多重分数Stratonovich积分的另一种构造(英文)

本文研究了当Hurst参数日小于1/2时关于分数布朗运动的随机积分问题.利用分数布朗运动的性质和卷积逼近的方法,获得了多重分数Stratonovich积分的另一种构造.     

一类由空间粗糙高斯噪声驱动分数阶动力学方程的性质研究（英文）

本文主要研究了一类空间粗糙高斯噪声驱动的分数阶动力学方程,其中高斯噪声关于时间是白色的,关于空间的相依结构由Hurst指数小于1/2的分数布朗运动的协方差刻画.基于Malliavin分析技巧,我们证明了该类方程温和解在Skorohod意义下的存在性.同时证明了其温和解矩的上、下界的估计.最后证明了其温和解关于时间和空间变量的H?lder连续性.     

基于混合次分数布朗运动环境下的欧式障碍期权定价

研究了基于混合次分数布朗运动环境下的欧式障碍期权的定价问题.考虑原生资产连续支付红利,运用Δ-对冲原理得到欧式下降敲出看涨障碍期权的显式解,以及欧式障碍期权看涨-看跌平价公式.最后进行数值模拟,通过控制变量法,研究了Hurst指数H、初始标的资产价格S、敲定价格K、障碍值S_B、无风险利率r、红利率q、波动率σ对期权价格的影响.与混合分数布朗运动相比,混合次分数布朗运动能更好地刻画金融资产价格的变动,因此本文得到的混合次分数布朗运动环境下欧式障碍期权定价公式更符合金融市场规律.     

混合型随机微分方程的传输不等式

该文探讨一类由Wiener过程和Hurst参数1/2＜H＜1分数布朗运动驱动的混合型随机微分方程.通过使用一些变换技巧和逼近方法,这类方程的强解在d2度量和一致度量d∞下的二次传输不等式被建立.     

股价遵循分数布朗运动的最大值期权定价模型

分数布朗运动由于具有自相似和长期相关等分形特性,已成为数理金融研究中更为合适的工具.通过假定股票价格服从几何分数布朗运动,构建了Ito分数Black--Scholes市场;接着在分数风险中性测度下,利用随机微分方程和拟鞅定价方法给出了分数Black-Scholes定价模型;进一步放松初始假定,讨论了多个标的情形的最大值期权定价问题.研究结果表明,与标准期权价格相比,分数期权价格要同时取决于到期日和Hurst参数.     

多重分数斯特拉托诺维奇积分的一个注记

本文研究了多重分数斯特拉托诺维奇积分,通过卷积逼近技巧和分数布朗运动的随机积分的性质,构造了当Hurst参数小于二分之一时的多重随机积分.这种方法是新的不同于文[8]中的构造方法.     

分数布朗运动驱动的幂型期权定价模型研究

股价运动分形特征的发现,说明布朗运动作为期权定价模型的初始假定存在缺陷.本文假定标的资产价格服从几何分数布朗运动,利用分数风险中性测度下的拟鞅(quasi-martingale)定价方法重新求解分数Black-Scholes模型,进而对幂型期权进行定价.结果表明,幂型期权结果包含了Black-Scholes公式和平方期权结果,且相比标准期权价格,分数期权价格要同时取决于到期日和Hurst参数H.     

分数布朗运动驱动的一类随机微分方程的弱解问题

本文,我们研究如下分数布朗运动驱动的一类随机微分方程的弱解问题Xt=x+BHt+∫t0b(s,Xs)ds,其中BH={BHt,0≤t≤T}是Hurst指数为H∈(0,1/2)∪(1/2,1)的分数布朗运动,b是Borel可测函数且满足线性增长条件|b(t,x)|≤(1+|x|)f(t),其中x∈R且0＜t＜T,f是非负...     

分数布朗运动驱动的一类随机微分方程的弱解问题（英文）

本文,我们研究如下分数布朗运动驱动的一类随机微分方程的弱解问题X_t=x+B_t~H+∫_0~t b(s,X_s)ds,其中B~H={B_t~H,0≤t≤T}是Hurst指数为H∈(0,1/2) ∪ (1/2,1)的分数布朗运动,b是Borel可测函数且满足线性增长条件|b(t,x)| ≤(1+|x|)f(t),其中x∈R且0tT,f是非负Borel函数.值得注意的是f是无界的,比如函数f(t)=(T-t)~(-β)或f(t)=t~(-α),对于一些0 α,β1无界.这个问题对于分数布朗运动驱动的随机微分方程来说是有意义的.     

分数布朗运动驱动的混合型随机微分方程的可行性（英文）

本文着重研究了在一些弱条件下,由分数布朗运动驱动的混合型随机微分方程的闭凸集K的可行性.通过近似论证,我们证明了在线性增长条件下,由Hurst参数为1/2 H <1的分数布朗运动驱动的一类混合型随机微分方程解的存在性.因此,我们可以运用一些转换技术,在一些弱假设的情况下,获得所考虑的混合系统的可行性结果.最后给出一个例子来说明所获结果的有效性.     

Method for Estimating the Hurst Exponent of Fractional Brownian Motion

Doklady Mathematics - Fractional Brownian motion is studied. Statistical estimators of the Hurst exponent are proposed, and their properties are examined. This stochastic process is widely used in...     

Wavelet analysis and covariance structure of some classes of non-stationary processes

Processes with stationary n-increments are known to be characterized by the stationarity of their continuous wavelet coefficients. We extend this result to the case of processes with stationary fractional increments and locally stationary processes. Then we give two applications of these properties. First, we derive the explicit covariance structure of processes with stationary n-increments. Second, for fractional Brownian motion, the stationarity of the fractional increments of order greater than the Hurst exponent is recovered.     

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.     

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

Abstract

We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.     

Comparative analysis of bacterial essential and nonessential genes with Hurst exponent based on chaos game representation

Essential genes are indispensable for the survival of an organism. Investigating features associated with gene essentiality is fundamental to the prediction and identification of essential genes with computational techniques. We use fractal theory approach to make comparative analysis of essential and nonessential genes in bacteria. The Hurst exponents of essential genes and nonessential genes available in the DEG database for 27 bacteria are calculated based on their gene chaos game representations. It is found that for most analyzed bacteria, weak negative correlation exists between Hurst exponent and gene length. Moreover, essential genes generally differ from nonessential genes in their Hurst exponent. For genes of similar length, the average Hurst exponent of essential genes is smaller than that of nonessential genes. The results of our work reveal that gene Hurst exponent is very probably useful gene feature for the algorithm predicting essential genes.     

Estimating the p-Variation Index of a Sample Function: An Application to Financial Data Set

In this paper we apply a real analysis approach to test continuous time stochastic models of financial mathematics. Specifically, fractal dimension estimation methods are applied to statistical analysis of continuous time stochastic processes. To estimate a roughness of a sample function we modify a box-counting method typically used in estimating fractal dimension of a graph of a function. Here the roughness of a function f is defined as the infimum of numbers p > 0 such that f has bounded p-variation, which we call the p-variation index of f. The method is also tested on estimating the exponent [1, 2] of a simulated symmetric -stable process, and on estimating the Hurst exponent H (0, 1) of a simulated fractional Brownian motion.     

Modelling NASDAQ Series by Sparse Multifractional Brownian Motion

The objective of this paper is to compare the performance of different estimators of Hurst index for multifractional Brownian motion (mBm), namely, Generalized Quadratic Variation (GQV) Estimator, Wavelet Estimator and Linear Regression GQV Estimator. Both estimators are used in the real financial dataset Nasdaq time series from 1971 to the 3rd quarter of 2009. Firstly, we review definitions, properties and statistical studies of fractional Brownian motion (fBm) and mBm. Secondly, a numerical artifact is observed: when we estimate the time varying Hurst index H(t) for an mBm, sampling fluctuation gives the impression that H(t) is itself a stochastic process, even when H(t) is constant. To avoid this artifact, we introduce sparse modelling for mBm and apply it to Nasdaq time series.     

Chaos and fractals in fish school motion

The once abstract notions of fractal patterns and processes now appear naturally and inevitably in various chaotic dynamical systems. The examples range from Brownian motion [1], [2], [3], [4], [5] to the dynamics of social relations [6]. In this paper, after introducing a certain hybrid mathematical model of the plankton–fish school interplay, we study the fractal properties of the model fish school walks. We show that the complex planktivorous fish school motion is dependent on the fish predation rate. A decrease in the rate is followed by a transition from low-persistent to high-persistent fish school walks, i.e., from a motion with frequent to a motion with few changes of direction. The low-persistent motion shows fractal properties for all time scales, whereas the high-persistent motion has pronounced multifractal properties for large-scale displacements.     

混合分数布朗运动下欧式期权模糊定价研究

本文采用混合分数布朗运动来刻画标的股票价格的动态变化,以此体现金融市场的长记忆性特征。在混合分数Black-Scholes模型的基础上,基于标的股票价格、无风险利率和波动率均是模糊数的假定下,构建了欧式期权模糊定价模型。其次,分析了金融市场长记忆性的度量指标Hurst指数H对欧式期权模糊定价模型的影响。最后,数值实验表明：考虑长记忆性特征得到的欧式期权模糊定价模型更符合实际。