UNIQUENESS OF INTEGRANDS IN QUANTUM STOCHASTIC INTEGRAL

As a general study for uniqueness of integrands in quantum martingale representation, we present a necessary and sufficient condition for uniqueness of integrands in a quantum stochastic integral. Also, several equivalent conditions to the necessary and sufficient condition are studied.

MALLIAVIN CALCULUS AND SKOROHOD INTEGRATION FOR QUANTUM STOCHASTIC PROCESSES

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space [Formula: see text] and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over [Formula: see text]. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For [Formula: see text], the divergence operator is shown to coincide with the Hudson–Parthasarathy quantum stochastic integral for adapted integrable processes and with the noncausal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.

Quantum stochastic integrals as belated integrals

Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.