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

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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.