Proceedings of the London Mathematical Society

Fock Space and the Poisson Process

Sureka Pathmanathan, G. F. Vincent-Smith

Published Nov 1, 2004• 0 citations

Abstract

Using the Wiener–Poisson isomorphism, we show that if (Ft)0 ⩽ t ⩽ 1 is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process (Xt)0 ⩽ t ⩽ 1, and if M ^ t is the operator corresponding to multiplication by M t = ∫ 0 t F s d X s , then for any regular self-adjoint quantum semimartingale J = ( J t ) 0 ⩽ t ⩽ 1 , the essentially self-adjoint quantum semimartingale ( M ^ t + J t ) 0 ⩽ t ⩽ 1 satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process to a measure space (M, M, μ) as an isometry I: L2 (M, M, μ) → L2(Ω, F, P) and give a new construction of the generalised Wiener–Poisson isomorphism WI: F+ (L2(M)) → L2 (Ω, F, 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. 2000 Mathematics Subject Classification 60G20, 60G35, 46L53, 81S25.

DOI: 10.1112/s0024611504014820

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Proceedings of the London Mathematical Society

Fock Space and the Poisson Process

Sureka Pathmanathan, G. F. Vincent-Smith

Published Nov 1, 2004• 0 citations

Publisher

Abstract

DOI: 10.1112/s0024611504014820

Related papers

The poisson process in quantum stochastic calculus

Shayanthan Pathmanathan

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. 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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.

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Mou-Hsiung Chang

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Stochastic Integration and Quantum Ito’s Formula

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Birkhäuser Basel eBooksJan 1, 19920 citations

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G. F. Vincent-Smith

The quantum Ito formula has so far been proved for regular (bounded) quantum semimartingales We give three different extensions to classes of essentially self-adjoint (unbounded) quantum semimartingales. The first extension is to quantum semimartingales whose integrands have chaos matrices controlled by certain scalar matrices The second extension is to perturbations of classical, possibly unbounded, quantum semimartingales by bounded quantum semimartingales The third extension is obtained by perturbing an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula by a regular quantum semimartingle.The quantum Ito formula has so far been proved for regular (bounded) quantum semimartingales We give three different extensions to classes of essentially self-adjoint (unbounded) quantum semimartingales. The first extension is to quantum semimartingales whose integrands have chaos matrices controlled by certain scalar matrices The second extension is to perturbations of classical, possibly unbounded, quantum semimartingales by bounded quantum semimartingales The third extension is obtained by perturbing an essentially self-adjoint quantum semimartingale satisfying the quantum Duhamel formula by a regular quantum semimartingle.

arXiv:math/0012183Dec 19, 20001 citation

A matrix formulation of quantum stochastic calculus

Alexander C. R. Belton

We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countably-infinite, direct-sum decomposition. A chaos matrix between two chaos spaces is a doubly-infinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to self-adjointness. This theory is used to re-formulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaos-matrix processes are defined using the Hitsuda-Skorokhod integral and Malliavin gradient,following Lindsay and Belavkin. A new way of defining adaptedness is developed and the consequent quantum product Ito formula is used to provide a genuine functional Ito formula for polynomials in a large class of unbounded processes, which include the Poisson process and Brownian motion. A new type of adaptedness, known as $\Omega$ -adaptedness, is defined. We show that quantum stochastic integrals of $\Omega$ -adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy: $U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+ G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $ where the coefficients are time-dependent, bounded, $\Omega$ -adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz. $(E,F,G,H)=(W-I,L,-WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$ where $W$ is unitary and $K$ self-adjoint, are necessary and sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.We develop the theory of chaos spaces and chaos matrices. A chaos space is a Hilbert space with a fixed, countably-infinite, direct-sum decomposition. A chaos matrix between two chaos spaces is a doubly-infinite matrix of bounded operators which respects this decomposition. We study operators represented by such matrices, particularly with respect to self-adjointness. This theory is used to re-formulate the quantum stochastic calculus of Hudson and Parthasarathy. Integrals of chaos-matrix processes are defined using the Hitsuda-Skorokhod integral and Malliavin gradient,following Lindsay and Belavkin. A new way of defining adaptedness is developed and the consequent quantum product Ito formula is used to provide a genuine functional Ito formula for polynomials in a large class of unbounded processes, which include the Poisson process and Brownian motion. A new type of adaptedness, known as $\Omega$-adaptedness, is defined. We show that quantum stochastic integrals of $\Omega$-adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy: $U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+ G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $ where the coefficients are time-dependent, bounded, $\Omega$-adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz. $(E,F,G,H)=(W-I,L,-WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$ where $W$ is unitary and $K$ self-adjoint, are necessary and sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.

Jan 1, 19987 citations

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.

\Omega

-adaptedness, is defined. We show that quantum stochastic integrals of

\Omega

-adapted processes are well-behaved; for instance, bounded processes have bounded integrals. We solve the appropriate modification of the evolution equation of Hudson and Parthasarathy: $U(t)=I+\int_{0}^{t}E(s)\mathrm{d}\Lambda(s)+F(s)\mathrm{d} A(s)+ G(s)U(s)\mathrm{d} A^{\dagger}(s)+H(s)U(s)\mathrm{d} s, $ where the coefficients are time-dependent, bounded,

\Omega

-adapted processes acting on the whole Fock space. We show that the usual conditions on the coefficients, viz. $(E,F,G,H)=(W-I,L,-WL^{*},iK+\mbox{$\frac{1}{2}$}LL^{*})$ where

W

is unitary and

K

self-adjoint, are necessary and sufficient conditions for the solution to be unitary. This is a very striking result when compared to the adapted case.