Weak convergence to the fractional Brownian sheet using martingale differences

We establish an invariance principle for the fractional Brownian sheet, starting from discrete random fields constructed from two-parameter strong martingales. This is an approximation in law of the fractional Brownian sheet in Skorohord space in the plane.     

An Itô Formula of Generalized Functionals and Local Time for Fractional Brownian Sheet

Abstract

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the generalized functionals for the fractional Brownian sheet with arbitrary Hurst parameters H ₁, H ₂ ∈ (0,1). As an application, we give the integral representations for two versions of local times of a fractional Brownian sheet, respectively.     

Weak convergence to the fractional Brownian sheet in Besov spaces

In this paper we study the problem of the approximation in law of the fractional Brownian sheet in the topology of the anisotropic Besov spaces. We prove the convergence in law of two families of processes to the fractional Brownian sheet: the first family is constructed from a Poisson procces in the plane and the second family is defined by the partial sums of two sequences of real independent fractional brownian motions.     

Parameter estimation for stochastic equations with additive fractional Brownian sheet

We study the maximum likelihood estimator for stochastic equations with additive fractional Brownian sheet. We use the Girsanov transform for the the two-parameter fractional Brownian motion, as well as the Malliavin calculus and Gaussian regularity theory.      

A Set-indexed Fractional Brownian Motion

We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no “really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.      

Differentiation formula in Stratonovich version for fractional Brownian sheet

We introduce two types of the Stratonovich stochastic integrals for two-parameter processes, and investigate the relationship of these Stratonovich integrals and various types of Skorohod integrals with respect to a fractional Brownian sheet. By using this relationship, we derive a differentiation formula in the Stratonovich sense for fractional Brownian sheet through Itô formula. Also the relationship between the two types of the Stratonovich integrals will be obtained and used to derive a differentiation formula in the Stratonovich sense. In this case, our proof is based on the repeated applications of differentiation formulas in the Stratonovich form for one-parameter Gaussian processes.     

Fractional Brownian Motion and Sheet as White Noise Functionals

In this short note, we show that it is more natural to look the fractional Brownian motion as functionals of the standard white noises, and the fractional white noise calculus developed by Hu and Фksendal follows directly from the classical white noise functional calculus. As examples we prove that the fractional Girsanov formula, the Ito type integrals and the fractional Black-Scholes formula are easy consequences of their classical counterparts. An extension to the fractional Brownian sheet is also briefly discussed.     

On the Equivalence of Multiparameter Gaussian Processes

Our purpose is to characterize the multiparameter Gaussian processes, that is Gaussian sheets, that are equivalent in law to the Brownian sheet and to the fractional Brownian sheet. We survey multiparameter analogues of the Hitsuda, Girsanov and Shepp representations. As an application, we study a special type of stochastic equation with linear noise.      

Example of a Gaussian Self-Similar Field With Stationary Rectangular Increments That Is Not a Fractional Brownian Sheet

We consider anisotropic self-similar random fields, in particular, the fractional Brownian sheet (fBs). This Gaussian field is an extension of fractional Brownian motion. It is well known that the fractional Brownian motion is a unique Gaussian self-similar process with stationary increments. The main result of this article is an example of a Gaussian self-similar field with stationary rectangular increments that is not an fBs. So we proved that the structure of self-similar Gaussian fields can be substantially more involved then the structure of self-similar Gaussian processes. In order to establish the main result, we prove some properties of covariance function for self-similar fields with rectangular increments. Also, using Lamperti transformation, we obtain properties of covariance function for the corresponding stationary fields.     

An Itô formula for a fractional Brownian sheet with arbitrary Hurst parameters

By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the fractional Brownian sheet with arbitrary Hurst parameters .

     

FRACTIONAL 2D-STOCHASTIC CURRENTS

Using multiple stochastic integrals and the stochastic calculus for the frac-tional Brownian sheet, we define and we analyze the 2D-fractional stochastic currents.     

Dimension properties of sample paths of self-similar processes

The Hausdorff dimensions of the image and the graph of random fields are given under general conditions. The results can be used to a wider class of self-similar random fields and processes, including Brownian motion, Brownian sheet, fractional Brownian motion, processes with stable or (α, β)-fractional stable components. Supported by the National Natural Science Foundation of China     

分数布朗单的幂函数随机积分逼近

本文研究了分数布朗单的逼近问题.利用Wiener积分,得到了分数布朗单的幂函数型随机积分逼近.     

Intersections and polar functions of fractional brownian sheets

Let B^H={B^H（t）,t∈R^N＋}be a real-valued（N,d）fractional Brownian sheet with Hurst index H=（H1,…,HN）.The characteristics of the polar functions for B^H are discussed.The relationship between the class of continuous functions satisfying Lipschitz condition and the class of polar-functions of B^H is obtained.The Hausdorff dimension about the fixed points and the inequality about the Kolmogorov’s entropy index for B^H are presented.Furthermore,it is proved that any two independent fractional Brownian sheets are nonintersecting in some conditions.A problem proposed by LeGall about the existence of no-polar continuous functions satisfying the Holder condition is also solved.     

Asymptotic behavior of the solution of heat equation driven by fractional white noise

This note deals with the asymptotic behavior of a weak solution of the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We study the solution given by the Feynman-Kac formula by the method of moments.     

Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators

We investigate the small ball problem for d-dimensional fractional Brownian sheets by functional analytic methods. For this reason we show that integration operators of Riemann–Liouville and Weyl type are very close in the sense of their approximation properties, i.e., the Kolmogorov and entropy numbers of their difference tend to zero exponentially. This allows us to carry over properties of the Weyl operator to the Riemann–Liouville one, leading to sharp small ball estimates for some fractional Brownian sheets. In particular, we extend Talagrand's estimate for the 2-dimensional Brownian sheet to the fractional case. When passing from dimension 1 to dimension d2, we use a quite general estimate for the Kolmogorov numbers of the tensor products of linear operators.     

Strassen-type Laws of Iterated Logarithm for a Fractional Brownian Sheet

Abstract

We study Strassen-type laws of iterated logarithm for a fractional Brownian sheet including that for small time, which imply most of the former laws of the iterated logarithm and Strassen's laws for one-parameter and two-parameter Wiener processes.     

Burgers' system with a fractional Brownian random force

We prove the existence of function solutions in mild and distributional sense to Burgers' equation, perturbed by a non-conservative random force given as the formal space–time derivative of a fractional Brownian sheet. In particular, the noise may be chosen to be fractional in time and white in space. We also provide basic regularity results. The methods involve both pointwise and vector-valued considerations.     

Functional limit theorems for generalized quadratic variations of Gaussian processes

In this paper, we establish functional convergence theorems for second order quadratic variations of Gaussian processes which admit a singularity function. First, we prove a functional almost sure convergence theorem, and a functional central limit theorem, for the process of second order quadratic variations, and we illustrate these results with the example of the fractional Brownian sheet (FBS). Second, we do the same study for the process of localized second order quadratic variations, and we apply the results to the multifractional Brownian motion (MBM).     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.