Time-Inconsistent Optimal Control Problems for Stochastic Differential Equations

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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.