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January 2004 Quantum stochastic calculus with maximal operator domains
Stéphane Attal, J. Martin Lindsay
Ann. Probab. 32(1A): 488-529 (January 2004). DOI: 10.1214/aop/1078415843


Quantum stochastic calculus is extended in a new formulation in which its stochastic integrals achieve their natural and maximal domains. Operator adaptedness, conditional expectations and 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. Free from exponential vector domains, our stochastic integrals may be satisfactorily composed yielding quantum Itô formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery---one closely related to classical Itô calculus; the other to noncausal stochastic analysis and Malliavin calculus. Our theory extends both of these approaches and may be viewed as a synthesis of the two. The main application given here is existence and uniqueness for the Attal--Meyer equations for implicit definition of quantum stochastic integrals.


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Stéphane Attal. J. Martin Lindsay. "Quantum stochastic calculus with maximal operator domains." Ann. Probab. 32 (1A) 488 - 529, January 2004.


Published: January 2004
First available in Project Euclid: 4 March 2004

zbMATH: 1053.81053
MathSciNet: MR2040790
Digital Object Identifier: 10.1214/aop/1078415843

Primary: 81S25

Keywords: chaotic representation property , Fock space , Itô calculus , Malliavin calculus , noncausal , noncommutative probability , Quantum stochastic

Rights: Copyright © 2004 Institute of Mathematical Statistics

Vol.32 • No. 1A • January 2004
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