Universal Malliavin Calculus in Fock and L\'{e}vy-It\^{o} Spaces
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 -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.
Keywords
Cite
@article{arxiv.0808.2593,
title = {Universal Malliavin Calculus in Fock and L\'{e}vy-It\^{o} Spaces},
author = {David Applebaum},
journal= {arXiv preprint arXiv:0808.2593},
year = {2008}
}
Comments
Submitted to the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)