Infinite dimensional analysis, viscosity solutions and applications

无限维分析、粘度解决方案和应用

• 批准号: 0856485
• 负责人: Andrzej Swiech
• 金额: $ 19.33万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-08-15 至 2013-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0856485&HistoricalAwards=false
• 关键词: Infinite; dimensional; analysis; viscosity; solutions

The research of the project is focused on fully nonlinear first- and second-order partial differential equations (PDE) in infinite dimensional spaces and applications thereof. Primary examples of such PDE are equations of Hamilton-Jacobi-Bellman-type (HJB) that are associated with optimal control of deterministic and stochastic PDE. The theory of PDE in infinite dimensional spaces has been studied from the point of view of mild, regular, weak, and viscosity solutions and has established itself over the last two decades as one of the modern tools of infinite dimensional analysis. In particular, in addition to optimal control of PDE, the theory of viscosity solutions of such equations has found applications in areas such as the study of large deviations of infinite dimensional diffusions, the theory of bond markets and other aspects of mathematical finance, and stochastic invariance. In this project the principal investigator focuses on two emerging areas of infinite dimensional PDE that are wide open. The first is integro-PDE in Hilbert spaces. The goal is to develop a viscosity solution theory for fully nonlinear first- and second-order integro-PDE in Hilbert spaces and study its various applications. The interest in such equations comes primarily from their association with infinite dimensional jump-diffusion processes, in particular with stochastic PDE driven by Levy processes. Regarding applications, the theory will be used to develop a PDE approach to large deviations for solutions of stochastic PDE with small Levy noise intensity and to study infinite dimensional Black-Scholes and Black-Scholes-Barenblatt integro-PDE coming from the theory of bond markets driven by impulsive noise. A second area of emphasis is PDE in the space of probability measures, which finally seems open for development owing to recent advances in the theory of mass transport and abstract gradient and Hamiltonian flows. This new area is extremely interesting and important, and the project will concentrate on its applications to large deviations and statistical mechanics. Other problems contained in the project include the use of HJB equations to obtain necessary and sufficient conditions for optimality for optimal control of PDE and investigations into maximum principles.The project contains a pioneering program of research that is aimed at the development of new tools in partial differential equations and infinite dimensional analysis. The proposed research spans areas as diverse as nonlinear partial differential equations, functional analysis, probability, stochastic processes, stochastic partial differential equations, mathematical finance, optimal control, game theory, statistical mechanics, mass transport, and calculus of variations. In particular, it will provide analytical techniques for the study of infinite dimensional jump diffusion processes that are used in stochastic modeling of various phenomena in which random and violent events can occur. Beyond mathematics, the project should have impact and stimulate research in fields such as engineering, physics, finance, and economics. The broader impacts of the project will also include attracting and training graduate students and postdoctoral scholars.

该项目的研究重点是无限维空间中的完全非线性一阶和二阶偏微分方程及其应用。这种PDE的主要例子是与确定性和随机PDE的最优控制相关联的Hamilton-Jacobi-Bellman型（HJB）方程。无限维空间中的偏微分方程理论已经从温和、正则、弱和粘性解的观点进行了研究，并且在过去的二十年中已经成为无限维分析的现代工具之一。特别是，除了最优控制的偏微分方程，理论的粘性解决方案，这样的方程已发现的应用领域，如研究大偏差的无限维扩散，理论的债券市场和其他方面的数学金融，和随机不变性。在这个项目中，主要研究人员集中在两个新兴领域的无限维偏微分方程是完全开放的。第一种是Hilbert空间中的积分偏微分方程。目标是建立Hilbert空间中完全非线性一阶和二阶积分偏微分方程的粘性解理论，并研究其各种应用。在这样的方程的兴趣主要来自于他们的协会与无限维跳跃扩散过程，特别是随机PDE驱动的Levy过程。在应用方面，该理论将被用来发展一个偏微分方程的方法，以大偏差的随机偏微分方程的解与小Levy噪声强度和研究无限维Black-Scholes和Black-Scholes-Barenblatt积分偏微分方程来自脉冲噪声驱动的债券市场理论。第二个领域的重点是偏微分方程在空间的概率措施，这最终似乎开放的发展，由于最近的进展，在理论的质量运输和抽象梯度和哈密顿流。这个新的领域是非常有趣和重要的，该项目将集中在其应用大偏差和统计力学。该项目中包含的其他问题包括使用HJB方程获得PDE最优控制的最优性的必要和充分条件以及最大值原理的调查。该项目包含一个旨在开发偏微分方程和无限维分析新工具的开创性研究计划。拟议的研究范围包括非线性偏微分方程、泛函分析、概率论、随机过程、随机偏微分方程、数学金融、最优控制、博弈论、统计力学、质量运输和变分法。特别是，它将提供分析技术，用于研究无穷维跳跃扩散过程，用于随机建模的各种现象，其中随机和暴力事件可能发生。除了数学之外，该项目还应在工程、物理、金融和经济等领域产生影响并促进研究。该项目更广泛的影响还将包括吸引和培训研究生和博士后学者。