Quantum stochastic calculus with maximal operator domains.

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Quantum stochastic calculus with maximal operator domains. / Lindsay, J. Martin; Attal, Stéphane.
In: Annals of Probability, Vol. 32, No. 1a, 01.01.2004, p. 488-529.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Lindsay, JM & Attal, S 2004, 'Quantum stochastic calculus with maximal operator domains.', Annals of Probability, vol. 32, no. 1a, pp. 488-529. https://doi.org/10.1214/aop/1078415843

APA

Lindsay, J. M., & Attal, S. (2004). Quantum stochastic calculus with maximal operator domains. Annals of Probability, 32(1a), 488-529. https://doi.org/10.1214/aop/1078415843

Vancouver

Lindsay JM, Attal S. Quantum stochastic calculus with maximal operator domains. Annals of Probability. 2004 Jan 1;32(1a):488-529. doi: 10.1214/aop/1078415843

Author

Lindsay, J. Martin ; Attal, Stéphane. / Quantum stochastic calculus with maximal operator domains. In: Annals of Probability. 2004 ; Vol. 32, No. 1a. pp. 488-529.

Bibtex

@article{3808dfaac38d46449374d83805d9300c,

title = "Quantum stochastic calculus with maximal operator domains.",

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 It{\^o} formulas for operator products as sums of stochastic integrals. The calculus has seen two reformulations since its discovery—one closely related to classical It{\^o} 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.",

keywords = "Quantum stochastic, Fock space, It{\^o} calculus, noncausal, chaotic representation property, Malliavin calculus, noncommutative probability",

author = "Lindsay, {J. Martin} and St{\'e}phane Attal",

note = "RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics",

year = "2004",

month = jan,

day = "1",

doi = "10.1214/aop/1078415843",

language = "English",

volume = "32",

pages = "488--529",

journal = "Annals of Probability",

publisher = "Institute of Mathematical Statistics",

number = "1a",

}

RIS

TY - JOUR

T1 - Quantum stochastic calculus with maximal operator domains.

AU - Lindsay, J. Martin

AU - Attal, Stéphane

N1 - RAE_import_type : Journal article RAE_uoa_type : Pure Mathematics

PY - 2004/1/1

Y1 - 2004/1/1

N2 - 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.

AB - 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.

KW - Quantum stochastic

KW - Fock space

KW - Itô calculus

KW - noncausal

KW - chaotic representation property

KW - Malliavin calculus

KW - noncommutative probability

U2 - 10.1214/aop/1078415843

DO - 10.1214/aop/1078415843

M3 - Journal article

VL - 32

SP - 488

EP - 529

JO - Annals of Probability

JF - Annals of Probability

IS - 1a

ER -

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