Mathematical Sciences: Research on Optimal Stochastic Control and Related Topics

该项目包括几个子项目，代表了相当多的 在随机控制领域，随机 过程和非线性滤波。 弗莱明教授提出的随机控制研究 主要讨论了与存在性和性质有关的问题 哈密顿-雅可比方程的解一起未决 问题是，给定哈密顿函数， 扩散过程的相应随机控制问题 导致这个功能。最近，弗莱明和苏甘尼 介绍了一个新的公式的随机微分对策， 基于一种新的确定性方法，微分游戏， 通过替换控制器， 一个游戏的问题。弗莱明还打算研究一些 的开放问题有关的粘性解决方案的非线性第一- 阶偏微分方程在有待解决的问题中 被认为是理论的不连续粘度的解决方案，和 凸分析对偶定理在 近似哈密顿-雅可比方程的值函数。 随机过程的研究主要集中在退出问题上 Freidlin-Wentsel理论弗莱明引入了随机 控制方法来解决这个问题，并应用了渐近展开 动态规划方程后得到的对数 转型这使他能够计算退出概率， 扩张。这些展开式对于罕见的 通信信道中的过载。 在非线性滤波问题上，Fleming建议使用一些 渐近技术来描述分析的结构， 用于输出中的小强度噪声的非线性滤波器 方程 随机控制是应用数学的一个分支， 工程系统理论，处理动态控制 系统在涉及随机输入和测量噪声的情况下。 本研究所用的数学工具是： 随机过程，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.