814 research outputs found
Infinite Dimensional Quantum Information Geometry
We present the construction of an infinite dimensional Banach manifold of
quantum mechanical states on a Hilbert space H using different types of small
perturbations of a given Hamiltonian. We provide the manifold with a flat
connection, called the exponential connection, and comment on the possibility
of introducing the dual mixture connection.Comment: Proceedings of the Disordered and Complex Systems, King's College,
London, 10-14 July 2000 (satellite meeting of the ICMP2000
Dual Connections in Nonparametric Classical Information Geometry
We construct an infinite-dimensional information manifold based on
exponential Orlicz spaces without using the notion of exponential convergence.
We then show that convex mixtures of probability densities lie on the same
connected component of this manifold, and characterize the class of densities
for which this mixture can be extended to an open segment containing the
extreme points. For this class, we define an infinite-dimensional analogue of
the mixture parallel transport and prove that it is dual to the exponential
parallel transport with respect to the Fisher information. We also define
{\alpha}-derivatives and prove that they are convex mixtures of the extremal
(\pm 1)-derivatives
Wiener Chaos and the Cox-Ingersoll-Ross model
In this we paper we recast the Cox--Ingersoll--Ross model of interest rates
into the chaotic representation recently introduced by Hughston and Rafailidis.
Beginning with the ``squared Gaussian representation'' of the CIR model, we
find a simple expression for the fundamental random variable X. By use of
techniques from the theory of infinite dimensional Gaussian integration, we
derive an explicit formula for the n-th term of the Wiener chaos expansion of
the CIR model, for n=0,1,2,.... We then derive a new expression for the price
of a zero coupon bond which reveals a connection between Gaussian measures and
Ricatti differential equations.Comment: 27 page
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