Abstract
In this paper, we define expectation of f∈E, i.e. E(f)=f(?), accordingto Wiener-Ito-Segal isomorphic relation between Guichardet-Fock space F and Wienerspace W. Meanwhile, we prove a moment identity for the Skorohod integrals aboutvacuum state.
Highlights
The quantum stochastic calculus [1] [2] developed by Hudson and Parthasarathy is essentially a noncommutative extension of classical Ito stochastic calculus
It is well known that Guichardet-Fock space F and Wiener space W are WienerIto-Segal isomorphic
Fixing a complex separable Hilbert space η, Guichardet-Fock space tensor product η ⊗ L2 (Γ), which we identify with the space of square-integrable functions L2 (Γ;η), and is denoted by F
Summary
The quantum stochastic calculus [1] [2] developed by Hudson and Parthasarathy is essentially a noncommutative extension of classical Ito stochastic calculus. In this theory, annihilation, creation, and number operator processes in boson Fock space play the role of “quantum noises”, [3] which are in continuous time. In 2002, Attal [4] discussed and extended quantum stochastic calculus by means of the Skorohod integral of anticipation processes and the related gradient operator on Guichardet-Fock spaces. Privault [5] [6] developed a Malliavin-type theory of stochastic calculus on Wiener spaces and showed its several interesting applications.
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