Fractional Bilinear Stochastic Equations with the Drift in the First Fractional Chaos

Abstract

In the paper we compute the explicit form of the fractional chaos decomposition of the solution of a fractional stochastic bilinear equation with the drift in the fractional chaos of order one and initial condition in a finite fractional chaos. The large deviations principle is also obtained for the one-dimensional distributions of the solution of the equation perturbed by a small noise.     

The Regularity of Stochastic Convolution Driven by Tempered Fractional Brownian Motion and Its Application to Mean-field Stochastic Differential Equations

In this paper, some properties of a stochastic convolution driven by tempered fractional Brownian motion are obtained. Based on this result, we get the existence and uniqueness of stochastic mean-field equation driven by tempered fractional Brownian motion. Furthermore, combining with the Banach fixed point theorem and the properties of Mittag-Leffler functions, we study the existence and uniqueness of mild solution for a kind of time fractional mean-field stochastic differential equation driven by tempered fractional Brownian motion.     

带跳的分数倒向重随机微分方程及相应的随机积分偏微分方程

本文首次把Poisson随机测度引入分数倒向重随机微分方程,基于可料的Girsanov变换证明由Brown运动、Poisson随机测度和Hurst参数在(1/2,1)范围内的分数Brown运动共同驱动的半线性倒向重随机微分方程解的存在唯一性.在此基础上,本文定义一类半线性随机积分偏微分方程的随机黏性解,并证明该黏性解由带跳分数倒向重随机微分方程的解唯一地给出,对经典的黏性解理论作出有益的补充.     

分数阶微分方程的一些新结果

第一部分,介绍分数阶导数的定义和著名的Mittag—Leffler函数的性质．第二部分,利用单调迭代方法给出了具有2序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性和唯一性．第三部分,利用上下解方法和Schauder不动点定理给出了具有2序列Riemann—Liouville分数阶导数微分方程周期边值问题解的存在性．第四部分,利用Leray—Schauder不动点定理和Banach压缩映像原理建立了具有n序列Riemann—Liouville分数阶导数微分方程初值问题解的存在性、唯一性和解对初值的连续依赖性．第五部分,利用锥上的不动点定理给出了具有Caputo分数阶导数微分方程边值问题,在超线性（次线性）条件下C310,11正解存在的充分必要条件．最后一部分,通过建立比较定理和利用单调迭代方法给出了具有Caputo分数阶导数脉冲微分方程周期边值问题最大解和最小解的存在性．     

分数布朗运动驱动下z一致连续的BSDE解的存在性与唯一性

本文研究一类由分数布朗运动驱动的一维倒向随机微分方程解的存在性与唯一性问题,在假设其生成元满足关于y Lipschitz连续,但关于z一致连续的条件下,通过应用分数布朗运动的Tanaka公式以及拟条件期望在一定条件下满足的单调性质,得到倒向随机微分方程的解的一个不等式估计,应用Gronwall不等式得到了一个关于这类方程的解的存在性与唯一性结果,推广了一些经典结果以及生成元满足一致Lipschitz条件下的由分数布朗运动驱动的倒向随机微分方程解的结果.     

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>~~     

分数阶微分方程边值问题研究简介

分数阶微积分是一个古老而又新颖的课题,近30年来,由于在包括分形现象在内的物理、工程等诸多应用学科领域应用的拓展,激发了科研人员对分数阶微积分的巨大热情。分数阶微分方程现在已应用于分数物理学、混沌与湍流、粘弹性力学与非牛顿流体力学、高分子材料的解链、自动控制理论、化学物理、随机过程和反常扩散等许多科学领域。分数阶微分方程边值问题是非线性常微分方程理论研究中一个活跃而成果丰硕的领域。本文讨论了分数阶微分方程边值问题的一些理论,介绍了作者的著作《分数阶微分方程边值问题理论及应用》的基本内容。     

分数维Vasicek利率模型下的欧式期权定价公式

假定股票价格和利率的运动过程服从几何分数维布朗运动,利用风险对冲技术,分数维布朗运动随机分析理论与偏微分方程方法,得到了分数维Vasicek随机利率下欧式期权所满足的定价方程,获得了波动率是对间函数的情形下欧式看涨和看跌期权的一般定价公式以及它们的平价公式.     

关于带有三点边值的非线性分数阶积分微分方程的解的存在性和惟一性

本文考虑一个带有三点边值的非线性分数阶积分微分方程,并分别用Krasnoselskii不动点定理和Banach压缩映像原理建立了其解的存在性和惟一性的充分条件.     

A New Generalized Gronwall Inequality with a Double Singularity and Its Applications to Fractional Stochastic Differential Equations

Abstract

In many cases, the existence and uniqueness of the solution of a differential equation are proved using fixed point theory. In this paper, we utilize the theory of operators and ingenious techniques to investigate the well-posedness of mild solution to semilinear fractional stochastic differential equations. We first discuss some properties of a class of Volterra integral operators and then establish a new generalized Gronwall integral inequality with a double singularity. Finally, we use the properties and integral inequality to study the well-posedness of mild solution to the semilinear fractional stochastic differential equations. One sees that it is concise and effectiveness using the previous results to investigate the well-posedness of the mild solution.     

Large deviation principle for the fourth-order stochastic heat equations with fractional noises

In this paper, we shall study a fourth-order stochastic heat equation driven by a fractional noise, which is fractional in time and white in space. We will discuss the existence and uniqueness of the solution to the equation. Furthermore, the regularity of the solution will be obtained. On the other hand, the large deviation principle for the equation with a small perturbation will be established through developing a classical method.     

Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter H>1/2). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motions and the corresponding underlying standard Brownian motions. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need to develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motions while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way.     

分数阶微分方程边值问题解的存在性

应用Gteen函数将分数阶微分方程边值问题可转化为等价的积分方程．近来此方法被应用于讨论非线性分数阶微分方程边值问题解的存在性．讨论非线性分数阶微分方程边值问题,应用Green函数,将其转化为等价的积分方程,并设非线性项满足Caratheodory条件,利用非紧性测度的性质和M6nch’s不动点定理证明解的存在性．     

混合分数布朗运动驱动的幂期权定价模型

假设标的资产遵循由混合分数布朗运动驱动的随机微分方程,建立了混合分数布朗运动环境下的金融数学模型.利用拟鞅方法,获得了欧式幂期权定价公式的解析式及其平价公式.最后阐述了分数布朗运动只是混合布朗运动的一种特殊情形.     

On a stochastic fractional partial differential equation with a fractional noise

We will prove the existence, uniqueness and regularity of the solution for a stochastic fractional partial differential equation driven by an additive fractional space–time white noise. Moreover, the absolute continuity of the solution is also obtained.     

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.     

多维分数布朗运动环境下的最优投资组合问题

在分数布朗运动环境下,讨论了单资产多噪声情形下的最优投资组合问题.假定标的资产价格遵循多维分数布朗运动驱动的常系数随机微分方程,在给定效用函数分别为幂函数和对数效用函数条件下,得到了最优投资组合问题的显式解.     

Fractional stochastic differential equation with discontinuous diffusion

In this article, we study a class of stochastic differential equations driven by a fractional Brownian motion with H > 1/2 and a discontinuous coefficient in the diffusion. We prove existence and uniqueness for the solution of these equations. This is a first step to define a fractional version of the skew Brownian motion.     

分数跳——扩散环境下的巨灾期权定价

在假设巨灾指数服从分数跳-扩散的条件下,利用保险精算方法给出了有N个独立跳跃源的分数跳-扩散过程下巨灾期权的定价.     

Time-Changed Processes Governed by Space-Time Fractional Telegraph Equations

We study global and local stabilities of the stationary zero solution to certain infinite-dimensional stochastic differential equations. The stabilities are in terms of fractional powers of the linear part of the drift. The abstract results are applied to semilinear stochastic partial differential equations with non-Lipschitzian drift terms and, in particular, to some specific models of population dynamics. We also expose the stabilizing effect of noise on the otherwise unstable zero solution

As a basic tool we use the Forward Inequality, a generalization of Kolmogorov's forward equation; it is an application of Lyapunov's second method with a sequence of Lyapunov functionals