Quasi-free stochastic integral representation theorems over the CCR

1988 ◽  
Vol 104 (2) ◽  
pp. 383-398 ◽  
Author(s):  
Ivan F. Wilde
Keyword(s):  
Hilbert Space ◽  
Integral Representation ◽  
Stochastic Integral ◽  
Stochastic Integrals ◽  
Representation Theorems ◽  
Martingale Representation ◽  
Stochastic Integral Representation ◽  
Vector Valued ◽  
General Vector

AbstractIt is shown that each vector in the Hilbert space of certain quasi-free representations of the CCR can be written uniquely in terms of quantum stochastic integrals. As a consequence, we obtain general vector-valued and operator-valued boson quantum martingale representation theorems.

scholarly journals Quantum stochastic integrals as belated integrals

1992 ◽  
Vol 34 (2) ◽  
pp. 165-173
Author(s):  
Chris Barnett ◽  
J. M. Lindsay ◽  
Ivan F. Wilde
Keyword(s):  
Hilbert Space ◽  
Brownian Motion ◽  
Stochastic Integral ◽  
Stochastic Integrals ◽  
Itô Integral ◽  
Ito Integral ◽  
Finite Sum ◽  
Vector Valued ◽  
Quantum Stochastic Integral

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.

INTEGRAL REPRESENTATION OF QUANTUM MARTINGALES

2005 ◽  
Vol 08 (01) ◽  
pp. 55-72 ◽  
Author(s):  
UN CIG JI ◽  
KALYAN B. SINHA
Keyword(s):  
Integral Representation ◽  
Wide Class ◽  
Stochastic Integral ◽  
Integral Kernel ◽  
Integral Operators ◽  
Second Quantization ◽  
Kernel Operator ◽  
Stochastic Integral Representation ◽  
Singular Kernels

A stochastic integral representation in terms of generalized integral kernel operator is proved for a wide class of quantum martingales which includes regular martingales and the martingales determined by the second quantization of integral operators containing singular kernels.

The convolution metric dg

1985 ◽  
Vol 98 (3) ◽  
pp. 533-540 ◽  
Author(s):  
J. E. Yukich
Keyword(s):  
Integral Representation ◽  
Gaussian Process ◽  
Probability Density ◽  
Stochastic Integral ◽  
Empirical Measure ◽  
Probability Measures ◽  
Mild Conditions ◽  
Iterated Logarithm ◽  
Stochastic Integral Representation ◽  
Image Position

SummaryWe introduce and study a new metric on denned bywhere is the space of probability measures on ℝk and where g: ℝk→ is a probability density satisfying certain mild conditions. The metric dg, relatively easy to compute, is shown to have useful and interesting properties not enjoyed by some other metrics on . In particular, letting pn denote the nth empirical measure for P, it is shown that under appropriate conditions satisfies a compact law of the iterated logarithm, converges in probability to the supremum of a Gaussian process, and has a useful stochastic integral representation.

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