Mathematical Sciences: Stochastic Differential Equations And Their Applications In Singular-Regular Stochastic Control

本项目的第一部分涉及两种类型 随机微分方程：方程 不连续路径和反射边界条件，以及 前向-后向方程，可能有不连续的路径。 这类随机微分方程的性质，例如 解的存在唯一性、可解性和 解决方案对数据的连续或可测量的依赖性， 将被调查。这项研究将提供基本的 本项目第二部分的工具， 非线性奇异正则随机控制问题 扩散奇异正则模型的特点是： 包括可测量控制和测量型控制。在 在某些情况下，对国家进程施加了限制， 导致了所谓的反射型模型。本部分的重点 的研究是对问题的根本利益， 控制理论与应用，如特征值 函数和最优策略，并导出充分和/或 最优控制的必要条件。 该研究项目将开发一个统一的框架， 一个广泛的一类随机控制问题， 经典的随机控制理论与新发展的 奇异随机控制理论后者是一个快速 扩大研究领域，提出了许多具有挑战性的理论 偏微分方程和随机变量的相关问题 分析.这项研究的主要应用是 耗散动力系统的最优控制问题 不确定性具体的预期应用是 位置/速度控制策略，用于在以下条件下操作的飞机 不确定的天气干扰;金融经济学问题 例如期权定价、消费/投资优化， 交易成本，最佳控制存储或库存类型 系统，除其他外。

The first part of this project is concerned with two types of stochastic differential equations: equations with discontinuous paths and reflecting boundary conditions, and forward-backward equations, possibly with discontinuous paths. Properties of such stochastic differential equations, such as existence and uniqueness of solutions, solvability properties and continuous, or measurable, dependence of solutions on the data, will be investigated. This study will provide the fundamental tools for the second part of this project, which is concerned with singular-regular stochastic control problems for nonlinear diffusions. The singular-regular model is distinguished by involving both measurable controls and measure-type controls. In some cases, constraints on the state process are imposed, which leads to so-called reflecting type models. The focus of this part of the research is on issues which are of fundamental interest in control theory and applications, such as characterizing the value function and optimal policy and deriving sufficient and/or necessary conditions on the optimal control. This research project will develop a unified framework for a broad class of stochastic control problems by combining classical stochastic control theory with the newly developed singular stochastic control theory. The latter is a rapidly expanding area of research and poses many challenging theoretical issues related to partial differential equations and stochastic analysis. The principal applications of this research are to optimal control problems for dissipative dynamical systems under uncertainty. Specific anticipated applications are to position/speed control policies for an aircraft operating under uncertain weather disturbances; to financial economics problems such as option pricing, consumption/investment optimization with transaction costs, optimal control of storage or inventory type systems, among others.