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Fock Space and the Poisson Process
Author(s) -
Pathmanathan S.,
VincentSmith G. F.
Publication year - 2004
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611504014820
Subject(s) - mathematics , semimartingale , fock space , poisson distribution , measure (data warehouse) , bounded function , isomorphism (crystallography) , compound poisson process , space (punctuation) , pure mathematics , combinatorics , discrete mathematics , mathematical analysis , poisson process , quantum mechanics , linguistics , chemistry , philosophy , database , crystal structure , crystallography , statistics , physics , computer science
Using the Wiener–Poisson isomorphism, we show that if ( F t ) 0 ⩽ t ⩽ 1 is a real, bounded, predictable process adapted to the filtration of a compensated Poisson process ( X t ) 0 ⩽ t ⩽ 1 , and ifM ^ 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 ⩽ 1satisfies the quantum Ito formula. We also introduce a generalisation of the Poisson process to a measure space ( M , M, μ) as an isometry I : L 2 ( M , M, μ) → L 2 (Ω, F, P) and give a new construction of the generalised Wiener–Poisson isomorphism W I : F + ( L 2 ( M )) → L 2 (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.