Fock Space and the Poisson Process

Abstract

Using the Wiener–Poisson isomorphism, we show that if $(F_t)_{0 \leq t \leq 1}$ is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, and if $\hat{M}_t$ is the operator corresponding to multiplication by $M_t = \int_0^t F_s dX_s$, then for any regular self-adjoint quantum semimartingale $J = (J_t)_{0 \leq t \leq 1}$, the essentially self-adjoint quantum semimartingale $(\hat{M}_t + J_t)_{0 \leq t \leq 1}$ satisfies the quantum Ito formula.

We also introduce a generalisation of the Poisson process to a measure space $(M, \mathcal{M} , \mu)$ as an isometry $I : L^2 (M, \mathcal{M}, \mu) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ and give a new construction of the generalised Wiener–Poisson isomorphism ${\mathcal{W}_I} : \mathfrak{F}_+ (L^2(M)) \to L^2 (\Omega, \mathcal{F}, {\mathbb{P}} )$ using exponential vectors. Using C*-algebra theory, given any measure space we construct a canonical generalised Poisson process. Unlike other constructions, we make no a priori use of Poisson measures.