô space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general Lévy process. In particular we establish maximal domains of these operators and obtain the Itô-Skorohod isometry on its maximal domain.

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David Applebaum. 2008-08-19. Universal Malliavin Calculus in Fock and Lévy-Itô Spaces. https://arxiv.org/abs/0808.2593

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