Stochastic Calculus of Variations and Stochastic Analysis with Fractal Noises

随机变分演算和分形噪声随机分析

• 批准号: 0604207
• 负责人: David Nualart
• 金额: $ 17万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604207&HistoricalAwards=false
• 关键词: Stochastic; Calculus; Variations; Analysis; Fractal

The project aims to establish new results in three diffeent topics of stochastic analysis. First, a new approach for estimating the negative moments of the solutions to linear stochastic partial differential equations of parabolic type will be developed. These estimates will allow us to derive the regularity of the probability law of the solution at a finite number of points using the techniques of Malliavin Calculus. A second objective is to further develop the stochastic calculus with respect to the fractional Brownian motion and related processes using both Malliavin Calculus and path-wise techniques. Our third goal is to establish chaotical central limit theorems for the asymptotic behavior of different types of functionals of a Gaussian process. Examples of these functionals include the self-intersection local time of the fractional Brownian motion, and power variation and related functionals of stochastic integrals.Stochastic analysis is a modern area in mathematics which aims to study ordinary and partial differential equations perturbed by a random noise. These equations play a central role as models in many areas in physics and economics. In order to derive important properties of the solutions, like to compute the probability that the solution takes values in some interval, one needs to apply suitable mathematical techniques like the Ito Calculus and the Malliavin Calculus. We aim to make substantial contributions to the development of these thecniques and their applications to stochastic partial differential equations. On the other hand, while the classical input noise used has independent increments, motivated by some applications in hydrology, telecommunications and mathematical finance, there has been a recent interest in input noises possessing a long memory property like fractional Brownian motion. The development of a stochastic calculus with respect to these long memory processes is also one of the aims of this project.

该项目旨在建立随机分析的三个重要主题的新成果。首先，我们将发展一个新的方法来估计线性随机抛物型偏微分方程解的负矩。这些估计将允许我们使用Malliavin演算的技术在有限数量的点处导出解的概率律的正则性。 第二个目标是进一步发展的随机微积分相对于分数布朗运动和相关过程使用Malliavin演算和路径明智的技术。我们的第三个目标是建立一个高斯过程的不同类型的泛函的渐近行为的混沌中心极限定理。这些泛函的例子包括分数布朗运动的自相交局部时，以及随机积分的幂变分和相关泛函。随机分析是数学中的一个现代领域，旨在研究受随机噪声扰动的常微分方程和偏微分方程。 这些方程在物理学和经济学的许多领域中作为模型发挥着核心作用。 为了得到解的重要性质，比如计算解在某个区间取值的概率，需要应用适当的数学技巧，比如伊藤演算和马利亚文演算。我们的目标是为这些理论的发展及其在随机偏微分方程中的应用做出实质性的贡献。 另一方面，虽然经典的输入噪声具有独立的增量，在水文，电信和数学金融中的一些应用的动机，有最近的兴趣在输入噪声具有长记忆特性，如分数布朗运动。关于这些长记忆过程的随机微积分的发展也是这个项目的目标之一。