Pascal white noise calculus

The Fock transform recently introduced by the authors in a previous paper is applied to investigate convergence of generalized functional sequences of a discrete-time normal martingale M . A necessary and sufficient condition in terms of the Fock transform is obtained for such a sequence to be strong convergent. A type of generalized martingales associated with M are introduced and their convergence theorems are established. Some applications are also shown.

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“…Proof. Obviously, condition (1) implies condition (2). We now verify the inverse implication relation.…”

Section: -Generalized Martingales and Their Convergence Theoremsmentioning

confidence: 77%

“…It is known that both Brownian motion and Poisson martingale are continuous-time normal martingales. There are theories of white noise analysis for some other continuous-time processes (see, e.g., [1,2,4,11,15]).…”

Section: Introductionmentioning

confidence: 99%

Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales

Wang

Chen

2015

Journal of Function Spaces

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“…As is well known, generalized functions (functionals) have wide application in mathematical physics (see, e.g. [1,3,5,12,10,15,6] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Spectral Integrals of Bernoulli Generalized Functionals

Zhang,

Wang,

Zhang

et al. 2021

Preprint

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Let S ⊂ L 2 ⊂ S * be the Gel'fand triple over the Bernoulli space, where elements of S * are called Bernoulli generalized functionals. In this paper, we define integrals of Bernoulli generalized functionals with respect to a spectral measure (projection operator-valued measure) in the framework of S ⊂ L 2 ⊂ S * , and examine their fundamental properties. New notions are introduced, several results are obtained and examples are also shown.

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“…e white noise distribution theory originally aiming to extend Itô theory keeping contact with Lévy's stochastic variational calculus [1] has been developed to an efficient infinite dimensional calculus with considerable applications to quantum physics, infinite dimensional harmonic analysis, infinite dimensional differential equations, quantum stochastic calculus, and mathematical finance, see, e.g., [2][3][4][5][6][7][8] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

An Analytic Characterization of $(p, q)$ -White Noise Functionals

Riahi

Ettaieb

Chammam

et al. 2020

Journal of Mathematics

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In this paper, a characterization theorem for the S -transform of infinite dimensional distributions of noncommutative white noise corresponding to the p , q -deformed quantum oscillator algebra is investigated. We derive a unitary operator U between the noncommutative L 2 -space and the p , q -Fock space which serves to give the construction of a white noise Gel’fand triple. Next, a general characterization theorem is proven for the space of p , q -Gaussian white noise distributions in terms of new spaces of p , q -entire functions with certain growth rates determined by Young functions and a suitable p , q -exponential map.

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Pascal white noise calculus

Cited by 16 publications

References 14 publications

Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales

Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales

Spectral Integrals of Bernoulli Generalized Functionals

An Analytic Characterization of $(p, q)$ -White Noise Functionals

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Pascal white noise calculus

Cited by 16 publications

References 14 publications

Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales

Convergence Theorems for Generalized Functional Sequences of Discrete-Time Normal Martingales

Spectral Integrals of Bernoulli Generalized Functionals

An Analytic Characterization of p , q -White Noise Functionals

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An Analytic Characterization of $(p, q)$ -White Noise Functionals