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Infinite Dimensional Analysis Quantum Probability and Related Topics

UNIQUENESS OF INTEGRANDS IN QUANTUM STOCHASTIC INTEGRAL

Un Cig Ji, Kalyan B. Sinha

Published Dec 1, 2006• 4 citations
Publisher

Abstract

As a general study for uniqueness of integrands in quantum martingale representation, we present a necessary and sufficient condition for uniqueness of integrands in a quantum stochastic integral. Also, several equivalent conditions to the necessary and sufficient condition are studied.
DOI
10.1142/s0219025706002561

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Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.
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.