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。时间不一致性是指决策者的偏好随着时间的推移而变化的现象。仔细的研究表明，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会更重视即时效用，而后者则是由于决策者在估计与其决策相关的风险时，在不同时间有不同的主观意见。本研究从随机最优控制理论的角度，定量地研究一般的时间不一致问题，目的是获得这些问题的时间一致均衡解。所获得的结果应有助于更好地了解时间不一致问题，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。期望本计画所发展之理论，能应用于资产定价、风险管理、资源配置及生产规划。研究生将作为该项目的一部分接受培训。 在时间不一致性问题中，时间偏好可以通过贴现（可以是指数或非指数的）来描述，而风险偏好可以通过期望算子的选择来描述，例如经典期望或其各种非线性版本。在这种情况下，贝尔曼的最优性原理成立，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的时间不保持最优。这个项目的目的是开发通用的工具，寻找时间不一致的随机最优控制问题的时间一致的平衡策略（而不是时间不一致的最优控制）。具体的问题要调查涉及成本泛函依赖于初始对和条件期望，递归成本泛函，以及扭曲的概率问题。预计该项目将提供一个更好的理解最优控制问题的时间不一致性，并开发的理论将显着地促进数学最优控制领域。此外，该项目将在应用和其他数学领域产生重大影响，如随机分析，数学金融，微分博弈和偏微分方程。该奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.