Estimating the Parameters of a Fractional Brownian Motion by Discrete Variations of its Sample Paths

This paper develops a class of consistent estimators of the parameters of a fractional Brownian motion based on the asymptotic behavior of the k-th absolute moment of discrete variations of its sampled paths over a discrete grid of the interval [0,1]. We derive explicit convergence rates for these types of estimators, valid through the whole range 0 < H < 1 of the self-similarity parameter. We also establish the asymptotic normality of our estimators. The effectiveness of our procedure is investigated in a simulation study. This revised version was published online in June 2006 with corrections to the Cover Date.     

Fractional Generalized Random Fields of Variable Order

Abstract

We study the class of random fields having their reproducing kernel Hilbert space isomorphic to a fractional Sobolev space of variable order on ? ⁿ . Prototypes of this class include multifractional Brownian motion, multifractional free Markov fields, and multifractional Riesz–Bessel motion. The study is carried out using the theory of generalized random fields defined on fractional Sobolev spaces of variable order. Specifically, we consider the class of generalized random fields satisfying a pseudoduality condition of variable order. The factorization of the covariance operator of the pseudodual allows the definition of a white-noise linear filter representation of variable order. In the ordinary case, the Hölder continuity, in the mean-square sense, of the class of random fields introduced is proved, and its mean-square Hölder spectrum is defined in terms of the variable regularity order of the functions in the associated reproducing kernel Hilbert space. The pseudodifferential representation of variable order of the resulting class of multifractal random fields is also defined. Some examples of pseudodifferential models of variable order are then given.     

多参数分数Lévy过程局部时的存在性和联合连续性

首先引进了一类比Xiao和Zhang研究过的Gauss随机场更为一般的多参数Lévy过程．然后给出并证明了此过程的一种分解,并利用这一分解,证明了该过程的局部时的存在性和联合连续性．     

Statistical Inference with Fractional Brownian Motion

We give a test between two complex hypothesis; namely we test whether a fractional Brownian motion (fBm) has a linear trend against a certain non-linear trend. We study some related questions, like goodness-of-fit test and volatility estimation in these models.     

Lower classes for fractional Brownian motion

We characterize the lower classes of fractional Brownian motion by an integral test.Work partially supported by an NSF grant. Equipe d'Analyse, Tour 46, U.A. at C.N.R.S. n^o 754, Université Paris VI, 4 place Jussieu, 75230 Paris Cedex 05, and Department of Mathematics, 231 West 18th Avenue, Columbus, Ohio 43210.     

An Isometric Approach to Generalized Stochastic Integrals

The possibility to extend the classical Ito's construction of stochastic integrals is studied. This construction can be applied to fractional Brownian motions with Hurst index H(0, 1/2). A change of variables formula for fractional Brownian motions in terms of the stochastic integrals is given.     

Hitting probabilities and fractal dimensions of multiparameter multifractional Brownian motion

The main goal of this paper is to study the sample path properties for the harmonisable-type N-parameter multifractional Brownian motion, whose local regularities change as time evolves. We provide the upper and lower bounds on the hitting probabilities of an （N, d）-multifractional Brownian motion. Moreover, we determine the Hausdorff dimension of its inverse images, and the Hausdorff and packing dimensions of its level sets.     

Integration with Respect to Fractional Local Time with Hurst Index 1/2 < H < 1

Let Open image in new window be the weighted local time of fractional Brownian motion B ^H with Hurst index 1/2?H?Open image in new window As an application, we investigate the weighted quadratic covariation \([f\big(B^H\big),B^H]^{(W)}\) defined by

$ \left[f\big(B^H\big),B^H\right]^{(W)}_t:=\lim_{n\to \infty}2H\sum_{k=0}^{n-1} k^{2H-1}\left\{f\big(B^H_{t_{k+1}}\big)-f\big(B^H_{t_{k}}\big)\right\} \left(B^H_{t_{k+1}}-B^H_{t_{k}}\right), $

where the limit is uniform in probability and t _k ?=?kt/n. We show that it exists and

Open image in new window

provided f is of bounded p-variation with \(1\leq p<\frac{2H}{1-H}\). Moreover, we extend this result to the time-dependent case. These allow us to write the fractional Itô formula for new classes of functions.

     

关于分数布朗运动预测的研究

本文对赫斯特参数H∈(1/2,1)的分数布朗运动的预测过程的样本轨道性质进行了讨论.利用布朗运动的随机积分理论,建立了一个重要的不等式,证明了(Z)的图集的Hausdorff维数等于1,得出了预测过程与分数布朗运动本身有显著不同特征的结论.     

Time-Localization of Random Distributions on Wiener Space II: Convergence,Fractional Brownian Density Processes

For a random element X of a nuclear space of distributions on Wiener space C([0,1],R ^d ), the localization problem consists in projecting X at each time t[0,1] in order to define an S(R ^d )-valued process X={X(t),t[0,1]}, called the time-localization of X. The convergence problem consists in deriving weak convergence of time-localization processes (in C([0,1],S(R ^d )) in this paper) from weak convergence of the corresponding random distributions on C([0,1],R ^d ). Partial steps towards the solution of this problem were carried out in previous papers, the tightness having remained unsolved. In this paper we complete the solution of the convergence problem via an extension of the time-localization procedure. As an example, a fluctuation limit of a system of fractional Brownian motions yields a new class of S(R ^d )-valued Gaussian processes, the fractional Brownian density processes.     

Optimality conditions for fractional variational problems with dependence on a combined Caputo derivative of variable order

We establish necessary optimality conditions for variational problems with a Lagrangian depending on a combined Caputo derivative of variable fractional order. The endpoint of the integral is free, and thus transversality conditions are proved. Several particular cases are considered illustrating the new results.