The Poisson Process in Quantum Stochastic Calculus

Given a compensated Poisson process $(X_t)_{t \geq 0}$ based on $(\Omega, \mathcal{F}, \mathbb{P})$, the Wiener-Poisson isomorphism $\mathcal{W} : \mathfrak{F}_+(L^2 (\mathbb{R}_+)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$ is constructed. We restrict the isomorphism to $\mathfrak{F}_+(L^2 [0,1])$ and prove some novel properties of the Poisson exponentials $\mathcal{E}(f) := \mathcal{W}(e(f))$. A new proof of the result $\Lambda_t + A_t + A^{\dagger}_t = \mathcal{W}^{-1}\widehat{X_t} \mathcal{W}$ is also given. The analogous results for $\mathfrak{F}_+(L^2 (\mathbb{R}_+))$ are briefly mentioned. The concept of a compensated Poisson process over $\mathbb{R}_+$ is generalised to any measure space $(M, \mathcal{M}, \mu)$ as an isometry $I : L^2(M, \mathcal{M}, \mu) \to L^2 (\Omega,\mathcal{F}, \mathbb{P})$ satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, $\mathcal{W}_I : \mathfrak{F}_+(L^2(M)) \to L^2 (\Omega, \mathcal{F}, \mathbb{P})$, using Charlier polynomials. Two alternative constructions of $\mathcal{W}_I$ are also provided, the first using exponential vectors and then deducing the connection with Charlier polynomials, and the second using the theory of reproducing kernel Hilbert spaces. Given any measure space $(M, \mathcal{M}, \mu)$, we construct a canonical generalised Poisson process $I : L^2 (M, \mathcal{M}, \mu) \to L^2(\Delta, \mathcal{B}, \mathbb{P})$, where $\Delta$ is the maximal ideal space, with $\mathcal{B}$ the completion of its Borel $\sigma$-field with respect to $\mathbb{P}$, of a $C^*$-algebra $\mathcal{A} \subseteq \mathfrak{B}(\mathfrak{F}_+(L^2(M)))$. The Gelfand transform $\mathcal{A} \to \mathfrak{B}(L^2(\Delta))$ is unitarily implemented by the Wiener-Poisson isomorphism $\mathcal{W}_I: \mathfrak{F}_+(L^2(M)) \to L^2(\Delta)$. This construction only uses operator algebra theory and makes no a priori use of Poisson measures. A new Fock space proof of the quantum Ito formula for $(\Lambda_t + A_t + A^{\dagger}_t)_{0 \leq t \leq 1}$ is given. If $(F_{\ \! \! t})_{0 \leq t \leq 1}$ is a real, bounded, predictable process with respect to a compensated Poisson process $(X_t)_{0 \leq t \leq 1}$, we show that if $M_t = \int_0^t F_s dX_s$, then on $\mathsf{E}_{\mathrm{lb}} := \mathrm{linsp} \{ e(f) : f \in L^2_{\mathrm{lb}}[0,1] \}$, $\mathcal{W}^{-1} \widehat{M_t} \mathcal{W} = \int_0^t \mathcal{W}^{-1} \widehat{F_s} \mathcal{W} (d\Lambda_s + dA_s + dA^{\dagger}_s),$ and that $(\mathcal{W}^{-1} \widehat{M_t} \mathcal{W})_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if $(J_t)_{0 \leq t \leq 1}$ is a regular self-adjoint quantum semimartingale, then $(\mathcal{W} \widehat{M_t} \mathcal{W}^{-1} + J_t)_{0 \leq t \leq 1}$ is an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula, and hence the quantum Ito formula. The equivalent result for the sum of a Brownian and Poisson martingale, provided that the sum is essentially self-adjoint with core $\mathsf{E}_{\mathrm{lb}}$, is also proved.