分数Stratonovich随机积分的收敛速度

本文研究了分数布朗运动的随机积分的收敛性.利用卷积逼近的方法,我们得到了相应的收敛速度.     

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.     

非Lipschitz条件下混合型随机微分方程解的矩估计北大核心CSCD

本文的研究对象为非Lipschitz条件下混合分数布朗运动驱动的随机微分方程.混合分数布朗运动是布朗运动和分数布朗运动的线性组合.通过证明和混合分数布朗运动有关的伊藤公式,借助Malliavin积分理论,本文证明在非Lipschitz条件下,由混合分数布朗运动驱动的随机微分方程解的矩估计和连续性.     

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>~~     

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.     

分数布朗运动模型中贝叶斯估计的渐近正态性

本文研究了由分数布朗运动驱动的线性随机微分方程中贝叶斯估计的渐近趋势．利用分数布朗运动的随机积分理论和Girsanov公式，得到了在平方损失函数下贝叶斯估计的渐近正态性．     

一类特殊的Riemann-Liouville分数阶积分的几何意义

主要研究阶数等于0.5的Riemann-Liouville (RL)分数阶积分的几何意义.通过变量替换,证明了0.5阶RL积分等价于沿着圆弧的曲线积分.对于同一个积分,如果沿着曲线观察,此积分是曲线积分;如果沿着坐标轴观察,此积分是分数阶积分.此结论说明,分数阶微积分提供了一种观察客观世界的新角度.另外,应用分数阶理论求解了一个关于转动惯量的问题,此实例说明分数阶现象是广泛存在的.     

随机利率下的Erlang(2)风险模型

本文主要对索赔记数过程是Erlang(2)过程,随机利率为一个L啨vy过程的风险模型进行了讨论.首先导出了破产概率满足的积分方程,估计了其上下界,然后针对随机利率为布朗运动以及漂移布朗运动的情况导出了破产概率满足的具体积分方程,最后讨论了罚金函数,并写出了罚金函数满足的积分方程以及在特殊情况下满足的积分微分方程.     

n重分数布朗运动的Strassen泛函重对数律

本文给出了由两个不同的分数布朗运动组成的重分数布朗运动的Strassen型泛函重对数律和局部Strassen型泛函重对数律.我们的结果也适用于由两个布朗运动组成的重布朗运动及由一个分数布朗运动和一个布朗运动组成的重过程.最后将上述结果推广到n重分数布朗运动中.推广了已有文献的相应结果.     

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.     

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.     

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.     

Cylindrical fractional Brownian motion in Banach spaces

In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen–Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.     

Regularity of Backward Stochastic Volterra Integral Equations in Hilbert Spaces

This article investigates backward stochastic Volterra integral equations in Hilbert spaces. The existence and uniqueness of their adapted solutions is reviewed. We establish the regularity of the adapted solutions to such equations by means of Malliavin calculus. For an application, we study an optimal control problem for a stochastic Volterra integral equation driven by a Hilbert space-valued fractional Brownian motion. A Pontryagin-type maximum principle is formulated for the problem and an example is presented.     

Integral Representation with Adapted Continuous Integrand with Respect to Fractional Brownian Motion

We show that if a random variable is a final value of an adapted Hölder continuous process, then it can be represented as a stochastic integral with respect to fractional Brownian motion, and the integrand is an adapted process, continuous up to the final point.     

分数Brown 运动驱动的非 Lipschitz随机微分方程

讨论了一类带分数Brown 运动的非Lipschitz 增长的随机微分方程适应解的存在唯一性。关于分数 Brown 运动的随机积分有多种定义,本文使用一种广义 Stieltjes积分定义方法,利用这种积分的性质,建立了一类由标准 Brown 运动和一个 Hurst 指数H ∈(1/2,1)的分数Brown 运动共同驱动的、系数为非Lipschitz 增长的随机微分方程适应解的存在唯一性定理。     

On the two-parameter fractional Brownian motion and Stieltjes integrals for Hölder functions

We extend the Stieltjes integral to Hölder functions of two variables and prove an existence and uniqueness result for the corresponding deterministic ordinary differential equations and also for stochastic equations driven by a two-parameter fractional Brownian motion.     

Stochastic analysis, rough path analysis and fractional Brownian motions

 In this paper we show, by using dyadic approximations, the existence of a geometric rough path associated with a fractional Brownian motion with Hurst parameter greater than 1/4. Using the integral representation of fractional Brownian motions, we furthermore obtain a Skohorod integral representation of the geometric rough path we constructed. By the results in [Ly1], a stochastic integration theory may be established for fractional Brownian motions, and strong solutions and a Wong-Zakai type limit theorem for stochastic differential equations driven by fractional Brownian motions can be deduced accordingly. The method can actually be applied to a larger class of Gaussian processes with covariance functions satisfying a simple decay condition. Received: 11 May 2000 / Revised version: 20 March 2001 / Published online: 11 December 2001     

Some linear fractional stochastic equations

Using the multiple stochastic integrals, we prove an existence and uniqueness result for a linear stochastic equation driven by the fractional Brownian motion with any Hurst parameter. We study both the one- and two-parameter cases. When the drift is zero, we show that in the one-parameter case the solution is an exponential—thus positive—function while in the two-parameter setting the solution is negative on a non-negligible set.