Geometric Properties of Fractional Brownian Sheets

Let be an (N, d)-fractional Brownian sheet with Hurst index H = (H₁,...,H_N) ∈ (0, 1)_N. Our objective of the present article is to characterize the anisotropic nature of B_H in terms of H. We prove the following results: (1) B_H is sectorially locally nondeterministic. (2) By introducing a notion of "dimension" for Borel measures and sets, which is suitable for describing the anisotropic nature of B_H, we determine for an arbitrary Borel set Moreover, when B_α is an (N, d)-fractional Brownian sheet with index 〈α〉 = (α,..., α) (0 < α < 1), we prove the following uniform Hausdorff dimension result for its image sets: If N ≤ αd, then with probability one,

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.     

Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets

Let B^H = {B^H(t ), t ∈ ℝ^N} be an (N, d)-fractional Brownian sheet with index H = (H₁, . . . , H_N) ∈ (0, 1)^N. The uniform and local asymptotic properties of B^H are proved by using wavelet methods. The Hausdorff and packing dimensions of the range B^H ([0, 1]^N), the graph Gr B^H ([0, 1]^N) and the level set are determined.     

An Invariance Principle for Triangular Arrays

Let A _{n, i} be a triangular array of sign-symmetric exchangeable random variables satisfying nE(A ² _{n, i} )1, nE(A ⁴ _{n, i} )0, n ² E(A ² _{n, 1} A ² _{n, 2})1. We show that ^[nt] _i=1 A _ni, 0t1, converges to Brownian motion. This is applied to show that if A is chosen from the uniform distribution on the orthogonal group O _n and X _n(t)=^[nt] _i=1 A _ii, then X _n converges to Brownian motion. Similar results hold for the unitary group.     

Large Deviation Principle for Fractional Brownian Motion with Respect to Capacity

Potential Analysis - We show that the fractional Brownian motion (fBM) defined via the Volterra integral representation with Hurst parameter $Hgeq frac {1}{2}$ is a quasi-surely defined Wiener...     

Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter H>1/2). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motions and the corresponding underlying standard Brownian motions. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need to develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motions while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way.     

Maximal Inequalities for Fractional Brownian Motion: An Overview

We give an overview of some maximal inequalities and limit theorems for the tail probabilities for the supremum of a fractional Brownian motion.     

Fractional Brownian Flows

We consider a stochastic flow on ? ⁿ driven by a fractional Brownian motion with Hurst parameter \(H\in(\frac{1}{2},1)\) and study a tangent flow and the growth of the Hausdorff measure of sub-manifolds of ? ⁿ as they evolve under the flow.The main result is a bound on the rate of (global) growth in terms of the (local) Hölder norm of the flow.     

分式Brown运动的极函数

设X(t)(t∈R^N)是指数为α的d维分式Brown运动。本文研究X（t）的极函数问题，得出了满足P｛t∈R^N\｛0｝，X（t）=f(t)=0｝的连续函数f组成的类的特征，解决了Legall提出的一个问题；并且得到了（N，N，2α）过程的不动点的Hausdorff维数。     

On Fractional Brownian Processes

We use Liouville spaces in order to prove the existence of some different fractional -Brownian motion ( 0 < 1 ), or fractional ( , )-Brownian sheets. There are also applications to the Wiener stochastic integral with respect to these -Brownian.     

Oscillatory Fractional Brownian Motion

We introduce oscillatory analogues of fractional Brownian motion, sub-fractional Brownian motion and other related long range dependent Gaussian processes, we discuss their properties, and we show how they arise from particle systems with or without branching and with different types of initial conditions, where the individual particle motion is the so-called c-random walk on a hierarchical group. The oscillations are caused by the ultrametric structure of the hierarchical group, and they become slower as time tends to infinity and faster as time approaches zero. A randomness property of the initial condition increases the long range dependence. We emphasize the new phenomena that are caused by the ultrametric structure as compared with results for analogous models on Euclidean space.     

An Optimal Series Expansion of the Multiparameter Fractional Brownian Motion

We derive a series expansion for the multiparameter fractional Brownian motion. The derived expansion is proven to be rate optimal. This work is supported in part by the Foundation for Knowledge and Competence Development and Sparbanksstiftelsen Nya.     

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.      

Functional Limit Theorems for Multiparameter Fractional Brownian Motion

We prove a general functional limit theorem for multiparameter fractional Brownian motion. The functional law of the iterated logarithm, functional Lévy’s modulus of continuity and many other results are its particular cases. Applications to approximation theory are discussed.      

A Nonconventional Invariance Principle for Random Fields

In Kifer and Varadhan (Ann Probab, to appear), we obtained a nonconventional invariance principle (functional central limit theorem) for sufficiently fast mixing stochastic processes with discrete and continuous time. In this article, we derive a nonconventional invariance principle for sufficiently well-mixing random fields.     

A Note on the Fractional Integrated Fractional Brownian Motion

We characterize the lower classes of the fractional integrated fractional Brownian motion by an integral test.     

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.     

Dimensional Properties of Fractional Brownian Motion

Let B^α = {B^α（t）,t E R^N} be an （N,d）-fractional Brownian motion with Hurst index α∈ （0, 1）. By applying the strong local nondeterminism of B^α, we prove certain forms of uniform Hausdorff dimension results for the images of B^α when N 〉 αd. Our results extend those of Kaufman for one-dimensional Brownian motion.     

On Fractional Brownian Motion and Wavelets

Given a fractional Brownian motion (fBm) with Hurst index H ? (0,1){H\in(0,1)} , we associate with this a special family of representations of Cuntz algebras related to frequency domains and wavelets. Vice versa, we consider a pair of Haar wavelets satisfying some compatibility conditions, and we construct the covariance functions of fBm with a fixed Hurst index. The Cuntz algebra representations enter the picture as filters of the associated wavelets. Extensions to q-dependent covariance functions leading to a corresponding fBm process will also be discussed.