Universal Malliavin Calculus in Fock and Lévy-Itô Spaces

Details Universal Malliavin Calculus in Fock and Lévy-Itô Spaces Universal Malliavin Calculus in Fock and Lévy-Itô Spaces

2008-08-19

arxiv.org/abs/0808.2593v1

Mathematics

Probability

Submitted to the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.i

Scientific paper

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 It\^{o}-Skorohod isometry is established. We then outline a new proof of It\^{o}'s chaos expansion of complex L\'{e}vy-It\^{o} space in terms of multiple Wiener-L\'{e}vy integrals based on Brownian motion and a compensated Poisson random measure. The duality transform now identifies L\'{e}vy-It\^{o} space as a Fock space. We can then easily obtain key properties of the gradient and divergence of a general L\'{e}vy process. In particular we establish maximal domains of these operators and obtain the It\^{o}-Skorohod isometry on its maximal domain.

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