On the relationship between Wick calculus and stochastic integration in the Lévy white noise analysis

2019 ◽  
Author(s):  
Maria M. Frei ◽  
Nikolai A. Kachanovsky
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Stochastic Integration ◽  
White Noise Analysis ◽  
Lévy White Noise ◽  
Wick Calculus ◽  
The Relationship

scholarly journals Interconnection between Wick multiplication and integration on spaces of nonregular generalized functions in the Lévy white noise analysis

2019 ◽  
Vol 11 (1) ◽  
pp. 70-88
Author(s):  
N.A. Kachanovsky ◽  
T.O. Kachanovska
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Stochastic Equations ◽  
Generalized Functions ◽  
Wick Product ◽  
Pettis Integral ◽  
White Noise Analysis ◽  
Lévy White Noise

We deal with spaces of nonregular generalized functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to describe a relationship between Wick multiplication and integration on these spaces. More exactly, we show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); and prove a theorem about a representation of the extended stochastic integral via the Pettis integral from the Wick product of the original integrand by a Lévy white noise. As examples of an application of our results, we consider some stochastic equations with Wick type nonlinearities.

Stochastic integration via white noise analysis

1997 ◽  
Vol 30 (1) ◽  
pp. 317-328 ◽  
Author(s):  
Hui-Hsiung Kuo
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Stochastic Integration ◽  
White Noise Analysis

scholarly journals On Wick calculus on spaces of nonregular generalized functions of Levy white noise analysis

2018 ◽  
Vol 10 (1) ◽  
pp. 114-132
Author(s):  
N.A. Kachanovsky
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Holomorphic Functions ◽  
Generalized Functions ◽  
Leibniz Rule ◽  
Wick Product ◽  
Stochastic Integrals ◽  
White Noise Analysis ◽  
Actual Problem ◽  
Lévy White Noise

Development of a theory of test and generalized functions depending on infinitely many variables is an important and actual problem, which is stipulated by requirements of physics and mathematics. One of  successful approaches to building of such a theory consists in introduction of spaces of the above-mentioned functions in such a way that the dual pairing between test and generalized functions is generated by integration with respect to some probability measure. First it was the Gaussian measure, then it were realized numerous generalizations. In particular, important results can be obtained if one uses the Levy white noise measure, the corresponding theory is called the Levy white noise analysis. In the Gaussian case one can construct spaces of test and generalized functions and introduce some important operators (e.g., stochastic integrals and derivatives) on these spaces by means of a so-called chaotic representation property (CRP): roughly speaking, any square integrable random variable can be decomposed in a series of repeated Itos stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP, but there are different generalizations of this property. In this paper we deal with one of the most useful and challenging generalizations of the CRP in the Levy analysis, which is proposed by E.W. Lytvynov, and with corresponding spaces of nonregular generalized functions. The goal of the paper is to introduce a natural product (a Wick product) on these spaces, and to study some related topics. Main results are theorems about properties of the Wick product and of Wick versions of holomorphic functions. In particular, we prove that an operator of stochastic differentiation satisfies the Leibniz rule with respect to the Wick multiplication. In addition we show that the Wick products and the Wick versions of holomorphic functions, defined on the spaces of regular and nonregular generalized functions, constructed by means of Lytvynov's generalization of the CRP, coincide on intersections of these spaces. Our research is a contribution in a further development of the Levy white noise analysis.

scholarly journals Wick calculus on spaces of regular generalized functions of Levy white noise analysis

2018 ◽  
Vol 10 (1) ◽  
pp. 82-104 ◽  
Author(s):  
M.M. Frei
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
Holomorphic Functions ◽  
Generalized Functions ◽  
Leibniz Rule ◽  
Wick Product ◽  
Stochastic Integrals ◽  
White Noise Analysis ◽  
Wick Calculus

Many objects of the Gaussian white noise analysis (spaces of test and generalized functions, stochastic integrals and derivatives, etc.) can be constructed and studied in terms of so-called chaotic decompositions, based on a chaotic representation property (CRP): roughly speaking, any square integrable with respect to the Gaussian measure random variable can be decomposed in a series of Ito's stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP (except the Gaussian and Poissonian particular cases). Nevertheless, there are different generalizations of this property. Using these generalizations, one can construct different spaces of test and generalized functions. And in any case it is necessary to introduce a natural product on spaces of generalized functions, and to study related topics. This product is called a Wick product, as in the Gaussian analysis. The construction of the Wick product in the Levy analysis depends, in particular, on the selected generalization of the CRP. In this paper we deal with Lytvynov's generalization of the CRP and with the corresponding spaces of regular generalized functions. The goal of the paper is to introduce and to study the Wick product on these spaces, and to consider some related topics (Wick versions of holomorphic functions, interconnection of the Wick calculus with operators of stochastic differentiation). Main results of the paper consist in study of properties of the Wick product and of the Wick versions of holomorphic functions. In particular, we proved that an operator of stochastic differentiation is a differentiation (satisfies the Leibniz rule) with respect to the Wick multiplication.

scholarly journals Operators of stochastic differentiation on spaces of nonregular generalized functions of Levy white noise analysis

2016 ◽  
Vol 8 (1) ◽  
pp. 83-106
Author(s):  
N.A. Kachanovsky
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
Generalized Functions ◽  
Test Functions ◽  
White Noise Analysis ◽  
Lévy White Noise ◽  
Stochastic Derivative ◽  
Further Development

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study some properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. During recent years the operators of stochastic differentiation were introduced and studied, in particular, in the framework of the Meixner white noise analysis, in the same way as on spaces of regular test and generalized functions and on spaces of nonregular test functions of the Levy white noise analysis. In the present paper we make the next natural step: introduce and study operators of stochastic differentiation on spaces of nonregular generalized functions of the Levy white noise analysis (i.e., on spaces of generalized functions that belong to the so-called nonregular rigging of the space of square integrable with respect to the measure of a Levy white noise functions). In so doing, we use Lytvynov's generalization of the chaotic representation property. The researches of the present paper can be considered as a contribution in a further development of the Levy white noise analysis. 

scholarly journals On operators of stochastic differentiation on spaces of regular test and generalized functions of Lévy white noise analysis

2014 ◽  
Vol 6 (2) ◽  
pp. 212-229 ◽  
Author(s):  
M.M. Dyriv ◽  
N.A. Kachanovsky
Keyword(s):  
White Noise ◽  
Stochastic Integral ◽  
Noise Analysis ◽  
Gaussian White Noise ◽  
Stochastic Equations ◽  
Generalized Functions ◽  
White Noise Analysis ◽  
Unbounded Operators ◽  
Lévy White Noise ◽  
Stochastic Derivative

The operators of stochastic differentiation, which are closely related with the extended Skorohod stochastic integral and with the Hida stochastic derivative, play an important role in the classical (Gaussian) white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and of solutions of stochastic equations with Wick-type nonlinearities. In this paper we introduce and study bounded and unbounded operators of stochastic differentiation in the Levy white noise analysis. More exactly, we consider these operators on spaces from parametrized regular rigging of the space of square integrable with respect to the measure of a Levy white noise functions, using the Lytvynov's generalization of the chaotic representation property. This gives a possibility to extend to the Levy white noise analysis and to deepen the corresponding results of the classical white noise analysis.

White-noise analysis in retinal physiology

1986 ◽  
Vol 4 ◽  
pp. S141-S152
Author(s):  
Masanori Sakuranaga ◽  
Yu-Ichiro Ando ◽  
Ken-Ichi Naka
Keyword(s):  
White Noise ◽  
Noise Analysis ◽  
White Noise Analysis

A Poisson Structure in White-Noise Analysis

2009 ◽  
Author(s):  
R. Léandre ◽  
Piotr Kielanowski ◽  
S. Twareque Ali ◽  
Anatol Odzijewicz ◽  
Martin Schlichenmaier ◽  
...  
Keyword(s):  
White Noise ◽  
Poisson Structure ◽  
Noise Analysis ◽  
White Noise Analysis

WHITE NOISE ANALYSIS: SOME APPLICATIONS IN COMPLEX SYSTEMS, BIOPHYSICS AND QUANTUM MECHANICS

2012 ◽  
Vol 26 (29) ◽  
pp. 1230014 ◽  
Author(s):  
CHRISTOPHER C. BERNIDO ◽  
M. VICTORIA CARPIO-BERNIDO
Keyword(s):  
Quantum Mechanics ◽  
Probability Density Function ◽  
Complex Systems ◽  
White Noise ◽  
Noise Analysis ◽  
Social Systems ◽  
White Noise Analysis ◽  
The Past ◽  
White Noise Calculus ◽  
With Memory

The white noise calculus originated by T. Hida is presented as a powerful tool in investigating physical and social systems. Combined with Feynman's sum-over-all histories approach, we parameterize paths with memory of the past, and evaluate the corresponding probability density function. We discuss applications of this approach to problems in complex systems and biophysics. Examples in quantum mechanics with boundaries are also given where Markovian paths are considered.

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