跳扩散模型下的欧式障碍期权的定价

本文在标的资产价格服从跳扩散模型的假设下,运用Girsanov定理获得了价格过程的等价鞅测度,用期权定价的鞅方法得出障碍期权的定价公式.     

跳扩散模型下的复合期权定价

首先建立股票价格的跳过程为Poisson过程,跳跃高度服从对数正态分布时股票价格过程的随机微分方程,利用测度变换的Girsanov定理,找到等价鞅测度,利用鞅方法,用较简单的数学推导得到了股票价格服从跳扩散过程的欧式期权以及复合期权的定价公式.     

Vasicek随机利率和纯生跳扩散模型下的期权定价

在Vasicek随机利率模型且股票价格服从纯生跳扩散过程的情形下,利用测度变换的Girsanov定理找到定价鞅测度,推导出了有连续红利支付的且影响股票价格的标准Brown运动与影响利率的标准Brown运动相关时欧式股票期权的定价公式,最后给出此定价模型的一些特例以及算例.     

基于鞅方法的分数Brown运动模型的期权定价

本文利用基础资产价格过程的逼近过程,研究了一类Hurst指数属于(1/2,1)的分数Brown.运动模型,通过逼近过程的鞅性,获得了FBM市场的等价鞅测度通过鞅测度变换获得了FBM下的期权定价控制方程和欧式期权的解析公式,改进了部分已有的结果.     

利用分数阶Carleson测度刻画BMO~α_-鞅空间

我们利用分数阶Carleson测度给出了BMOα-鞅空间的一个刻画.该刻画表明一个鞅f属于BM Oα-鞅空间当且仅当与该鞅的鞅差有关的某个张量测度为有界α-Carleson测度.利用这个刻画,我们给出了鞅变换算子、均方算子在BMOα-鞅空间上的有界性的非常简洁的证明.此外,我们还证明了在鞅背景下与分数阶Carleson测度有关的Carleson不等式.最后,我们利用分数阶Carleson测度给出了UMD空间的一个刻画.     

跳-扩散模型下的再装期权定价

本文建立股票价格的跳过程为Poisson过程,跳跃高度服从对数正态分布时股票价格过程的随机微分方程,在风险中性的假设下找到等价鞅测度,利用鞅方法,用较简单的数学推导得到了股票价格服从跳-扩散过程的欧式再装期权定价公式.     

一个条件数学期望公式在期权定价中的应用

首先证明一个条件数学期望公式,然后建立股票价格的跳过程为Poisson过程,跳跃高度为常数时股票价格过程的随机微分方程,在风险中性的假设下,找等价鞅测度.利用鞅方法和已证明的条件数学期望公式,用较简单的数学推导得到了股票价格股从跳—扩散过程的欧式期权以及复合期权的定价公式.     

随机市场系数条件下不连续股价的M-V投资组合选择

在假定市场系数为随机过程并且股票价格服从跳跃扩散过程的市场条件下应用鞅方法讨论一个M-V模型的最优投资组合选择问题.通过引进凹函数U(x)以及等价鞅测度,应用鞅方法以及贝叶斯定理得到了最优投资策略以及有效边界表达式.     

跳-扩散模型下相对收益过程带停时的均值-方差随机控制

针对跳扩散模型下鞅测度不唯一的问题,利用识别定理和Riccati方程研究了跳扩散模型下带停时的均值-方差随机控制问题,得到了相对收益过程最优投资策略的显式解及相应的最优停时,并且给出了在最优停止时间的均值方差有效边界.     

跳-扩散幂型支付的期权定价公式

假设股票价格服从跳扩散过程,并且参数为时间函数的条件下,利用等价鞅测度变换方法得到了幂型支付的欧式期权的定价公式.并且将其推广到有N个独立跳跃源的定价模型中.     

On the Integration by Parts and Characterization of a Fractional Diffusion Process北大核心CSCD

本文研究分数扩散过程和其分部积分公式的关系.首先利用Bismut方法给出拉回公式,进而得到分数扩散过程的分部积分公式。反过来,证明了分数扩散过程可由其分部积分公式唯一刻画.     

控制系统中的分形

本文将整数维与分形的Hausdorff测度引入并应用于控制系统,同时也介绍了准自相似集这个新概念,证明了这种集合的存在性与唯一性.并将计算自相似集维数的公式推广到准自相似集,在此基础上,说明了控制系统的可达集可以具有分数维.表明在分析非线性系统可控性与可观性时,分形几何学也将是一种有意义的工具.     

随机利率下分数跳-扩散Ornstein-Uhlenbeck期权定价模型

假设股票价格遵循分数布朗运动和复合泊松过程驱动的随机微分方程,短期利率服从HullWhite模型,建立了随机利率情形下的分数跳-扩散Ornstein-Uhlenbeck期权定价模型,利用价格过程的实际概率测度和公平保费原理,得到了欧式看涨期权定价的解析表达式,推广了Black-Scholes模型.     

Existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion,discontinuous coefficients,and a partly degenerate diffusion operator

We obtain estimates for functionals of solutions of stochastic differential equations with standard and fractional Brownian motion. We prove a theorem on the existence of weak solutions of stochastic differential equations with standard and fractional Brownian motion, discontinuous coefficients, and a partly degenerate diffusion operator.     

Estimates for the density of functionals of SDEs with irregular drift

We obtain upper and lower bounds for the density of a functional of a diffusion whose drift is bounded and measurable. The argument consists of using Girsanov’s theorem together with an Itô–Taylor expansion of the change of measure. One then applies Malliavin calculus techniques in a non-trivial manner so as to avoid the irregularity of the drift. An integration by parts formula for this set-up is obtained.     

Analysis of the time fractional nonlinear diffusion equation from diffusion process

Under investigation in this paper is a time fractional nonlinear diffusion equation which can be utilized to express various diffusion processes. The symmetry of this considered equation has been obtained via fractional Lie group approach with the sense of Riemann-Liouville (R-L) fractional derivative. Based on the symmetry, this equation can be changed into an ordinary differential equation of fractional order. Moreover, some new invariant solutions of this considered equation are found. Lastly, utilising the Noether theorem and the general form of Noether type theorem, the conservation laws are yielded to the time fractional nonlinear diffusion equation, respectively. Our discovery that there are no conservation laws under the general form of Noether type theorem case. This result tells us the symmetry of this equation is not variational symmetry of the considered functional. These rich results can give us more information to interpret this equation.     

A continuous/discrete fractional Noether’s theorem

We prove a fractional Noether’s theorem for fractional Lagrangian systems invariant under a symmetry group both in the continuous and discrete cases. This provides an explicit conservation law (first integral) given by a closed formula which can be algorithmically implemented. In the discrete case, the conservation law is moreover computable in a finite number of steps.     

Operators associated with the Jacobi semigroup

The aim of this paper is to introduce some operators induced by the Jacobi differential operator and associated with the Jacobi semigroup, where the Jacobi measure is considered in the multidimensional case.In this context, we introduce potential operators, fractional integrals, fractional derivates, Bessel potentials and give a version of Carleson measures.We establish a version of Meyer’s multiplier theorem and by means of this theorem, we study fractional integrals and fractional derivates.Potential spaces related to Jacobi expansions are introduced and using fractional derivates, we give a characterization of these spaces. A version of Calderon’s Reproduction Formula and a version of Fefferman’s theorem are given.Finally, we present a definition of Triebel–Lizorkin spaces and Besov spaces in the Jacobi setting.     

On an αth-order fractional Radon transform and a wave type of equation

The Fourier slice theorem holds for the classical Radon transform. In this paper, we consider a fractional Radon transform for which a sort of Fourier slice theorem also holds, and then present an inversion formula. The fractional Radon transform is shown to be characterized by the multi-dimensional case of a wave type of equation in analogy to the classical Radon transform.     

分数布朗运动环境下的期权定价与测度变换

研究分数B-S市场中的期权定价问题.通过选取不同的资产作为计价单位及相应的测度变换,将经典模型中的计价单位变换方法推广到分数布朗运动市场环境,既丰富了分数期权定价的拟鞅方法,也得到了分数期权定价公式的新的推导方法.