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的随机微分方程：具有不连续路径和反映边界条件的方程，以及可能具有不连续路径的正-反方程。这类随机微分方程的性质，如解的存在唯一性、可解性和解对数据的连续或可测依赖性，将被研究。本研究将为研究非线性扩散的奇异规则随机控制问题的第二部分提供基础工具。奇异规则模型的特点是既包括可测量控制又包括测量型控制。在某些情况下，对状态过程施加了约束，这导致了所谓的反射类型模型。这部分研究的重点是控制理论和应用中具有根本意义的问题，如价值函数和最优策略的表征以及最优控制的充分和必要条件的推导。本研究项目将经典随机控制理论与新近发展的奇异随机控制理论相结合，为广泛的随机控制问题建立一个统一的框架。后者是一个迅速发展的研究领域，提出了许多与偏微分方程和随机分析有关的具有挑战性的理论问题。本研究的主要应用是不确定耗散动力系统的最优控制问题。具体的预期应用是在不确定天气干扰下飞行的飞机的位置/速度控制政策；金融经济学问题，如期权定价、交易成本下的消费/投资优化、存储或库存类型系统的最优控制等。

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.