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

Dyriv, M. M.; Kachanovsky, N. A.

doi:10.15330/cmp.6.2.212-229

Cited by 5 publications

(22 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In order to consider many problems of the Lévy white noise analysis, in terms of Lytvynov's generalization of the CRP, it is necessary to know an explicit formula for the scalar products (·, ·) ext . Such a formula is calculated in [38]; in another record form (more convenient for some calculations) it is given in, e.g., [13,15,16].…”

Section: Lytvynov's Generalization Of the Crpmentioning

confidence: 99%

“…It is clear that F ∈ (H −τ ) (F ∈ (D ′ )) if and only if F can be presented in form (12) and norm (13) is finite for some q ∈ N q 0 (τ) (for some τ ∈ T and some q ∈ N q 0 (τ) ).…”

Section: Lemma 2 For Each τ ∈ T There Existsmentioning

confidence: 99%

“…−τ,C , k ∈ Z + , be the kernels from decomposition (12) for F. Since by Schwartz's theorem for some q ∈ Z + F ∈ (H −τ ) − q , by (13) for each k we have |F…”

Section: Remarkmentioning

confidence: 99%

“…These analogs are introduced and studied (in a real case) in [30]. They have properties similar to properties of "classical" operators of stochastic differentiation [13], and can be accepted as operators of stochastic differentiation on the spaces of nonregular generalized functions. Now we'll recall the definition of such operator of order 1, and will show that this operator satisfies the Leibniz rule with respect to the Wick multiplication ♦.…”

Section: Interconnection Between the Wick Calculus And Operators Of Smentioning

confidence: 99%

“…In a moment these spaces are well studied. In particular, the extended stochastic integral and the Hida stochastic derivative on them are introduced and studied in [14,25], operators of stochastic differentiation -in [12,13,16], some elements of a Wick calculusin [15]. But, as in the Gaussian analysis, in connection with some problems of the mathematical physics and of the stochastic analysis (in particular, of the theory of stochastic equations with Wick-type nonlinearities), it is necessary to introduce into consideration so-called spaces of nonregular test and generalized functions in terms of Lytvynov's generalization of the CRP [25], and to study operators and operations on these spaces.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Kachanovsky

2018

Carpathian Math. Publ.

Self Cite

View full text Add to dashboard Cite

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 Lévy white noise measure, the corresponding theory is called the Lévy 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 Itô's stochastic integrals from nonrandom functions. In the Lévy 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 Lévy 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 Lévy white noise analysis.

show abstract

Section: Lytvynov's Generalization Of the Crpmentioning

confidence: 99%

Section: Lemma 2 For Each τ ∈ T There Existsmentioning

confidence: 99%

“…−τ,C , k ∈ Z + , be the kernels from decomposition (12) for F. Since by Schwartz's theorem for some q ∈ Z + F ∈ (H −τ ) − q , by (13) for each k we have |F…”

Section: Remarkmentioning

confidence: 99%

Section: Interconnection Between the Wick Calculus And Operators Of Smentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Kachanovsky

2018

Carpathian Math. Publ.

Self Cite

View full text Add to dashboard Cite

show abstract

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

Frei

Kachanovsky

2019

European Journal of Mathematics

View full text Add to dashboard Cite

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.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 5 publications

References 6 publications

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

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

On the relationship between Wick calculus and stochastic integration 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

Contact Info

Product

Resources

About