Quantum Stochastic Integral Representations on Interacting Fock Space

2014 ◽  
Vol 28 (3) ◽  
pp. 1007-1027 ◽  
Author(s):  
Yuanbao Kang ◽  
Caishi Wang
Keyword(s):  
Fock Space ◽  
Stochastic Integral ◽  
Integral Representations ◽  
Stochastic Integral Representations ◽  
Interacting Fock Space ◽  
Quantum Stochastic Integral

Asymptotic Distributions of the Least-Squares Estimators and Test Statistics in the Near Unit Root Model with Non-Zero Initial Value and Local Drift and Trend

1994 ◽  
Vol 10 (5) ◽  
pp. 937-966 ◽  
Author(s):  
Seiji Nabeya ◽  
Bent E. Sørensen
Keyword(s):  
Characteristic Function ◽  
Asymptotic Distribution ◽  
Unit Root ◽  
Stochastic Integral ◽  
Integral Representations ◽  
Characteristic Functions ◽  
General Interest ◽  
Initial Value ◽  
Stochastic Integral Representations ◽  
Near Unit Root

This paper considers the distribution of the Dickey-Fuller test in a model with non-zero initial value and drift and trend. We show how stochastic integral representations for the limiting distribution can be derived either from the local to unity approach with local drift and trend or from the continuous record asymptotic results of Sørensen [29]. We also show how the stochastic integral representations can be utilized as the basis for finding the corresponding characteristic functions via the Fredholm approach of Nabeya and Tanaka [16,17], This “link” between those two approaches may be of general interest. We further tabulate the asymptotic distribution by inverting the characteristic function. Using the same methods, we also find the characteristic function for the asymptotic distribution for the Schmidt-Phillips [26] unit root test. Our results show very clearly the dependence of the various tests on the initial value of the time series.

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.

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