Ergodic Theory of Decisions Under Partial Information

本课题研究部分信息下的序贯决策问题。在这样的问题中，决策必须按顺序及时做出。决策者可以访问遍历马尔可夫过程的观察部分，但她的决策成本也由该过程的未观察部分决定。决策者的目标是通过选择一种仅适用于观察所产生的过滤的策略来最小化长期平均成本。该项目的目标是刻画最优决策的基本性质，如路径最优性和遍历定理，并研究某些可在实践中有效实现的次优决策算法。部分观测随机系统广泛存在于信号处理、通信、导航与跟踪、机器人学、数据同化与预报、生物信息学、物理学、经济学和金融学等多个学科。在这些问题中，人们面临着两大困难：利益系统动态的不确定性和信息的有限。这个项目旨在了解在只有部分信息可用的情况下，如何在不确定的环境中做出最佳决策。发展一种描述最优决策性质的数学理论可以帮助阐明在有限信息下做出决策所面临的基本限制，并导致可用于应用的作出近最优决策的实用算法。

This project is concerned with sequential decision problems under partial information. In such problems, decisions must be made sequentially in time. The decision maker has access to an observed component of an ergodic Markov process, but the cost of her decisions is determined also by the unobserved part of the process. The decision maker aims to minimize the long time average cost by the choice of a strategy that is adapted to the filtration generated by the observations only. The goal of the project is to characterize the fundamental properties of optimal decisions, such as pathwise optimality and ergodic theorems, and to investigate certain sub-optimal decision algorithms which can be efficiently implemented in practice. The research will build on and extend recent advances in the ergodic theory of nonlinear filters.Partially observed stochastic systems are ubiquitous in a wide range of disciplines including signal processing, communications, navigation and tracking, robotics, data assimilation and forecasting, bioinformatics, physics, economics and finance. In such problems, one is faced with two major difficulties: uncertainty in the dynamics of the system of interest, and the limited availability of information. This project aims to understand how one can best make decisions in an uncertain environment when only partial information is available. The development of a mathematical theory characterizing the properties of optimal decisions can help elucidate the fundamental limitations one faces in making decisions with limited information, as well as lead to practical algorithms for making near-optimal decisions which can be used in applications.