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We prove the Parseval theorem and the Plancherel Theorem among the analytic Feynman integral and the Fourier Feynman transform and the convolution for the function F : B ν → C in the Class F (B ν) : F ( x → ) = ∫ H exp { i ∑ j = 1 ν (h , x j) ∼ } d μ (h) , μ ∈ M (H) on the Product Abstract Wiener Space (H , B ν , m ν) , where B ν = B × B × ⋯ × B (ν times) and M (H) is the space of a complex valued countably additive measure μ defined on B (H) , the Borel class of a real separable infinite dimensional Hilbert space H. [ABSTRACT FROM AUTHOR]

In this paper, we introduce Wiener integrals with respect to the generalized Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an application, we investigate the corresponding stochastic differential equations with the generalized Hermite process as a driving noise, we prove the existence and the uniqueness of the solution, and we give a generalization of the Hermite Ornstein–Uhlenbeck process and the Hermite-driving Vasicek process. [ABSTRACT FROM AUTHOR]

In this paper, we establish a rotation theorem for the Wiener integral associated with bounded operator on abstract Wiener spaces. To do this, we introduce a class of the angle preserving operators defined on the abstract Wiener space B. [ABSTRACT FROM AUTHOR]

This article investigates the controllability for neutral stochastic delay functional integro-differential equations driven by a fractional Brownian motion, with Hurst parameter lesser than $ 1/2 $. We employ the theory of resolvent operators developed by [ 10 ] combined with the Banach fixed point theorem to establish sufficient conditions to prove the desired result. [ABSTRACT FROM AUTHOR]

This article explains the history and mathematics of Brownian motion. [ABSTRACT FROM AUTHOR]

Hermite processes are self-similar processes with stationary increments, the Hermite process of order 1 is fractional Brownian motion and the Hermite process of order 2 is the Rosenblatt process. In this paper we consider a class of time-dependent neutral stochastic functional differential equations with finite delay driven by Rosenblatt process with index H ∈ (1/2, 1) which is a special case of a self-similar process with long-range dependence. More precisely, we prove the existence and uniqueness of mild solutions by using stochastic analysis and a fixed-point strategy. Finally, an illustrative example is provided to demonstrate the effectiveness of the theoretical result. [ABSTRACT FROM AUTHOR]

We prove that the Wiener integral, the analytic Wiener integral and the analytic Feynman integral of the first variation of F(x)=exp{∫0Tθ(t,x(t))dt} successfully exist under the certain condition, where θ(t,u)=∫Rexp{iuv}dσt(v) is a Fourier–Stieltjes transform of a complex Borel measure σt∈M(R) and M(R) is a set of complex Borel measures defined on R. We will find this condition. Moreover, we prove that the change of scale formula for Wiener integrals about the first variation of F(x) sucessfully holds on the Wiener space.

Using the method of Girsanov transformation, we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution of fractional stochastic functional differential equations driven by fractional Brownian motion. The case of neutral fractional stochastic functional differential equations are also investigated. [ABSTRACT FROM AUTHOR]

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t], and define a random vector Z : C[0, t] → R by $${Z_n}\left( x \right) = \left( {x\left( 0 \right) + a\left( 0 \right),\int_o^{{t_1}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_1}} \right),...,\int_0^{{t_n}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_n}} \right)} } } \right)$$ , where a ∈ C[0, t], h ∈ L [0, t], and 0 < t <... < t ≤ t is a partition of [0, t]. Using simple formulas for generalized conditional Wiener integrals, given Z we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions F in a Banach algebra which corresponds to Cameron-Storvick's Banach algebra S. Finally, we express the generalized analytic conditional Feynman integral of F as a limit of the non-conditional generalized Wiener integral of a polygonal function using a change of scale transformation for which a normal density is the kernel. This result extends the existing change of scale formulas on the classical Wiener space, abstract Wiener space and the analogue of the Wiener space C[0, t]. [ABSTRACT FROM AUTHOR]

We introduce a generalized convolution product (F 8 G)α,β for integral transform Fγ,η for functionals defined on K[0,T] , the space of complex valued continuous functions on [0,T] that vanish at zero. We study some interesting properties of our generalized convolution product and establish various relationships that exist among the generalized convolution product, the integral transform and the first variation for functionals defined on K[0,T] . We also discuss the associativity of the generalized convolution product. [ABSTRACT FROM AUTHOR]

We obtain a maximal inequality for sub-fractional Brownian motion with Hurst indexanalogous to the Burkholder–Davis–Gundy inequality for fractional Brownian motion derived by Novikov and Valkeila [Statist. Probab. Lett.44(1999):47–54] and an integral inequality for Wiener integrals with respect to a sub-fractional Brownian motion with Hurst index. [ABSTRACT FROM AUTHOR]

We investigate the behavior of a first variation of a Fourier transformof a measure of the form :upon the Fourier-Feynman transform and analytic Feynman integral and the convolution, whereand {h1,h2,…,hn} is the orthonormal class of elements inHon the abstract Wiener space (H,B,m). [ABSTRACT FROM PUBLISHER]

Let C[0,t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0,t], and define a random vector Zn:C[0,t]→Rn by...where 0

We use the spectral representation to describe the random electromagnetic fields, which are coupled by Maxwell's equations with a random source term. The covariance matrix functions of the random electromagnetic fields are expressed in terms of the spectral matrix function of the random source term and the Green function of Maxwell's equations. When the random source term is Gaussian, the electromagnetic fields exhibit diffusive scaling limits in the sense of distributions. [ABSTRACT FROM AUTHOR]

The Hermite random field has been introduced as a limit of some weighted Hermite variations of the fractional Brownian sheet. In this work we define it as a multiple integral with respect to the standard Brownian sheet and introduce Wiener integrals with respect to it. As an application we study the wave equation driven by the Hermite sheet. We prove the existence of the solution and we study the regularity of its sample paths, the existence of the density and of its local times. [ABSTRACT FROM AUTHOR]

In this paper we obtain a change of scale formula for Wiener integrals on abstract Wiener spaces. This formula is shown to hold for many classes of functions of interest in Feynman integration theory and quantum mechanics.

Abstract: Using the path-integral framework to cast the pricing problem of the outside barrier Asian option into a Wiener functional integral form, we show that, after the introduction of a law-equivalent process and transformation of the new system, the deviation from the Monte Carlo price is seen to be widely reduced. Bypassing the path-partitioning step, we show that our results behave nicely with respect to increasing correlation. After putting forward empirical evidence of this improvement, we extend the scope to a double knock-out outside barrier, and derive there an original formula. In the latter setting, we propose a simple scheme to reduce the relative error due to a nearby knock-out barrier. [Copyright &y& Elsevier]

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Abstract: We prove that for any second order stochastic process X with stationary increments with continuous paths and continuous variance function, there exists a tempered measure μ (for which we give an explicit expression) related with the domain of the Wiener integral with respect to X as follows: the space of tempered distributions f such that the Fourier transform of f is square integrable with respect to μ is always a dense subset of the domain of the Wiener integral. Moreover, we provide sufficient conditions on μ in order that the domain of the integral is exactly this space of distributions. We apply our results to the fractional Brownian motion. In particular, it is proved that the domain of the Wiener integral with respect to the fractional Brownian motion with Hurst parameter contains distributions that are not given by locally integrable functions, this fact was suggested by Pipiras and Taqqu (2000) in . We have also considered the example of the process given by Ornstein and Uhlenbeck as a model for the position of a Brownian particle. [Copyright &y& Elsevier]

Abstract: Quasi-invariance under translation is established for the σ-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor [J. Najnudel, B. Roynette, M. Yor, A remarkable σ-finite measure on related to many Brownian penalisations, C. R. Math. Acad. Sci. Paris 345 (8) (2007) 459–466]. For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor [T. Funaki, Y. Hariya, M. Yor, Wiener integrals for centered Bessel and related processes. II, ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 225–240 (electronic)], plays a key role. [Copyright &y& Elsevier]

We investigate the behaviour of the first variation of certain cylinder-type functions: [image omitted] under the Fourier-Feynman transform, where [image omitted] is a Fourier transform of the measure μ on Rn of the form [image omitted] and {α1, α2, ..., αn} is an orthonormal class of L2[0, T] and [image omitted] and [image omitted] are in Rn. And we show that the first variation of [image omitted] behave nicely under Fourier-Feynman transforms. And we show that the Fourier-Feynman transform of the first variation of [image omitted] can be perfectly expressed as the limit of a sequence of Wiener integrals of the first variation on the Wiener space. [ABSTRACT FROM AUTHOR]

For the Wiener integral, one property of inversion is established. This property is used for the construction of nonparametric statistical estimation of the unknown logarithmic derivative for the distribution of random processes, which is observed in Wiener noise. [ABSTRACT FROM AUTHOR]

Abstract: The domain of the Wiener integral with respect to a sub-fractional Brownian motion , , , is characterized. The set is a Hilbert space which contains the class of elementary functions as a dense subset. If , any element of is a function and if , the domain is a space of distributions. [Copyright &y& Elsevier]

We investigate the behaviour of the first variation under the Fourier-Feynman transform, and we show that the Wiener integral of the first variation behaves successfully under the Lp-analytic Fourier-Feynman transform for 1≤p≤2 for cylinder functions of the form [image omitted] where f: Rn→C∈Lp (Rn), 1≤p≤2 and {h1, h2, ..., hn} is the orthonormal class of elements in H and the Lp-analytic Fourier-Feynman transform of the first variation can be perfectly expressed as the limit of Wiener integrals of the first variation. [ABSTRACT FROM AUTHOR]

We introduce Wiener integrals with respect to the Hermite process and we prove a non-central limit theorem in which this integral appears as limit. As an example, we study a generalization of the fractional Ornstein-Uhlenbeck process. [ABSTRACT FROM AUTHOR]

Abstract: We characterize the domain of the Wiener integral with respect to the fractional Brownian motion of any Hurst parameter on an interval . The domain is the set of restrictions to of the distributions of with support contained in . In the case any element of the domain is given by a function, but in the case this space contains distributions that are not given by functions. The techniques used in the proofs involve distribution theory and Fourier analysis, and allow to study simultaneously both cases and . [Copyright &y& Elsevier]

In this paper, we evaluate various analytic Feynman integrals of first variation, conditional first variation, Fourier-Feynman transform and conditional Fourier-Feynman transform of cylinder type functions defined over Wiener paths in abstract Wiener space. We also derive the analytic Feynman integral of the conditional Fourier-Feynman transform for the product of the cylinder type functions which define the functions in a Banach algebra introduced by Yoo, with n linear factors. [ABSTRACT FROM AUTHOR]

We establish the analytic Wiener and Feynman integral theory for Wiener integrals over paths on C 0 ( B ), the space of abstract Wiener space valued continuous functions on [0, T ], for certain cylinder-type functions of the form: where f ??? L p (R mn ), 1?=? p ?=?8, and 0?=? s 0 ?=? s 1 ?=?&CenterDot;?&CenterDot;?&CenterDot;?=? s n ?=? T is a partition of [0, T ]. In addition, we establish some relationships between analytic Feynman integrals, Fourier–Feynman transforms and Wiener integrals and we prove the change of scale formula for Wiener integrals over paths on C 0 ( B ) in abstract Wiener spaces. [ABSTRACT FROM AUTHOR]

The purpose of this article is to establish some relationships between L 1 -analytic Fourier-Feynman transforms and Wiener integrals for certain bounded functions F in the Fresnel class &Fouriertrf;( B ) of the form: F ( x ) = ∫ H exp{i( h , x ) &nvsim; }dμ( h ), μ &Element; M( H ), on the abstract Wiener space. [ABSTRACT FROM AUTHOR]

We establish some relationships between L p -analytic Fourier-Feynman transforms and Wiener integrals over paths on C 0 ( B ), the space of abstract Wiener space valued continuous functions on [0, T ], for certain cylinder-type functions of the form where f ∈ L p ( R mn ), 1 ≤ p ≤ 2 and 0 = s 0 ≤ s 1 ≤ ··· ≤ s n = T is a partition of [0, T ]. [ABSTRACT FROM AUTHOR]

We establish some relationships between Fourier-Feynman transforms and Wiener integrals for certain bounded cylinder functions of the form; F ( x ) = μ (( h 1 , x ) ∼ , …, ( h n , x ) ∼ ), x ∈ B , where μ: ℝ n → ℂ is the Fourier-transform of the complex-valued Borel Measure μ on &Bscr;(ℝ n ), the Borel σ-algebra of ℝ n with |μ | < ∞, on the abstract Wiener Space. [ABSTRACT FROM AUTHOR]