Time-Consistency Theory for Time-Inconsistent Stochastic Optimal Control Problems

决策问题在许多领域都会遇到，尤其是在经济学领域。时间不一致是一种现象，在这种现象中，由于各种因素，决策者的偏好会随着时间而改变。仔细研究表明，这主要有两个原因：决策者的时间偏好和风险偏好。前者是由于决策者可能更看重眼前效用，而后者是由于决策者在不同时间对与其决策相关的风险的估计有不同的主观意见。本课题从随机最优控制理论的角度，对一般时间不一致问题进行定量研究，以期得到这些问题的时间一致均衡解。获得的结果应该有助于更好地理解时间不一致问题，并为制定在实际情况下可接受的时间一致决策提供一些指导。期望在这个项目中发展的理论将适用于资产定价，风险管理，资源配置和生产计划。研究生将作为项目的一部分接受培训。在时间不一致问题中，时间偏好可以通过折现来描述，折现可以是指数或非指数的，而风险偏好可以通过期望算子的选择来描述，例如经典期望或各种非线性版本。经典的连续时间动力系统随机最优控制问题涉及指数折现和经典期望。在这种情况下，Bellman的最优性原理成立，这导致了最优控制的时间一致性，即对于给定的初始时间和状态对找到的最优控制将在之后保持最优。然而，当随机最优控制问题涉及非指数折现或非经典期望算子时，问题变得时间不一致，即在给定时间基于给定初始状态选择的最优控制在以后的时间不会保持最优。该项目旨在开发通用工具，用于寻找时间一致的均衡策略（而不是时间不一致的最优控制），以解决时间不一致的随机最优控制问题。要研究的具体问题包括依赖于初始对和条件期望的成本函数、递归成本函数以及具有扭曲概率的问题。期望本项目将提供对最优控制问题的时间不一致性的更好理解，并且所发展的理论将对数学最优控制领域做出重大贡献。此外，该项目将对应用和其他数学领域产生重大影响，如随机分析、数学金融、微分对策和偏微分方程。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Decision-making problems are encountered in many areas, most notably in economics. Time-inconsistency is a phenomenon in which the preferences of a decision maker change over time, due to various factors. Careful studies show that there are two main reasons for this: the decision makers' time preferences and their risk preferences. The former is due to the fact that decision makers may place more weight on the immediate utility, while the latter is due to the decision makers' different subjective opinions in estimating the risks associated to their decisions at various times. This project studies general time-inconsistent problems quantitatively, from the point of view of stochastic optimal control theory, with the goal of obtaining time-consistent equilibrium solutions to these problems. The results obtained should lead to a better understanding of the time-inconsistency issue and provide some guidance towards making time-consistent decisions that are acceptable in practical situations. The expectation is that the theories developed in this project will be applicable to asset pricing, risk management, resource allocation, and production planning. Graduate students will be trained as part of the project. In time-inconsistency problems, time preferences can be described mathematically by discounting, which may be exponential or non-exponential, while risk preferences can be described by the choice of expectation operators, such as the classical expectation or various nonlinear versions of it. Classical stochastic optimal control problems of continuous-time dynamical systems involve exponential discounting and classical expectations. In this case, Bellman's principle of optimality holds, which leads to time-consistency of optimal controls, that is, an optimal control found for a given initial pair of time and state will remain optimal afterwards. However, when a stochastic optimal control problem involves either a non-exponential discounting, or a non-classical expectation operator, the problem becomes time-inconsistent, namely, an optimal control selected at a given time based on the given initial state does not remain optimal at a later time. This project aims to develop general tools for finding time-consistent equilibrium strategies (rather than time-inconsistent optimal controls) for time-inconsistent stochastic optimal control problems. The specific problems to be investigated involve cost functionals depending on initial pair and conditional expectations, recursive cost functionals, as well as problems with distorted probability. It is expected that this project will provide a better understanding of the time-inconsistency of optimal control problems and that the theory developed will significantly contribute to the area of mathematical optimal control. Further, the project will have a significant impact in applications and in other areas of mathematics, such as stochastic analysis, mathematical finance, differential games, and partial differential equations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.