Quantum stochastic calculus with maximal operator domains
Article, Other literature type
English
OPEN
Attal, Stéphane
;
Lindsay, J. Martin
(2004)
 Publisher: The Institute of Mathematical Statistics

Journal:
(issn: 00911798)

Related identifiers:
doi: 10.1214/aop/1078415843

Subject:
Itô calculus  81S25  Quantum stochastic  noncausal  noncommutative probability  Fock space  chaotic representation property  Malliavin calculus
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 discoveryone 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 AttalMeyer equations for implicit definition of quantum stochastic integrals.

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