Mathematical Sciences: Topics in Stochastic Control and Diffusion Processes

在随机控制问题中，观察到一个随机现象，并利用该现象所遵循的概率规律知识和观测的过去历史来控制该现象。目标是最小化相关成本函数的期望值。虽然动态规划原理在理论上会产生最优策略，但在实践中，理论结果很少提供明确的可计算答案。本研究的主要目的之一是尽可能明确地确定由消费/投资问题引起的特定随机控制问题的最优政策形式。其他研究将处理最优停止问题和奇异随机控制之间的联系。后者涉及具有单个部分的控制动作。特别有趣的是所谓的“有限燃料”约束，其中控制器可用的工作量（燃料）是有限的。

In stochastic control problems, a random phenomenon is observed which is to be controlled using knowledge of the probabalistic law that the phenomenon obeys and the past history of the observations. The goal is to minimize the expected value of an associated cost function. Although the dynamic propramming principle will in theory yield an optimal strategy, in practice the theoretical results rarely provide explicitly computable answers. One of the main objects of this research is to determine, as explicitly as possible, the form of optimal policies in specific stochastic control problems arising from consumption/investment issues. Other research will deal with the connections between optimal stopping problems and singular stochastic control. The latter involves control actions having singular parts. Of particular interest are so-called 'finite fuel' constraints wherein the amount of effort (fuel) available to the controller is limited.