A generalization of the conservation integral

Starting from the scheme given by Hudson and Parthasarathy [7,11] we extend the conservation integral to the case where the underlying operator does not commute with the time observable. It turns out that there exist two extensions, a left and a right conservation integral. Moreover, Itô's formula demands for a third integral with two integrators. Only the left integral shows similar continuity properties to that derived in [11] used for extending the integral to more than simple integrands. In another approach we extend the previous notions for the integrals to much larger domains of definition and to much more processes, including anticipating ones. Similar to [5,10], we use the Skorohod integral and the Malliavin derivative acting on a symmetric Fock space [3,4]. It appears that this formulation unifies all three integrals in the double integrator one.