关于分数布朗运动预测的研究

本文对赫斯特参数H∈(1/2,1)的分数布朗运动的预测过程的样本轨道性质进行了讨论.利用布朗运动的随机积分理论,建立了一个重要的不等式,证明了(Z)的图集的Hausdorff维数等于1,得出了预测过程与分数布朗运动本身有显著不同特征的结论.     

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.     

分数布朗运动参数估计

为了估计分形布朗运动参数,作者进一步修改了Wornell算法.克服了Wornell算法要求样本数量大的缺陷.在仿真中,和lance等人修改的Wornell算法相比较,算法达到了更高的精度,并以均方根误差和标准差为指标说明了本文方法的优越性.     

分数布朗运动驱动的期权定价模型及其风险特征

分数布朗运动由于具有自相似或长记忆等分形特性,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型分数Black-Scholes市场;随后基于拟鞅定价方法,求解了分数风险中性测度下的期权定价模型;进而放松执行价格为固定值的假定,研究了股价和履行价共同受分数布朗运动驱动的期权定价模型。数值模拟研究表明,长记忆参数值越大,对投资者和券商的避险策略越有利。     

分数布朗运动的线性滤波

本文利用正态相关性定理给出了关于分数布朗运动的线性滤波最佳估计.     

广义混合分数布朗运动

本文研究了由多个分数布朗运动组合而成的广义混合分数布朗运动的性质.利用分数布朗运动的基本性质,获得了广义分数布朗运动的混合自相似性、马氏性、增量间的相关性、H(o|¨)lder连续性和α-可微性,推广了关于混合分数布朗运动的相应结论.     

混合分数布朗运动下永久美式期权的定价

假设标的资产由混合分数布朗运动驱动,利用分数It6公式得到了混合分数布朗运动环境下永久美式期权的Black-Scholes偏微分方程,并通过偏微分方程获得永久美式期权的定价公式.     

分数布朗运动下几何平均亚式权证的定价

假设股票价格变化过程服从几何分数布朗运动,建立了分数布朗运动下的亚式期权定价模型.利用分数-It-公式,推导出分数布朗运动下亚式期权的价值所满足的含有三个变量偏微分方程.然后,引进适当的组合变量,将其定解问题转化为一个与路径无关的一维微分方程问题.进一步通过随机偏微分方程方法求解出分数布朗运动下亚式期权的定价公式.最后利用权证定价原理对稀释效用做出调整后,得到分数布朗运动下亚式股本权证定价公式.<正>~~     

分数布朗运动环境下的幂期权定价

在等价鞅测度下,研究标的资产价格服从几何分数布朗运动的幂期权看涨、看跌定价公式及其平价公式.并与基于标准布朗运动的幂期权定价公式进行比较分析,进一步论证布朗运动只是分数布朗运动的一种特例,可基于分数布朗运动对原有的期权定价模型进行推广.     

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.     

Stochastic delay fractional evolution equations driven by fractional Brownian motion

In this paper, we consider a class of stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space. Sufficient conditions for the existence and uniqueness of mild solutions are obtained. An application to the stochastic fractional heat equation is presented to illustrate the theory. Copyright © 2014 John Wiley & Sons, Ltd.     

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.     

Synchronization for fractional FitzHugh–Nagumo equations with fractional Brownian motion

This paper is devoted to dynamics of the Caputo-type fractional FitzHugh–Nagumo equations (FHN) driven by fractional Brownian motion (fBm). The existence and uniqueness of mild solution for of the Caputo-type fractional FHN are established, and the exponential synchronization and finite-time synchronization for the stochastic FHN are provided. Finally, the numerical simulation of the synchronization for time-fractional FHN perturbed by fBm is provided; the effects of the order of time fractional derivative and Hurst parameter $H$ on synchronization are also revealed.     

On $L_p$-solution of fractional heat equation driven by fractional Brownian motion

In this paper, we study the fractional stochastic heat equation driven by fractional Brownian motions of the form $$ du(t,x)=\left(-(-\Delta)^{\alpha/2}u(t,x)+f(t,x)\right)dt +\sum\limits^{\infty}_{k=1} g^k(t,x)\delta\beta^k_t $$ with $u(0,x)=u_0$, $t\in[0,T]$ and $x\in\mathbb{R}^d$, where $\beta^k=\{\beta^k_t,t\in[0,T]\},k\geq1$ is a sequence of i.i.d. fractional Brownian motions with the same Hurst index $H>1/2$ and the integral with respect to fractional Brownian motion is Skorohod integral. By adopting the framework given by Krylov, we prove the existence and uniqueness of $L_p$-solution to such equation.     

Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

In this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.     

Stochastic integration for tempered fractional Brownian motion

Tempered fractional Brownian motion is obtained when the power law kernel in the moving average representation of a fractional Brownian motion is multiplied by an exponential tempering factor. This paper develops the theory of stochastic integrals for tempered fractional Brownian motion. Along the way, we develop some basic results on tempered fractional calculus.     

Packing measure of the sample paths of fractional Brownian motion

Let be a fractional Brownian motion of index in If , then there exists a positive finite constant such that with probability 1,

where and - is the -packing measure of .

     

Reinsurance control in a model with liabilities of the fractional Brownian motion type

We propose a model for reinsurance control for an insurance firm in the case where the liabilities are driven by fractional Brownian motion, a stochastic process exhibiting long-range dependence. The problem is transformed to a nonlinear programming problem, the solution of which provides the optimal reinsurance policy. The effect of various parameters of the model, such as the safety loading of the reinsurer and the insurer, the Hurst parameter, etc. on the optimal reinsurance program is studied in some detail. Copyright © 2007 John Wiley & Sons, Ltd.