Quantum stochastic calculus with maximal operator domains.
type: Item
instance of: scholarly article
title: Quantum stochastic calculus with maximal operator domains.
zbMATH Open document ID: 1053.81053
author: S. Attal
author: J. Martin Lindsay
published in: The Annals of Probability
publication date: 2004-09-15T00:00:00Z
review text: The purpose of this paper is to introduce a new formulation in quantum stochastic (QS) calculus and to unify the following two existent approaches: one closely related to classical Itô calculus; and the other to noncausal stochastic analysis and Malliavin calculus. The authors' extension of QS calculus with new formulation provides QS integrals with their natural maximal domains. The most remarkable features consist in the point that operator adaptedness and conditional expectations as well as stochastic integrals are all defined simply in terms of the orthogonal projections of the time filtration of Fock space, together with sections of the adapted gradient operator. Moreover, free from exponential vector domains, their newly defined stochastic integrals with respect to the basic processes of QS calculus enjoy quantum Itô formula for operator products and can be composed as sums of stochastic integrals. Lastly, as the main application, proved is existence and uniqueness for the Attal-Meyer equations for implicit definition of QS integrals. As for other related works, see e.g. \textit{J. M. Lindsay} and \textit{S. J. Wills} [Probab.\ Theory Related Fields 116, No.4, 505--543 (2000; Zbl 1079.81542)].
reviewed by: Isamu Dôku
Mathematics Subject Classification ID: 81S25
Mathematics Subject Classification ID: 46N50
Mathematics Subject Classification ID: 47N50
Mathematics Subject Classification ID: 60H05
zbMATH DE Number: 2100726
zbMATH Keywords: quantum stochastic calculus
zbMATH Keywords: fock space
zbMATH Keywords: Itô calculus
zbMATH Keywords: noncausal
zbMATH Keywords: quantum stochastic integrals
zbMATH Keywords: noncommutative probability
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1078415843
OpenAlex ID: W2082578377
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DOI: 10.1214/AOP/1078415843
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description: scientific article; zbMATH DE number 2100726