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

Frei, M.M.; Kachanovsky, N. A.

doi:10.1007/s40879-019-00317-8

Cited by 2 publications

(21 citation statements)

References 20 publications

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“…Also, the extended stochastic integral can be presented as a Pettis integral (or a formal Pettis integral -depending on the concrete situation) from the Wick product of the original integrand by the corresponding white noise. On the above-mentioned spaces of nonregular generalized functions in the Lévy analysis such results were obtained in [29], on the spaces of regular generalized functions -in [13].…”

Section: Introductionmentioning

confidence: 84%

“…β q ⊗ H C (such functions often arise in problems). Now for any measurable ∆ ⊆ Θ one can define ∆ f (u) dL u by formula (13). It is clear that similar generalization is possible for f : R + → (L 2 ) β and F : R + → (L 2 ) −β , here β ∈ [0, 1].…”

Section: The Extended Stochastic Integralmentioning

confidence: 96%

“…It is easy to show (see details in [12][13][14]) that such Wick monomials form orthogonal bases in the spaces (L 2 )…”

Section: The Extended Stochastic Integralmentioning

confidence: 99%

“…Like using CRP in the Gaussian analysis, one can use Lytvynov's generalization of CRP, in particular, in order to construct and study spaces of regular and nonregular test and generalized functions [20], introduce and investigate various operators and operations on these spaces, etc. Note that the extended stochastic integral and the Hida stochastic derivative on the spaces of regular test and generalized functions are introduced and studied in [11,20], operators of stochastic differentiation -in [9,10,14], some elements of a Wick calculus and its relationship with operators of stochastic differentiation and integration on the spaces of regular generalized functionsin [12,13]. As for the spaces of nonregular test and generalized functions -the corresponding results are presented in [20,[26][27][28][29].…”

Section: Introductionmentioning

confidence: 99%

“…The aim of the present paper is to introduce by analogy with [22] elements of the Wick calculus on the spaces of regular test functions of the Lévy analysis; to transfer the results of [13] to these spaces; and to consider some related topics (in particular, a Pettis integral over a set of infinite Lebesgue measure and a Wick product under the sign of this integral).…”

Section: Introductionmentioning

confidence: 99%

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On Wick calculus and its relationship with stochastic integration on spaces of regular test functions in the Lévy white noise analysis

Kachanovsky

2022

Carpathian Math. Publ.

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We deal with spaces of regular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to study properties of Wick multiplication and of Wick versions of holomorphic functions, and to describe a relationship between Wick multiplication and integration, on these spaces. More exactly, we establish that a Wick product of regular test functions is a regular test function; under some conditions a Wick version of a holomorphic function with an argument from the space of regular test functions is a regular test function; 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 with respect to a Lévy process; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the extended stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise. As an example of an application of our results, we consider an integral stochastic equation with Wick multiplication.

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Section: Introductionmentioning

confidence: 84%

Section: The Extended Stochastic Integralmentioning

confidence: 96%

“…It is easy to show (see details in [12][13][14]) that such Wick monomials form orthogonal bases in the spaces (L 2 )…”

Section: The Extended Stochastic Integralmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Wick calculus and its relationship with stochastic integration on spaces of regular test functions in the Lévy white noise analysis

Kachanovsky

2022

Carpathian Math. Publ.

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Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

Kachanovsky

2024

Carpathian Math. Publ.

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We deal with spaces of nonregular test functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our goal is to study properties of a natural multiplication $-$ a Wick multiplication on these spaces, and to describe the relationship of this multiplication with integration and stochastic differentiation. More exactly, we establish that the Wick product of nonregular test functions is a nonregular test function; show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of a generalized stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); obtain a representation of the generalized stochastic integral via formal Pettis integral from the Wick product of the original integrand by a Lévy white noise; and prove that the operator of stochastic differentiation of first order on the spaces of nonregular test functions satisfies the Leibnitz rule with respect to the Wick multiplication.

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

Cited by 2 publications

References 20 publications

On Wick calculus and its relationship with stochastic integration on spaces of regular test functions in the Lévy white noise analysis

On Wick calculus and its relationship with stochastic integration on spaces of regular test functions in the Lévy white noise analysis

Wick multiplication and its relationship with integration and stochastic differentiation on spaces of nonregular test functions in the Lévy white noise analysis

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