Collaborative Research: Time-Consistent Risk-Averse Control of Markov Systems

Ruszczynski，1312016，Dentcheva，1311978该项目研究具有风险厌恶的多维动态随机系统的最优控制。考虑了两种规避风险的方法：动态风险度量和随机排序。研究人员试图推进他们在风险厌恶离散时间模型方面的工作，并发展将风险模型纳入马尔可夫结构的连续时间最优控制问题的一般方法。与该项目相关的三个主要挑战。首先，研究人员开发了适当的数学工具，用于以时间一致的方式测量风险，这将适用于连续时间马尔可夫系统。其次，研究人员为涉及时间一致的风险动态模型的控制问题发展了最优化理论。这包括对控制模型的结构、解的存在性和解的性质的分析。在开发风险规避控制模型时，必须考虑第三个挑战：以有效的方式以数字方式解决问题的可能性。不确定环境下的决策和控制问题出现在许多领域：能源生产和分配、电信、保险和金融、物流、医药、安全和军事应用。在大多数情况下，决策必须随着时间的推移而做出：决策以及随机环境影响系统的演变，从而产生做出新决策的需要，等等。因此，必须设计一项政策，其中包括响应系统未来状态的规则。到目前为止，此类控制过程的大多数理论模型都是基于平均性能标准的。调查人员建议考虑风险，即发生非常不受欢迎的情况的可能性。该项目开发了量化动态系统中以连续方式发展的风险的数学模型。它还提供了在避险必不可少的情况下确定最佳政策的方法。

Ruszczynski, 1312016Dentcheva, 1311978 The project is concerned with optimal control of multi-dimensional dynamic stochastic systems with risk aversion. Two approaches to risk aversion are considered: dynamic risk measures and stochastic orders. The investigators seek to advance their work on risk-averse discrete-time models and to develop general methodology for incorporating risk models into continuous-time optimal control problems of Markov structure. Three major challenges are associated with this project. First, the investigators develop proper mathematical tools for measuring risk in a time-consistent manner that would be suitable for continuous-time Markov systems. Second, the investigators develop optimality theory for control problems involving time-consistent dynamic models of risk. This includes the analysis of the structure of the control models, existence of solutions, and properties of the solutions. When developing risk-averse control models the third challenge has to be taken into account: the possibility to solve the problems numerically in an efficient way. Decision and control problems under uncertainty arise in many areas: energy production and distribution, telecommunication, insurance and finance, logistics, medicine, security and military applications. In most cases decisions have to be made over time: decisions, as well as the random environment, influence evolution of a system, which creates the need to make new decisions, etc. Therefore, a policy has to be designed that incorporates rules for responding to future states of the system. So far, most theoretical models of such control processes have been based on average performance criteria. The investigators propose to take into account risk, that is, the possibility of occurrence of highly undesirable scenarios. The project develops mathematical models for quantifying risk in dynamical systems that evolve in a continuous way. It also provides methods to determine the best policy, when risk aversion is essential.