Quantum stochastic calculus with maximal operator domains.
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)].
scientific article; zbMATH DE number 2100726
scientific article