Search

In this article, we investigate the nonparametric Nadaraya-Watson estimator for the drift function of stochastic differential equations driven by fractional Brownian motion of the Hurst parameter H ∈ (1 / 4 , 1). The drift function is a one-sided dissipative Lipschitz that ensures the ergodic property for the stochastic differential equation. The explicit formula of the estimator is obtained by using the Wick product based on the discretely observed process, which is of the utmost importance for practical applications. With the proper bandwidth selectors, we derive the strong consistency of the proposed estimator, and the main tools are ergodic theory and Malliavin calculus. [ABSTRACT FROM AUTHOR]

This paper identifies certain interesting mathematical problems of stochastic quantization type in the modeling of Laser propagation through turbulent media. In some of the typical physical contexts, the problem reduces to stochastic Schrödinger equation with space–time white noise of Gaussian or Poisson or Lévy type. We identify their mathematical resolution via stochastic quantization. Nonlinear phenomena such as Kerr effect can be modeled by a stochastic nonlinear Schrödinger equation in the focusing case with space–time white noise. A treatment of stochastic transport equation, the Korteweg–De Vries equation as well as a number of other nonlinear wave equations with space–time white noise is also given. The main technique is the S-transform (we will actually use the closely related Hermite transform) which converts the stochastic partial differential equation (PDE) with space–time white noise to a deterministic PDE defined on the Hida–Kondratiev white noise distribution space. We then utilize the inverse S-transform/Hermite transform known as the characterization theorem combined with the infinite-dimensional implicit function theorem for analytic maps to establish local existence and uniqueness theorems for path-wise solutions of this class of problems. The particular focus of this paper on singular white noise distributions is motivated by practical situations where the refractive index fluctuations in the propagation medium in space and time are intense due to turbulence, ionospheric plasma turbulence, marine-layer fluctuations, etc. Since a large class of PDEs, that arise in nonlinear wave propagation, have polynomial-type nonlinearities, white noise distribution theory is an effective tool in studying these problems subject to different types of white noises. [ABSTRACT FROM AUTHOR]

The objective of this article is to introduce and study Itô type stochastic integrals with respect to tempered fractional Brownian motion (TFBM) of Hurst index H ∈ (1 2 , 1) and tempering parameter λ > 0 , by using the Wick product. The main tools are fractional calculus and Malliavin calculus. The Itô formula for this stochastic integral is established for the Itô type processes driven by TFBM. Based on this new Itô formula, we analyze the stability of stochastic differential equations driven by TFBM in the sense of p -th moment. A numerical example is given to illustrate our stability results. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In a recent paper, Lanconelli and Scorolli10 extended to the multidimensional case a Wong–Zakai-type approximation for Itô stochastic differential equations (SDEs) proposed by Øksendal and Hu7. The aim of this paper is to extend the latter result to system of SDEs of Itô type driven by fractional Brownian motion (fBm) like those considered by Hu6. This extension is not trivial since the covariance structure of the fBm precludes us from using the same approach as that used by Lanconelli and Scorolli. Instead we employ a truncated Cameron–Martin expansion as the approximation for the fBm. We are naturally led to the investigation of a semilinear hyperbolic system of evolution equations in several space variables that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the number of terms in the expansion goes to infinite. [ABSTRACT FROM AUTHOR]

We consider the (unique) mild solution u (t , x) of a one-dimensional stochastic heat equation on [ 0 , T ] × ℝ driven by time-homogeneous white noise in the Wick–Skorokhod sense. The main result of this paper is the computation of the spatial derivative of u (t , x) , denoted by ∂ x u (t , x) , and its representation as a Feynman–Kac type closed form. The chaos expansion of ∂ x u (t , x) makes it possible to find its (optimal) Hölder regularity especially in space. [ABSTRACT FROM AUTHOR]

The aim of this work is to first build the underlying theory behind fractional Brownian motion and applying fractional Brownian motion to financial market. By incorporating the Hurst parameter into geometric Brownian motion in order to characterize the long memory among disjoint increments, geometric fractional Brownian motion model is constructed to model S &P 500 stock price index. The empirical results show that the fitting effect of fractional Brownian motion model is better than ordinary Brownian motion. [ABSTRACT FROM AUTHOR]

We propose a novel small time approximation for the solution to the Zakai equation from nonlinear filtering theory. We prove that the unnormalized filtering density is well described over short time intervals by the solution of a deterministic partial differential equation of Kolmogorov type; the observation process appears in a pathwise manner through the degenerate component of the Kolmogorov's type operator. The rate of convergence of the approximation is of order one in the lenght of the interval. Our approach combines ideas from Wong-Zakai-type results and Wiener chaos approximations for the solution to the Zakai equation. The proof of our main theorem relies on the well-known Feynman-Kac representation for the unnormalized filtering density and careful estimates which lead to completely explicit bounds. [ABSTRACT FROM AUTHOR]

Stochastic parabolic equations with random potentials, where the Wick product is used to give sense to the product of generalized stochastic processes, are considered. The existence and uniqueness of solutions are proved via the chaos expansion method from white noise analysis and explicit estimates on the coefficients in the chaos expansion representation of the solutions are provided.

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.

We extend to the multidimensional case a Wong–Zakai-type theorem proved by Hu and Øksendal (1996) for scalar quasi-linear Itô stochastic differential equations (SDEs). More precisely, with the aim of approximating the solution of a quasilinear system of Itô's SDEs, we consider for any finite partition of the time interval [ 0 , T ] a system of differential equations, where the multidimensional Brownian motion is replaced by its polygonal approximation and the product between diffusion coefficients and smoothed white noise is interpreted as a Wick product. We remark that in the one dimensional case this type of equations can be reduced, by means of a transformation related to the method of characteristics, to the study of a random ordinary differential equation. Here, instead, one is naturally led to the investigation of a semilinear hyperbolic system of partial differential equations that we utilize for constructing a solution of the Wong–Zakai approximated systems. We show that the law of each element of the approximating sequence solves in the sense of distribution a Fokker–Planck equation and that the sequence converges to the solution of the Itô equation, as the mesh of the partition tends to zero. [ABSTRACT FROM AUTHOR]

Fractional-stochastic quadratic-cubic nonlinear Schrödinger equation (QC-NLSE) describing propagation of solitons through optical fibers is analyzed. Hermite transforms, white noise analysis and an improved computational method are used to investigate uncertain solutions for QC-NLSE. Specifically, Hermite transformation is applied to convert fractional-stochastic differential equations by Wick-type into deterministic fractional differential equations with an integral term. Furthermore, inverse Hermite transformation is employed to obtain stochastic solutions in the white noise space. Characteristics of presented equations are shown by using some specific values of physical arguments on obtained solutions.

In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions.

In this article, we consider the nonlinear stochastic Wick-type fractional Gardner equation and the generalized fractional Gardner equation with coefficient variables. We obtain new Wick-type explicit solutions of the nonlinear stochastic Wick-type fractional Gardner equation and new explicit solutions of the generalized fractional Gardner equation with coefficients variables by employing the improved system method. Additionally, dynamical behaviors of the obtained solutions are provided graphically under suitable functions of white noises and physical parameters. We show that the improved system method is powerful and effective to find explicit solutions of the considered equations. It is also recommended to use the improved system method to solve the fractional nonlinear partial differential equations as well as nonlinear partial differential equations.

In this article, we discuss the (2 + 1)-D coupled Korteweg–De Vries (KdV) equations whose coefficients are variables, and stochastic (2 + 1)-D C-KdV (C-KdV) equations with the χ-Wick-type product. White noise functional solutions (WNFS) are presented with the homogeneous equilibrium principle, Hermite transform (HT), and technicality via the F-expansion procedure. By means of the direct connection between the theory of hypercomplex systems (HCS) and white noise analysis (WNA), we establish non-Gaussian white noise (NGWN) by studying stochastic partial differential equations (PDEs) with NG-parameters. So, by using the F-expansion method we present multiples of exact and stochastic families from variable coefficients of travelling wave and stochastic NG-functional solutions of (2 + 1)-D C-KdV equations. These solutions are Jacobi elliptic functions (JEF), trigonometric, and hyperbolic forms, respectively.

We treat several classes of hyperbolic stochastic partial differential equations in the framework of white noise analysis, combined with Wiener–Itô chaos expansions and Fourier integral operator methods. The input data, boundary conditions and coefficients of the operators are assumed to be generalized stochastic processes that have both temporal and spatial dependence. We prove that the equations under consideration have unique solutions in the appropriate Sobolev–Kondratiev or weighted-Sobolev–Kondratiev spaces. Moreover, an explicit chaos form of the solutions is obtained, useful for calculating expectations, variances and higher order moments of the solution. [ABSTRACT FROM AUTHOR]

Variable coefficients andWick-type stochastic (2+1)-dimensional coupled KdV equa- tions are investigated. By using the F-expansion method, Hermite transform and white noise theory, the white noise functional solutions for Wick-type stochastic (2+1)- dimensional coupled KdV equations are obtained. The exact travelling wave solutions are expressed in terms of Jacobi elliptic (JEF), trigonometric and hyperbolic functions. [ABSTRACT FROM AUTHOR]

In this article, we have considered Wick-type stochastic Korteweg de Vries (KdV) equation with conformable derivatives. By the help of white noise analysis, Hermit transform and extended G'/G- expansion method, we have obtained exact travelling wave solutions of KdV equation with conformable derivatives. We have applied the inverse Hermit transform for stochastic soliton solutions and then we have shown how stochastic solutions can be presented as Brownian motion functional solutions by an application example. [ABSTRACT FROM AUTHOR]

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.

Let BH = {BtH, t ⩾ 0} be a fractional Brownian motion with Hurst index H ∈ (0, 1). Inspired by pathwise integrals and Wick product, in this paper, we consider the forward and symmetric Wick-Itô integrals with respect to BH as follows: ∫ 0 t u s ⋄ d − B s H = lim ε ↓ 0 1 ε ∫ 0 t u s ⋄ (B s + ε H − B s H) d s ∫ 0 t u s ⋄ d ° B s H = lim ε ↓ 0 1 2 ε ∫ 0 t u s ⋄ (B s + ε H − B (s − ε) ∨ 0 H) d s in probability, where ⋄ denotes the Wick product. We show that the two integrals coincide with divergence-type integral of BH for all H ∈ (0, 1). [ABSTRACT FROM AUTHOR]

Based on the distributions space on 𝒮 ℂ ′ (denoted by ℱ 𝜃 ∗ ( 𝒮 ′ ℂ)) which is the topological dual space of the space of entire functions with exponential growth of order 𝜃 and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in ℱ 𝜃 ∗ ( 𝒮 ′ ℂ). These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples. [ABSTRACT FROM AUTHOR]

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.

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.

This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [29-13]). Anti-Wick quantization and Segal–Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick and Weyl symbols of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder terms up to any orders of the expansion are controlled, still in the Fock space framework. [ABSTRACT FROM AUTHOR]

Fractional-stochastic quadratic-cubic nonlinear Schrödinger equation (QC-NLSE) describing propagation of solitons through optical fibers is analyzed. Hermite transforms, white noise analysis and an improved computational method are used to investigate uncertain solutions for QC-NLSE. Specifically, Hermite transformation is applied to convert fractional-stochastic differential equations by Wick-type into deterministic fractional differential equations with an integral term. Furthermore, inverse Hermite transformation is employed to obtain stochastic solutions in the white noise space. Characteristics of presented equations are shown by using some specific values of physical arguments on obtained solutions. [ABSTRACT FROM AUTHOR]

In this paper, we seek non-Gaussian stochastic solutions of X-Wick-type stochastic (3+1)-dimensional modified Benjamin-Bona-Mahony equations. Using the generalized modified tanh-coth method, the connection between hypercomplex system and transforming white noise theory, X-Wick product and X-Hermite transform, we generate a new set of exact travelling non-Gaussian wave solutions for the (3+1)-dimensional modified Benjamin-Bona-Mahony equations. This set contains solutions with non-Gaussian parameters of exponential, hyperbolic, and trigonometric types. [ABSTRACT FROM AUTHOR]

In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this space in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions. [ABSTRACT FROM AUTHOR]

Our purpose in this paper, is to derive the main properties of the generalized moment functions defined on some types of white noise spaces. A new version of Wick product on some spaces of generalized functions is introduced. Applying the direct connection between the theory of construction for hypercomplex systems and white noise analysis, we setup a framework to construct a lot of spaces of generalized functions connected with different examples of hypercomplex systems. [ABSTRACT FROM AUTHOR]

We consider a class of stochastic differential equations driven by a one-dimensional Brownian motion, and we investigate the rate of convergence for Wong–Zakai-type approximated solutions. We first consider the Stratonovich case, obtained through the pointwise multiplication between the diffusion coefficient and a smoothed version of the noise; then, we consider Itô equations where the diffusion coefficient is Wick-multiplied by the regularized noise. We discover that in both cases the speed of convergence to the exact solution coincides with the speed of convergence of the smoothed noise toward the original Brownian motion. We also prove, in analogy with a well-known property for exact solutions, that the solutions of approximated Itô equations solve approximated Stratonovich equations with a certain correction term in the drift. [ABSTRACT FROM AUTHOR]

The Cauchy representation formula enables to obtain a solution to a nonhomogeneous equation with the help of the linear homogeneous part solution and nonhomogeneities. In case of known asymptotics of the linear homogeneous part solution, we can establish some properties of behavior of a solution to nonhomogeneous equation. For diffusion equations the Cauchy formula was ascertained and successfully applied for different cases. In this paper, the Cauchy representation formula for a solution to a multidimentional affine stochastic differential equation with the Skorohod integral is established. Conditions for inclusion of the solution into generalized Wiener functional spaces are given. [ABSTRACT FROM AUTHOR]

We provide a particle picture representation for the non-symmetric Rosenblatt process and for Hermite processes of any order, extending the result of Bojdecki, Gorostiza and Talarczyk in [4]. We show that these processes can be obtained as limits of certain functionals of a system of particles evolving according to symmetric stable Lévy motions. In the case of k -Hermite processes the corresponding functional involves k -intersection local time of symmetric stable Lévy processes. [ABSTRACT FROM AUTHOR]

We deal with spaces of nonregular generalized functions in the L´evywhite noise analysis, which are constructed using Lytvynov’s generalization of a chaotic representation property. Our aim is to describe a relationship betweenWick 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 L´evy white noise. As examples of an application of our results, we consider some stochastic equations withWick type nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, we unify techniques of Pascal white noise analysis and harmonic analysis on configuration spaces establishing relations between the main structures of both ones. Fix a Random measure σ on a Riemannian manifold X , we construct on the space of finite compound configuration space Ω 0 the so-called Lebesgue–Pascal measure λ σ ̂ and as a consequence we obtain the Pascal measure μ σ ̂ on the compound configuration space Ω. Next, the natural realization of the symmetric Fock space over L 2 (σ ̂) as the space L 2 (λ σ ̂ ) leads to the unitary isomorphism 𝒥 λ μ between the space L 2 (λ σ ̂ ) and L 2 (μ σ ̂ ). Finally, in the first application we study some algebraic products, namely, the Borchers product on the Fock space, the Wick product on the Pascal space, and the ⋆-convolution on the Lebesgue–Pascal space and we prove that the Pascal white noise analysis and harmonic analysis are related through an equality of operators involving 𝒥 λ μ . The second application is devoted to solve the implementation problem. [ABSTRACT FROM AUTHOR]

This paper reports on the characterization of the quantum white noise (QWN) Gross Laplacian based on nuclear algebra of white noise operators acting on spaces of entire functions with θ-exponential growth of minimal type. First, we use extended techniques of rotation invariance operators, the commutation relations with respect to the QWN-derivatives and the QWN-conservation operator. Second, we employ the new concept of QWN-convolution operators. As application, we study and characterize the powers of the QWN-Gross Laplacian. As for their associated Cauchy problem it is solved using a QWN-convolution and Wick calculus. [ABSTRACT FROM AUTHOR]

We derive the Wick calculus for test and generalized functionals of noncommutative white noise corresponding to q-deformed commutation relations with q∈(-1,1). We construct a Gel’fand triple centered at the q-deformed Fock space in which both the test, nuclear space and its dual space are algebras with respect to the addition and the Wick multiplication. Furthermore, we prove a Våge-type inequality for the Wick product on the dual space. [ABSTRACT FROM AUTHOR]

A sharp inequality about the L p -norms of the Wick product generated by a Gamma probability distribution whose mean is half of a positive integer is presented. In order to bound the Wick product, we first smoothen its factors by applying some second quantization operators of a positive and less than one constant times the identity operator. [ABSTRACT FROM AUTHOR]

An integral representation of the Wick product for Gamma distributed random variables, with mean greater than , is presented first. We use this integral representation to prove a Hölder inequality for norms of Gamma Wick products. [ABSTRACT FROM AUTHOR]

In this manuscript, we determine the optimal approximation rate for Skorohod integrals of sufficiently regular integrands. This generalizes the optimal approximation results for Itô integrals. However, without adaptedness and the Itô isometry, new proof techniques are required. The main tools are a characterization via S-transform and a reformulation of the Wiener chaos decomposition in terms of Wick-analytic functionals. [ABSTRACT FROM AUTHOR]

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 L´evy white noise measure, the corresponding theory is called the L´evy 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 It ˆo’s stochastic integrals from nonrandom functions. In the L´evy 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 L´evy 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 (aWick 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 theWick multiplication. In addition we show that the Wick products and theWick 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 L´evy white noise analysis. [ABSTRACT FROM AUTHOR]

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 It ˆo’s stochastic integrals from nonrandom functions. In the L´evy 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 theWick product in the L´evy 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 theWickmultiplication. [ABSTRACT FROM AUTHOR]