Ergodic Theory of Decisions Under Partial Information

本课题研究部分信息条件下的顺序决策问题。在这类问题中，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可观察部分，但其决策的成本也由该过程的不可观察部分决定。决策者的目标是通过选择一种策略来最小化长时间的平均成本，这种策略只适应由观察产生的过滤。该项目的目标是描述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最优决策算法。该研究将建立并扩展非线性滤波器遍历理论的最新进展。部分观察随机系统在广泛的学科中无处不在，包括信号处理、通信、导航和跟踪、机器人、数据同化和预测、生物信息学、物理学、经济学和金融学。在这些问题中，人们面临着两个主要困难：利益系统动态的不确定性和信息的有限可用性。这个项目旨在了解当只有部分信息可用时，人们如何在不确定的环境中最好地做出决策。描述最优决策特性的数学理论的发展可以帮助阐明人们在有限信息下做出决策时面临的基本限制，并导致可以在应用中使用的近似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.