Time-Inconsistent Optimal Control Problems for Stochastic Differential Equations

计划通常是在时间一致的假设下制定的，即在一个时期开始时设计的最佳战略将在整个审议期间保持最佳状态。然而，几乎每个人都不得不承认，这种情况很少发生。事实上，通常情况下，在一个时期开始时制定的最优政策在之后不会保持最优；这种现象被称为时间不一致。研究表明，时间不一致的两个主要原因是人们的时间偏好和风险偏好。前者的一个例子是，如果没有强制执行的合同，人们可能会发现很难信守承诺。后者的一个例子是，不同群体的人对某一股票购买固有的风险会有不同的主观看法。主要由于这两种偏好，随着时间的推移，“最优”计划不可能保持最优。目前的项目是从最优控制的角度对这一问题进行定量研究。目标是在一定条件下，对时间不一致问题获得时间一致的均衡策略。我们期望，拟议理论的结果将增加我们对时间不一致问题的理解，并最终有助于做出更好的时间一致的决定。在数学上，经典的最优控制问题是时间一致的，即在给定的初始时间和状态对找到的最优控制将在此后对相应的初始对保持最优。当折扣是一般的，非指数的，和/或状态和控制的条件期望非线性地出现在性能指标中时，相应的最优控制问题将是时间不一致的。为了获得时间一致的开环均衡策略，我们将使用变分方法，并结合正反向随机微分方程理论。为了获得时间一致的闭环均衡策略，我们将修改动态规划原理，并采用/引入多人微分对策。该项目将丰富(随机)最优控制理论及相关领域。

Plans are most often made under the time-consistency assumption that optimal strategies designed at the beginning of a time period will remain optimal throughout the period under consideration. However, almost everyone has to admit that this is rarely the case. In fact, more often than not, an optimal policy made at the beginning of a period will not stay optimal thereafter; this phenomenon is called time-inconsistency. Studies show that the two main reasons for time-inconsistency are people's time-preferences and risk-preferences. An example of the former is that if there is no enforced contract, people may find it difficult to keep their promises. An example of the latter is that different groups of people will have different subjective views on the risk inherent in a certain stock purchase. Mainly due to these two types of preferences, the 'optimal' plan cannot stay optimal as time goes by. The current project is to quantitatively study such a problem from an optimal control point of view. The goal is to obtain time-consistent equilibrium strategies, under certain conditions, for time-inconsistent problems. We expect that the results of the proposed theory will increase our understanding of the time-inconsistency issue, and ultimately help in the making of better time-consistent decisions. Mathematically, classical optimal control problems are time-consistent in the sense that an optimal control found at a given initial pair of time and state will stay optimal thereafter for the corresponding initial pair. When the discount is general, not exponential, and/or the conditional expectations of the state and the control nonlinearly appear in the performance index, the corresponding optimal control problem will be time-inconsistent. To obtain time-consistent open-loop equilibrium strategies, we will use variational methods, together with theory of forward-backward stochastic differential equations. To obtain time-consistent closed-loop equilibrium strategies, we will modify dynamic programming principles, and adopt/introduce multi-person differential games. This project will enrich (stochastic) optimal control theory and related areas.