Nonlinear Second-Order PDE in Infinite Dimensional Spaces and Optimal Control of Stochastic PDE

无穷维空间中的非线性二阶偏微分方程与随机偏微分方程的最优控制。Andrzej Swiech佐治亚理工学院摘要该项目专注于无限维Hilbert空间中的完全非线性二阶偏微分方程（PDE）。这是一个相对较新的领域，近年来引起了很多关注。这些方程的主要例子是与随机偏微分方程的最优控制相关的Hamilton-Jacobi-Bellman（HJB）型方程。线性的，所谓的柯尔莫哥洛夫方程，给出了无限维扩散的分析描述，也属于这一类，并已被深入研究，主要是与来自数学物理学，流体动力学，期权定价和人口生物学的方程。该项目的研究将集中在与希尔伯特空间中粘性解的一般理论有关的几个基本问题，以及一些特别重要的方程的应用和研究。一般的问题包括工具的发展，如Perron的方法，半松弛极限的方法，有限维近似，和发展的粘性解理论Kolmogorov和HJB方程与三维随机Navier-Stokes（NS）方程。应用将包括随机偏微分方程（包括随机NS方程），大偏差，数学经济学和金融的最优控制。对HJB方程和随机NS方程的研究有助于回答有关三维随机NS方程本身的一些问题。此外，还将研究Hilbert空间中与无穷维跳跃扩散过程的新兴场有关的完全非线性积分偏微分方程。本研究将包括研究无限维积分Black-Scholes和积分Black-Scholes-Barenblatt方程与远期利率的Musiela模型的跳跃扩散版本中的期权定价。最后，我们将研究有限维完全非线性随机偏微分方程的粘性解，了解无限维Hilbert空间中的二阶偏微分方程的理论是发展动态规划方法来研究随机偏微分方程系统的最优控制的第一步。该项目包含一个研究计划，重点是为这些方程开发新的基本工具。它跨越不同的领域，如偏微分方程，泛函分析和算子理论，概率论，随机过程，随机偏微分方程，数学金融和控制理论。该研究将对非线性偏微分方程领域以外的几个数学领域产生重大影响，如随机偏微分方程的最优控制，大偏差，随机NS方程和数学金融。此外，它有可能刺激相关领域的研究，如工程，大气科学，金融和经济学，特别是流体动力学和债券市场理论。它还应有助于吸引和培养研究生和博士后学者。

Nonlinear Second-order PDE in infinite dimensional spaces and optimal control of Stochastic PDE. Andrzej SwiechGeorgia Institute of TechnologyAbstractThe project concentrates on fully nonlinear second-order partial differential equations (PDE) in infinite dimensional Hilbert spaces. This is a relatively new area which has attracted a lot of attention in recent years. Primary examples of such equations are equations of Hamilton-Jacobi-Bellman (HJB) type that are associated with optimal control of stochastic PDE. Linear, so called Kolmogorov equations, that give an analytic description of infinite dimensional diffusions, also fall into this category and have been studied intensively, primarily in connection with equations coming from mathematical physics, fluid dynamics, option pricing, and population biology. The research of the project will focus on several fundamental issues related to the general theory of viscosity solutions in Hilbert spaces, together with applications and the study of some particularly important equations. The general questions include the development of tools like Perron's method, the method of half-relaxed limits, finite dimensional approximations, and the development of a viscosity solution theory for Kolmogorov and HJB equations associated with three dimensional stochastic Navier-Stokes (NS) equations. Applications will include optimal control of stochastic PDE (including stochastic NS equations), large deviations, and mathematical economics and finance. The research on HJB equations associated with stochastic NS equations may help answer some open questions about three dimensional stochastic NS equations themselves. Moreover fully nonlinear integro-PDE in Hilbert spaces that are connected to the emerging field of infinite dimensional jump-diffusion processes will also be investigated. This research will include the study of infinite dimensional integro-Black-Scholes and integro-Black-Scholes-Barenblatt equations related to option pricing in the jump diffusion version of the Musiela model of forward rates. Finally questions related to viscosity solutions of finite dimensional fully nonlinear stochastic PDE will be studied.Understanding the theory of second-order PDE in infinite dimensional Hilbert spaces is the first step in the development of the dynamic programming approach to optimal control of systems governed by stochastic PDE. This project contains a program of research that focuses on the development of new basic tools for such equations. It spans areas as diverse as partial differential equations, functional analysis and operator theory, probability, stochastic processes, stochastic PDE, mathematical finance, and control theory. The research will have significant impact on several areas of mathematics outside the field of nonlinear PDE like optimal control of stochastic PDE, large deviations, stochastic NS equations, and mathematical finance. Moreover it has a potential to stimulate research in related fields like engineering, atmospheric sciences, finance, andeconomics, in particular in fluid dynamics, and the theory of bondmarkets. It should also help attract and train graduate students and postdoctoral scholars.