Quantum stochastic calculus with maximal operator domains
Stéphane Attal and J. Martin Lindsay
Source: Ann. Probab. Volume 32, Number 1A (2004), 488-529.
Abstract
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.
Primary Subjects: 81S25
Keywords: Quantum stochastic; Fock space; Itô calculus; noncausal; chaotic representation property; Malliavin calculus; noncommutative probability
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1078415843
Digital Object Identifier: doi:10.1214/aop/1078415843
Mathematical Reviews number (MathSciNet):
MR2040790
Zentralblatt MATH identifier:
02100726
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Mathematical Reviews (MathSciNet):
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Digital Object Identifier: doi:10.1007/s004400050194
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Mathematical Reviews (MathSciNet):
MR660187
Digital Object Identifier: doi:10.1016/0022-1236(82)90036-2
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Mathematical Reviews (MathSciNet):
MR1712583
Digital Object Identifier: doi:10.1007/s002200050682
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Mathematical Reviews (MathSciNet):
MR493402
Zentralblatt MATH:
0265.43008
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Mathematical Reviews (MathSciNet):
MR745686
Digital Object Identifier: doi:10.1007/BF01258530
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Mathematical Reviews (MathSciNet):
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Digital Object Identifier: doi:10.1007/BF01210951
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Mathematical Reviews (MathSciNet):
MR1151553
Digital Object Identifier: doi:10.1016/0022-1236(92)90126-4
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Mathematical Reviews (MathSciNet):
MR816708
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Mathematical Reviews (MathSciNet):
MR1012630
Digital Object Identifier: doi:10.1007/BF01219655
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Mathematical Reviews (MathSciNet):
MR1757598
Digital Object Identifier: doi:10.1007/s004400050261
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MR1222649
Zentralblatt MATH:
0773.60098
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Mathematical Reviews (MathSciNet):
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Mathematical Reviews (MathSciNet):
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Mathematical Reviews (MathSciNet):
MR1176275
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Mathematical Reviews (MathSciNet):
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Zentralblatt MATH:
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Mathematical Reviews (MathSciNet):
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Mathematical Reviews (MathSciNet):
MR1482934
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Mathematical Reviews (MathSciNet):
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G.F. Vincent-Smith, The Itô formula for quantum semimartingales, Proc. London Math. Soc. 75 (1997), 671--720.
Mathematical Reviews (MathSciNet):
MR1466665
Digital Object Identifier: doi:10.1112/S0024611597000476
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