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.

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 The Itô Integral with respect to an Infinite Dimensional Lévy Process: A Series Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-14 ◽  
Author(s):  
Stefan Tappe
Keyword(s):  
Hilbert Space ◽  
Lévy Process ◽  
Stochastic Integral ◽  
Levy Processes ◽  
Levy Process ◽  
Stochastic Integration ◽  
Itô Integral ◽  
Infinite Dimensional ◽  
Ito Integral ◽  
Alternative Construction

We present an alternative construction of the infinite dimensional Itô integral with respect to a Hilbert space valued Lévy process. This approach is based on the well-known theory of real-valued stochastic integration, and the respective Itô integral is given by a series of Itô integrals with respect to standard Lévy processes. We also prove that this stochastic integral coincides with the Itô integral that has been developed in the literature.

Hilbert transform of G-Brownian local time

2014 ◽  
Vol 14 (04) ◽  
pp. 1450006 ◽  
Author(s):  
Litan Yan ◽  
Qinghua Zhang ◽  
Bo Gao
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Hilbert Transform ◽  
Natural Extension ◽  
Itô Integral ◽  
Brownian Local Time ◽  
Ito Integral ◽  
Natural Result ◽  
The Hilbert Transform

Let B be a G-Brownian motion with quadratic process 〈B〉 under the G-expectation. In this paper, we consider the integrals [Formula: see text] We show that the integral diverges and the convergence [Formula: see text] exists in 𝕃2 for all a ∈ ℝ, t > 0. This shows that [Formula: see text] coincides with the Hilbert transform of the local time [Formula: see text] of G-Brownian motion B for every t. The functional is a natural extension to classical cases. As a natural result we get a sublinear version of Yamada's formula [Formula: see text] where the integral is the Itô integral under the G-expectation.

scholarly journals The Itô integral for Brownian motion in vector lattices: Part 1

2015 ◽  
Vol 423 (1) ◽  
pp. 797-819 ◽  
Author(s):  
Jacobus J. Grobler ◽  
Coenraad C.A. Labuschagne
Keyword(s):  
Brownian Motion ◽  
Itô Integral ◽  
Vector Lattices ◽  
Ito Integral

scholarly journals MALLIAVIN CALCULUS AND SKOROHOD INTEGRATION FOR QUANTUM STOCHASTIC PROCESSES

2001 ◽  
Vol 04 (01) ◽  
pp. 11-38 ◽  
Author(s):  
UWE FRANZ ◽  
RÉMI LÉANDRE ◽  
RENÉ SCHOTT
Keyword(s):  
Fock Space ◽  
Stochastic Integral ◽  
Real Hilbert Space ◽  
Sufficient Conditions ◽  
Stochastic Integrals ◽  
Bounded Operators ◽  
Divergence Operator ◽  
Derivation Operator ◽  
Quantum Stochastic Processes ◽  
Quantum Stochastic Integral

A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space [Formula: see text] and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over [Formula: see text]. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For [Formula: see text], the divergence operator is shown to coincide with the Hudson–Parthasarathy quantum stochastic integral for adapted integrable processes and with the noncausal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.

scholarly journals GIBBS MEASURES WITH DOUBLE STOCHASTIC INTEGRALS ON A PATH SPACE

2009 ◽  
Vol 12 (01) ◽  
pp. 135-152 ◽  
Author(s):  
VOLKER BETZ ◽  
FUMIO HIROSHIMA
Keyword(s):  
Brownian Motion ◽  
Interaction Energy ◽  
Quantum Electrodynamics ◽  
Stochastic Integral ◽  
Gibbs Measures ◽  
Stochastic Integrals ◽  
Infinite Volume ◽  
Nonrelativistic Quantum ◽  
Volume Limit ◽  
Infinite Volume Limit

We investigate Gibbs measures relative to Brownian motion in the case when the interaction energy is given by a double stochastic integral. In the case when the double stochastic integral is originating from the Pauli–Fierz model in nonrelativistic quantum electrodynamics, we prove the existence of its infinite volume limit.

On the path regularity of a stochastic process in a hilbert space, defined by the ito integral

Stochastics ◽  
1982 ◽  
Vol 6 (3-4) ◽  
pp. 315-322 ◽  
Author(s):  
G. Da prato ◽  
M. Iannelli ◽  
L. Tubaro
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Itô Integral ◽  
Ito Integral

THE RESTRICTION OF THE FRACTIONAL ITÔ INTEGRAL TO ADAPTED INTEGRANDS IS INJECTIVE

2005 ◽  
Vol 05 (01) ◽  
pp. 37-43 ◽  
Author(s):  
CHRISTIAN BENDER
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Integrability Condition ◽  
Skorohod Integral ◽  
Itô Integral ◽  
Ito Integral

We prove that the Wick–Itô–Skorohod integral with respect to a fractional Brownian motion is injective when it is restricted to a class of adapted integrands that satisfy an appropriate integrability condition.

scholarly journals The Itô integral for Brownian motion in vector lattices: Part 2

2015 ◽  
Vol 423 (1) ◽  
pp. 820-833 ◽  
Author(s):  
Jacobus J. Grobler ◽  
Coenraad C.A. Labuschagne
Keyword(s):  
Brownian Motion ◽  
Itô Integral ◽  
Vector Lattices ◽  
Ito Integral
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