Diagonal versions and quantum stochastic integrals on the symmetric Fock space with nonadapted integrands

Abstract

We develop a quantum stochastic calculus for nonadapted processes and the three standard integrators on the symmetric Fock space over an arbitrary Polish space. The integrable processes are restricted to a class of operators which possess some kernel description. Our approach is mainly based on the concept of diagonalized versions [16]. In the case of the real line we consider connections to the usual stochastic and quantum stochastic calculi and derive Ito-formulae. Besides the standard integrals with two integrands and one integrator these formulae lead to a new type of quantum stochastic integrals which has two integrators and three integrands.