On Gaussian Processes Equivalent in Law to Fractional Brownian Motion

We consider Gaussian processes that are equivalent in law to the fractional Brownian motion and their canonical representations. We prove a Hitsuda type representation theorem for the fractional Brownian motion with Hurst index H1/2. For the case H>1/2 we show that such a representation cannot hold. We also consider briefly the connection between Hitsuda and Girsanov representations. Using the Hitsuda representation we consider a certain special kind of Gaussian stochastic equation with fractional Brownian motion as noise.     

Stochastic heat equation driven by fractional noise and local time

The aim of this paper is to study the d-dimensional stochastic heat equation with a multiplicative Gaussian noise which is white in space and has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (0,1) in time. Two types of equations are considered. First we consider the equation in the Itô-Skorohod sense, and later in the Stratonovich sense. An explicit chaos expansion for the solution is obtained. On the other hand, the moments of the solution are expressed in terms of the exponential moments of some weighted intersection local time of the Brownian motion.     

The asymptotic behavior of the R/S statistic for fractional Brownian motion

This paper provides a proof of the fact that asymptotically the R/S statistic and the self-similarity index of fractional Brownian motion agree in the expectation sense. In particular for fractional Gaussian noise time series, the R/S statistic is an estimator of the self-similarity index H. We also show that two other methods for estimating H yield consistent estimators.     

Approximation of stationary solutions to SDEs driven by multiplicative fractional noise

In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such an equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian motion with Hurst parameter H>1/2$H>1/2$ and obtain some (functional) convergence properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.     

On fractional tempered stable motion

Fractional tempered stable motion (fTSm) is defined and studied. FTSm has the same covariance structure as fractional Brownian motion, while having tails heavier than Gaussian ones but lighter than (non-Gaussian) stable ones. Moreover, in short time it is close to fractional stable Lévy motion, while it is approximately fractional Brownian motion in long time. A series representation of fTSm is derived and used for simulation and to study some of its sample paths properties.     

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.     

Insider trading equilibrium in a market with memory

We consider the Kyle-Back model for insider trading, with the difference that the classical Brownian motion noise of the noise traders is replaced by the noise of a fractional Brownian motion B ^H with Hurst parameter ${H>\frac{1}{2}}$ (when ${H=\frac{1}{2}, B^H}$ coincides with the classical Brownian motion). Heuristically, for ${H>\frac{1}{2}}$ this means that the noise traders has some ??memory??, in the sense that any increment from time t on has a positive correlation with its value at t. (In other words, the noise trading is a persistent stochastic process). It also means that the paths of the noise trading process are more egular than in the classical Brownian motion case. We obtain an equation for the optimal (relative) trading intensity for the insider in this setting, and we show that when ${H\rightarrow\frac{1}{2}}$ the solution converges to the solution in the classical case. Finally, we discuss how the size of the Hurst coefficient H influences the optimal performance and portfolio of the insider.     

Hausdorff measures of the image, graph and level set of bifractional Brownian motion

Let BH,K = {BH,K(t), t ∈ R+} be a bifractional Brownian motion in Rd. This process is a selfsimilar Gaussian process depending on two parameters H and K and it constitutes a natural generalization of fractional Brownian motion (which is obtained for K = 1). The exact Hausdorff measures of the image, graph and the level set of BH,K are investigated. The results extend the corresponding results proved by Talagrand and Xiao for fractional Brownian motion.     

SPDE with generalized drift and fractional-type noise

We consider a stochastic partial differential equation involving a second order differential operator whose drift is discontinuous. The equation is driven by a Gaussian noise which behaves as a Wiener process in space and the time covariance generates a signed measure. This class includes the Brownian motion, fractional Brownian motion and other related processes. We give a necessary and sufficient condition for the existence of the solution and we study the path regularity of this solution.     

Couplings and Strong Approximations to Time-Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions

We define a time-dependent empirical process based on n i.i.d. fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for this process.     

Test to distinguish a Brownian motion from a Brownian bridge using Polya tree process

The problem of distinguishing a Brownian bridge from a Brownian motion, both with possible drift, on the closed unit interval, is investigated via a pair of hypothesis tests. The first, tests for observations obtained at n discrete time points to be arising from a Brownian bridge with drift by embedding the Brownian bridge into a mixture of Polya trees which represents the non-parametric alternative. The second test, tests in an identical manner, for the observations to be coming from a Brownian motion with drift. The Bayes factors for the two tests are derived and then combined to obtain the Bayes factor for the test to distinguish between the two Gaussian processes. The Tierney-Kadane approximation of the Bayes factor is derived with an error approximation of order O(n⁻⁴).     

A Fractional Donsker Theorem

We prove a Donsker-type approximation of the fractional Brownian motion which extends a result by Sottinen for the case H > 1/2 to the full range of Hurst parameters H ∈ (0, 1). The convergence is established by a Donsker-type theorem for Volterra Gaussian processes. The approximation is applied to weak convergence of fractional Wiener integrals.     

CONVERGENCE RATE OF AN APPROXIMATION TO MULTIPLE INTEGRAL OF FBM

In this article, we study the rate of convergence of the polygonal approximation to multiple stochastic integral Sp （f） of fractional Brownian motion of Hurst parameter H 〈 1/2 when the fractional Brownian motion is replaced by its polygonal approximation. Under different conditions on f and for different p, we obtain different rates.     

Exact asymptotics in eigenproblems for fractional Brownian covariance operators

Many results in the theory of Gaussian processes rely on the eigenstructure of the covariance operator. However, eigenproblems are notoriously hard to solve explicitly and closed form solutions are known only in a limited number of cases. In this paper we set up a framework for the spectral analysis of the fractional type covariance operators, corresponding to an important family of processes, which includes the fractional Brownian motion and its noise. We obtain accurate asymptotic approximations for the eigenvalues and the eigenfunctions. Our results provide a key to several problems, whose solution is long known in the standard Brownian case, but was missing in the more general fractional setting. This includes computation of the exact limits of $L^2$-small ball probabilities and asymptotic analysis of singularly perturbed integral equations, arising in mathematical physics and applied probability.     

Bahadur–Kiefer representations for time dependent quantile processes

We define a time dependent empirical process based on n independent fractional Brownian motions and describe strong approximations to it by Gaussian processes. They lead to strong approximations and functional laws of the iterated logarithm for the quantile or inverse of this empirical process. They are obtained via time dependent Bahadur–Kiefer representations.     

Strong approximations in a charged-polymer model

We study the large-time behavior of the charged-polymer Hamiltonian H _n of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian case], using strong approximations to Brownian motion. Our results imply, among other things, that in one dimension the process {H _[nt]}_0≤t≤1 behaves like a Brownian motion, time-changed by the intersection local-time process of an independent Brownian motion. Chung-type LILs are also discussed.     

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.     

Stein’s method on Wiener chaos

We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener–Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry–Esséen bounds in the Breuer–Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler’s formula for Ornstein–Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.     

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.      

Quadratic functionals of Brownian motion

Functionals of Brownian motion can be dealt with by realizing them as functionals of white noise. Specifically, for quadratic functionals of Brownian motion, such a realization is a powerful tool to investigate them. There is a one-to-one correspondence between a quadratic functional of white noise and a symmetric L²(R²)-function which is considered as an integral kernel. By using well-known results on the integral operator we can study probabilistic properties of quadratic or certain exponential functionals of white noise. Two examples will illustrate their significance.