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.
