Abstract
As stochastic gradient and Skorohod integral operators, is an adjoint pair of unbounded operators on Guichardet Spaces. In this paper, we define an adjoint pair of operator , where with being the conditional expectation operator. We show that (resp.) is essentially a kind of localization of the stochastic gradient operators (resp. Skorohod integral operators δ). We examine that and satisfy a local CAR (canonical ani-communication relation) and forms a mutually orthogonal operator sequence although each is not a projection operator. We find that is s-adapted operator if and only if is s-adapted operator. Finally we show application exponential vector formulation of QS calculus.
Highlights
The quantum stochastic calculus [4] [6] developed by Hudson and Parthasarathy is essentially a noncommutative extension of classical Ito stochastic calculus
The quantum stochastic calculus has been extended by Hitsuda is by means of the Hitsuda-Skorohod integral of anticipative process [3] [9] and the related gradient operator of Malliavin calculus
Maximality of operator domains is demon-strated for these QS integrals on Guichardet spaces
Summary
The quantum stochastic calculus [4] [6] 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”, [2] which are in continuous time. The quantum stochastic calculus has been extended by Hitsuda is by means of the Hitsuda-Skorohod integral of anticipative process [3] [9] and the related gradient operator of Malliavin calculus. In this noncausal formulation the action of each QS integral is defined explicitly on Fock space vectors, and the essential quantum. We show application exponential vector formulation of QS calculus
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