The semi-martingale property of the square of white noise integrators

The abstract commutation relations of the algebra of the square of white noise of Accardi, Lu, and Volovich are shown to be realized by operator processes acting on the Fock space of Accardi and Skeide which is very closely related to the Finite Difference Fock space of Boukas and Feinsilver. The processes are shown to satisfy the necessary conditions for inclusion in the framework of the representation free quantum stochastic calculus of Accardi,Fagnola,and Quaegebeur. The connection between the Finite-Difference operators and the creation, annihilation, and conservation operators on usual symmetric Boson Fock space is further studied