AMC-SS: Collaborative Research: Dynamic Stochastic Optimization with Stochastic Ordering Constraints and Risk Functionals

该研究旨在为风险规避下的动态随机系统优化问题建立数学基础。我们的目标是开发一种方法，能够捕获结果的整个分布，包括小概率但高后果的事件，而不仅仅是平均性能。避险行为将以两种互补的方式建模。第一种方法是基于随机顺序，将系统的随机结果与一些基准结果进行比较。这两种结果都被认为是时间的随机函数。第二种方法是使用动态风险量度来评估系统控制政策的质量。对于这两类模型，我们将分析由此产生的优化问题，发展最优性条件，并提出寻找最优风险规避策略的方法。研究的动机是解决涉及风险的动态决策问题。现有的随机最优控制技术着眼于系统的平均性能，不足以分析和解决小概率事件可能是关键的问题。该项目将提供新的数学工具和数值方法来形式化和解决此类控制问题。研究成果将适用于医学、军事问题、供应链管理、计算机科学、电信、保险和金融等许多涉及重大不确定性和风险的领域。

The proposed research aims at building mathematical foundations of optimization of dynamic stochastic systems under risk aversion. Our goal is to develop approaches which will capture the entire distribution of outcomes, including events of small probability but high consequences, rather than just the average performance. Risk aversion will be modeled in two complementary ways. The first approach is based on stochastic orders comparing the random outcomes of the system to some benchmark outcomes. Both outcomes are considered as random functions of time. The second approach is based on the use of dynamic risk quantifiers in the assessment of the quality of the control policy of the system. For both classes of models we shall analyze the resulting optimization problems, develop conditions of optimality, and propose methods for finding optimal risk-averse strategies.The research is motivated by the need to address dynamic decision problems involving risk. The existing techniques of stochastic optimal control address the average performance of the system and are not sufficient to analyze and solve problems in which events with low probability may be crucial. The project will provide new mathematical tools and numerical methods to formalize and solve such control problems. The results of the research will be applicable to medicine, military problems, supply-chain management, computer science, telecommunication, insurance, and finance, and many other areas involving substantial uncertainty and risk.