Singularity of Fractional Brownian Motions with Different Hurst Indices

Abstract

We prove that the probability measures generated by two fractional Brownian motions with different Hurst indices are singular with respect to each other.     

$H<1/6$时次分数布朗运动赋权立变差的一个注记

本文中我们利用Malliavin计算的技巧研究了$H<1/6$时次分数布朗运动赋权立变差的$L^2$收敛性.     

Identification of the Hurst Index of a Step Fractional Brownian Motion

We propose a semi-parametric estimator for a piece-wise constant Hurst coefficient of a step fractional Brownian motion (SFBM). For the applications, we want to detect abrupt changes of the Hurst index (which represents long-range correlation) for a Gaussian process with a.s. continuous paths. The previous model of multifractional Brownian motion give a.s. discontinuous paths at change times of the Hurst index. Thus, we first propose a new kind of Fractional Brownian Motion, the SFBM and prove some (Hölder) continuity results. After, we propose an estimator of the piecewise constant Hurst parameter and prove its consistency.     

Prediction of Fractional Brownian Motion-Type Processes

Abstract

We introduce a class of continuous-time Gaussian processes with stationary increments via moving-average representation with good MA coefficient. The class includes fractional Brownian motion with Hurst index less than 1/2 as a typical example. It also includes processes which have different indices corresponding to the local and long-time properties, repsectively. We derive some basic properties of the processes, and, using the results, we establish a prediction formula for them. The prediction kernel in the formula is given explicitly in terms of MA and AR coefficients.     

次分数布朗运动局部时的研究

设X^H={X^H(t),t∈R+}是一个取值于R^d参数为H的次分数布朗运动.本文给出了X^H在单参数情况下局部时的Holder条件和尾概率估计.同时,还给出了X^H在多参数情况下局部时的存在性及L^2表示.     

Power Variation of Subfractional Brownian Motion and Application

In this paper, we consider the power variation of subfractional Brownian motion. As an application, we introduce a class of estimators for the index of a subfractional Brownian motion and show that they are strongly consistent.     

Singularity of generalized grey Brownian motions with different parameters

In this note we prove that the probability measures generated by two generalized grey Brownian motions with different parameters are singular with respect to each other. This result can be interpreted as an extension of the Feldman–Hájek dichotomy of Gaussian measures to a family of non-Gaussian measures.     

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.     

Reaction-Diffusion Equations with Polynomial Drifts Driven by Fractional Brownian Motions

A reaction-diffusion equation on [0, 1] ^d with the heat conductivity κ > 0, a polynomial drift term and an additive noise, fractional in time with H > 1/2, and colored in space, is considered. We have shown the existence, uniqueness and uniform boundedness of solution with respect to κ. Also we show that if κ tends to infinity, then the corresponding solutions of the equation converge to a process satisfying a stochastic ordinary differential equation.     

Stochastic evolution equations with fractional Brownian motion

In this paper linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion are studied. A necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed; separate proofs are required for the cases of Hurst parameter above and below 1/2. The particular case of the Laplacian on the circle is discussed in detail. Mathematics Subject Classification (2000): 60H15, 60G15     

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.     

Intersection Local Time for Two Independent Fractional Brownian Motions

Let B ^H and be two independent, d-dimensional fractional Brownian motions with Hurst parameter H∈(0,1). Assume d≥2. We prove that the intersection local time of B ^H and

Hurst指数属于(1/3,1/2)的分数模型的期权定价

本文建立了由一类分数Brown运动驱动的新的随机微分方程模型,当基础资产价格运动服从该随机微分方程时,推导出了欧式期权的解析公式.     

Stochastic Burgers' equation driven by fractional Brownian motion

In this paper, we consider the stochastic Burgers' equation driven by a genuine cylindrical fractional Brownian motion with Hurst parameter . We first prove the regularities of the solution to the linear stochastic problem corresponding to the stochastic Burgers' equation. Then we obtain the local and global existence and uniqueness results for the stochastic Burgers' equation.     

On the Reproducing Kernel Hilbert Spaces Associated with the Fractional and Bi-Fractional Brownian Motions

We present decompositions of various positive kernels as integrals or sums of positive kernels. Within this framework we study the reproducing kernel Hilbert spaces associated with the fractional and bi-fractional Brownian motions. As a tool, we define a new function of two complex variables, which is a natural generalization of the classical Gamma function for the setting we consider. D. Alpay thanks the Earl Katz family for endowing the chair which supports his research.     

Simulation of First-Passage Times for Alternating Brownian Motions

The first-passage-time problem for a Brownian motion with alternating infinitesimal moments through a constant boundary is considered under the assumption that the time intervals between consecutive changes of these moments are described by an alternating renewal process. Bounds to the first-passage-time density and distribution function are obtained, and a simulation procedure to estimate first-passage-time densities is constructed. Examples of applications to problems in environmental sciences and mathematical finance are also provided.AMS 2000 Subject Classification: 60J65, 60G40, 93E30     

Stochastic elastic equation driven by fractional Brownian motion

In this paper, we consider the stochastic elastic equation driven by a cylindrical fractional Brownian motion. The regularities of the solution to the linear stochastic problem corresponding to the stochastic elastic equation are proved. Then, we obtain the existence of the solution using the Picard iteration.