Large Deviations of a Storage Process with Fractional Brownian Motion as Input

We study probabilities of large extremes of the storage process Y(t) = sup _t (X() - X(t) - c( - t)), where X(t) is the fractional Brownian motion. We derive asymptotic behavior of the maximum tail distribution for the process on fixed or slowly increased intervals by a reduction the problem to a large extremes problem for a Gaussian field.     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.     

以分数布朗运动为输入过程的存储过程生成的上穿点过程

本文研究了以分数布朗运动为输入过程的存储过程上穿高水平u形成的点过程的渐近泊松特性,结果表明当分数布朗运动参数H∈(0,1/2),u→∞时,该点过程弱收敛到泊松过程.     

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.     

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.     

Identifying the Anisotropical Function of a <Emphasis Type="Italic">d</Emphasis>-Dimensional Gaussian Self-similar Process with Stationary Increments

We perform the estimation of the anisotropical function of a Gaussian self-similar process with stationary increments. Article note: In final form 31 May 2005     

Small Ball Estimates for Gaussian Processes under Sobolev Type Norms

A sharp small ball estimate under Sobolev type norms is obtained for certain Gaussian processes in general and for fractional Brownian motions in particular. New method using the techniques in large deviation theory is developed for small ball estimates. As an application the Chung's LIL for fractional Brownian motions is given in this setting.     

Estimating the Diffusion Coefficient for Diffusions Driven by fBm

When the Hurst coefficient of a fBm B _t ^H is greater than 1/2, it is possible to define a stochastic integral with respect to B _t ^H as the pathwise limit of Riemann sums. In this article we consider diffusion equations of the type X_t = x₀ + ₀ ^T (X_s) dB_s ^H. We then construct a simple-to-use estimator of the diffusion coefficient (x), based on the number of crossings of level x of the process X _t. We then study consistency in probability of this estimator and calculate convergence rates in probability.     

Continuity in Law with Respect to the Hurst Parameter of the Local Time of the Fractional Brownian Motion

We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter H converges weakly to that of the local time of , when H tends to H ₀.      

Conditions for singularity for measures generated by two fractional psuedo-diffusion processes

We derive sufficient conditions under which the probability measures generated by two fractional psuedo-diffusion processes are singular with respect to each other.     

Estimation of change point for switching fractional diffusion processes

We study the asymptotic distribution of the maximum likelihood estimator (MLE) for the change point for fractional diffusion processes as the noise intensity tends to zero. It was shown that the rate of convergence here is higher than the rate of convergence of the distribution of the MLE in classical parametric models dealing with independent identically distributed observations with finite and positive Fisher information.     

Free Jacobi Process

In this paper, we define and study two parameters dependent free processes (λ,θ) called free Jacobi, obtained as the limit of its matrix counterpart when the size of the matrix goes to infinity. The main result we derive is a free SDE analogous to that satisfied in the matrix setting, derived under injectivity assumptions. Once we did, we examine a particular case for which the spectral measure is explicit and does not depend on time (stationary). This allows us to determine easily the parameters range ensuring our injectivity requirements so that our result applies. Then, we show that under an additional condition of invertibility at time t=0, this range extends to the general setting. To proceed, we set a recurrence formula for the moments of the process via free stochastic calculus.     

Small and Large Scale Asymptotics of some Lévy Stochastic Integrals

We provide general conditions for normalized, time-scaled stochastic integrals of independently scattered, Lévy random measures to converge to a limit. These integrals appear in many applied problems, for example, in connection to models for Internet traffic, where both large scale and small scale asymptotics are considered. Our result is a handy tool for checking such convergence. Numerous examples are provided as illustration. Somewhat surprisingly, there are examples where rescaling towards large times scales yields a Gaussian limit and where rescaling towards small time scales yields an infinite variance stable limit, and there are examples where the opposite occurs: a Gaussian limit appears when one converges towards small time scales and an infinite variance stable limit occurs when one converges towards large time scales.      

分式Brownian运动的多重相交局部时

本文研究了分式布朗运动的多重相交局部时的问题.利用白噪声分析的方法,获得了分式布朗运动的多重相交局部时的展开式.进行适当的截取,展开式在白噪声广义泛函意义下存在,并给出它们的核函数.推广了布朗运动的多重相交局部时.     

Statistical Analysis of the Fractional Ornstein–Uhlenbeck Type Process

We consider the fractional analogue of the Ornstein–Uhlenbeck process, that is, the solution of a one-dimensional homogeneous linear stochastic differential equation driven by a fractional Brownian motion in place of the usual Brownian motion. The statistical problem of estimation of the drift and variance parameters is investigated on the basis of a semimartingale which generates the same filtration as the observed process. The asymptotic behaviour of the maximum likelihood estimator of the drift parameter is analyzed. Strong consistency is proved and explicit formulas for the asymptotic bias and mean square error are derived. Preparing for the analysis, a change of probability method is developed to compute the Laplace transform of a quadratic functional of some auxiliary process. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Note on the Local Time of Fractional Brownian Motion

Asymptotic behavior of the local time at the origin of q-dimensional fractional Brownian motion is considered when the index approaches the critical value 1/q. It is proved that, under a suitable (temporally inhomogeneous) normalization, it converges in law to the inverse of an extremal process which appears in the extreme value theory.     

一类多维参数高斯过程的弱逼近

本文研究了一类多维参数高斯过程的弱极限问题.在一般情况下,利用泊松过程得到了此类过程的弱极限定理,此多维参数高斯过程可表示为确定的核函数关于维纳过程的随机积分,且包含多维参数的分数布朗运动.