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AIP conference proceedings

Quantum Stochastic Processes

Joseph Spring, Alexander I. Lvovsky

Published Jan 1, 2009• 2 citations
Publisher

Abstract

We consider quantum analogues of n‐parameter stochastic processes, associated integrals and martingale properties extending classical results obtained in [1, 2, 3], and quantum results in [4, 5, 6, 7, 8, 9, 10].
DOI
10.1063/1.3131370

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Martingales are fundamental stochastic process used to model the concept of fair game. They have a multitude of applications in the real world that include, random walks, Brownian motion, gamblers fortunes and survival analysis, Just as commutative integration theory may be realised as a special case of the more general non-commutative theory for integrals, so too, we find classical probability may be realised as a limiting, special case of quantum probability theory. In this thesis we are concerned with the development of multiparameter quantum stochastic integrals extending non-commutative constructions to the general n parameter case, these being multiparameter quantum stochastic integrals over the positive n - dimensional plane, employing martingales as integrator. The thesis extends previous analogues of type one, and type two stochastic integrals, for both Clifford and quasi free representations. As with one and two dimensional parameter sets, the stochastic integrals constructed form orthogonal, centred L2 - martingales, obeying isometry properties. We further explore analogues for weakly adapted processes, properties relating to the resulting quantum stochastic integrals, develop analogues to Fubini’s theorem, and explore applications for quantum stochastic integrals in a security setting.
Classical stochastic integration is based upon a probability space involving a filtration of sigma‐algebras. This construction lends itself to non‐commutative quantum analogues based for example, on a Hilbert space, a filtration of von Neumann algebras and gage. We recall a non‐commutative construction for the two parameter case, these being integrals in the plane, resulting in type one and type two stochastic integrals which are orthogonal, centred L2 — martingales, obeying isometry properties and develop the construction to obtain an Ito‐Clifford Wong‐Zakai martingale representation.