Mathematical Sciences: Research on Optimal Stochastic Control and Related Topics

这个项目包括几个子项目，代表了在随机控制、随机过程和非线性滤波领域相当广泛的研究计划。Fleming教授提出的随机控制研究主要集中在Hamilton-Jacobi方程解的存在性和性质问题上。一个突出的问题是，给定哈密顿函数，是否存在导致该函数的扩散过程的相应随机控制问题。最近，Fleming和Souganidis基于微分对策的一种新的确定性方法引入了随机微分对策的新公式，该公式通过用博弈问题代替控制问题来解决该问题。弗莱明还打算研究一些与非线性一阶偏微分方程的粘度解有关的开放问题。要考虑的开放问题包括不连续粘性解的理论，以及应用凸分析的对偶定理来近似Hamilton-Jacobi方程的值函数。随机过程的研究主要集中在Freidlin-Wentsel理论的出口问题上。Fleming引入了随机控制方法来解决这个问题，并对经过对数变换得到的动态规划方程进行了渐近展开。这使他能够计算出膨胀的退出概率。这些扩展对于通信通道中罕见的过载问题很有意义。在非线性滤波问题上，Fleming提出用渐近技术对输出方程中小强度噪声的非线性滤波器结构进行解析描述。随机控制是应用数学和工程系统理论的一个分支，研究涉及随机输入和测量噪声的动态系统的控制。本研究涉及的数学工具是随机过程理论、非线性偏微分方程、渐近方法和数值方法。

This project includes several subprojects and represents quite a vast research program in the area of stochastic control, stochastic processes and nonlinear filtering. The research in stochastic control proposed by Professor Fleming is focused on the questions related to the existence and properties of solutions to the Hamilton-Jacobi equation. One outstanding question is whether given the Hamiltonian function there is a corresponding stochastic control problem for diffusion processes leading to this function. Recently, Fleming and Souganidis introduced a new formulation of a stochastic differential game, based on a new deterministic approach to differential games, which leads to the solution of this problem by replacing the control problem by a game problem. Fleming also intends to work on a number of open problems related to viscosity solutions for nonlinear first- order partial differential equations. Among open problems to be considered are the theory of discontinuous viscosity solutions, and the application of duality theorems of convex analysis to approximate the value function of the Hamilton-Jacobi equation. The research in stochastic processes centers on the exit problem of Freidlin-Wentsel theory. Fleming introduced the stochastic control approach to this problem and applied an asymptotic expansion to the dynamic programming equation obtained after a logarythmic transformation. This enables him to compute exit probabilities from the expansion. These expansions are of interest for problems of rare overloads in communications channels. On the nonlinear filtering problem, Fleming proposes to use some asymptotic techniques to describe analytically the structure of the nonlinear filter for small intensities of noise in the output equation. Stochastic control is a branch of applied mathematics and engineering system theory that deals with control of dynamical systems in situations involving random inputs and measurement noise. The mathematical tools involved in this research are the theory of stochastic processes, nonlinear partial differential equations, asymptotic methods and numerical methods.