# The poisson process in quantum stochastic calculus

Pathmanathan, S. (2002) The poisson process in quantum stochastic calculus. PhD thesis, University of Oxford.

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## Abstract

Given a compensated Poisson process based on , the Wiener-Poisson isomorphism is constructed. We restrict the isomorphism to and prove some novel properties of the Poisson exponentials . A new proof of the result is also given. The analogous results for are briefly mentioned.

The concept of a compensated Poisson process over is generalised to any measure space as an isometry satisfying certain properties. For such a generalised Poisson process we recall the construction of the generalised Wiener-Poisson isomorphism, , using Charlier polynomials. Two alternative constructions of 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 , we construct a canonical generalised Poisson process , where is the maximal ideal space, with the completion of its Borel -field with respect to , of a -algebra . The Gelfand transform is unitarily implemented by the Wiener-Poisson isomorphism . 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 is given. If is a real, bounded, predictable process with respect to a compensated Poisson process , we show that if , then on , and that is an essentially self-adjoint quantum semimartingale. We prove, using the classical Ito formula, that if is a regular self-adjoint quantum semimartingale, then 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 , is also proved.

Item Type: Thesis (PhD) D - G > Functional analysisO - Z > Probability theory and stochastic processes Functional Analysis Group 46 Eprints Administrator 11 Mar 2004 29 May 2015 18:15

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