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

Georgia Tech Research Corporation, Atlanta GA

Investigators

Abstract

Nonlinear Second-order PDE in infinite dimensional spaces and optimal control of Stochastic PDE. Andrzej Swiech Georgia Institute of Technology Abstract The 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, and economics, in particular in fluid dynamics, and the theory of bondmarkets. It should also help attract and train graduate students and postdoctoral scholars.