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

决策问题在许多领域都会遇到，最明显的是在经济学领域。时间不一致是指决策者的偏好因各种因素而随着时间的推移而发生变化的现象。仔细的研究表明，造成这种情况的主要原因有两个：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直接效用，而后者是由于决策者在估计不同时间决策的风险时主观意见不同所致。本课题从随机最优控制理论的角度，对一般的时间不一致问题进行定量研究，以求得到这些问题的时间一致均衡解。所取得的结果应有助于更好地理解时间不一致问题，并为作出在实际情况中可以接受的时间一致的决定提供一些指导。期望该项目中开发的理论将适用于资产定价、风险管理、资源配置和生产计划。作为该项目的一部分，研究生将接受培训。在时间不一致问题中，时间偏好可以通过贴现来描述，贴现可以是指数的，也可以是非指数的，而风险偏好可以通过选择期望算子来描述，例如经典的期望或它的各种非线性版本。连续时间动力系统的经典随机最优控制问题涉及指数折扣和经典期望。在这种情况下，Bellman的最优性原理成立，这导致最优控制的时间一致性，即对于给定的初始时间和状态对，找到的最优控制在之后将保持最优。然而，当随机最优控制问题涉及非指数贴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稍后时间不保持最优。该项目旨在开发通用工具来为时间不一致的随机最优控制问题寻找时间一致的均衡策略(而不是时间不一致的最优控制)。要研究的具体问题涉及依赖于初始对和条件期望的代价泛函、递归代价泛函以及具有扭曲概率的问题。预计该项目将更好地理解最优控制问题的时间不一致性，所发展的理论将对数学最优控制领域作出重大贡献。此外，该项目将在应用程序和其他数学领域产生重大影响，如随机分析、数学金融、微分游戏和偏微分方程式。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.