Quantum Stochastic Integral Representations on Interacting Fock Space

Abstract

A quantum stochastic integration theory on interacting Fock space (IFS) has been established in Crismale (Commun Stoch Anal 1(2):321–341, 2007). In this paper, we firstly put forward a family of left–right conditional expectations \(E_{t}\), for \(t\in R_{+}\), on the von Neumann algebra \(\mathcal {V}\) generated by the creation, annihilation and gauge operators acting on IFS over \({L}^{2}(R_{+})\). Next, we develop a generalized quantum stochastic integral. Finally, we prove that any process \(\Xi \in (\mathcal {V}_{t})_{t\in R_{+}}\) admits a quantum stochastic integral representation.