dc.description.abstract | 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. | en_US |