Optimal Control Problems with Time-Inconsistency and Related Topics

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This project is to establish a general theory of continuous-time optimal control problems (both deterministic and stochastic), for which the state equation and the cost functional are parameterized by the initial time and/or the initial state. For this type of problem, an optimal control/strategy, if it exists, will also depend on the initial time and/or the initial state. Therefore, in general, it will no longer be optimal immediately after the initial time. This feature is referred to as the time-inconsistency. To find time-consistent solutions to the time-inconsistent problem, the investigator will introduce a method of multi-person hierarchical differential games. It is expected that the limiting game as the number of players goes to infinity should lead to a time-consistent equilibrium control/strategy to the original problem. With such a general approach, this project will carefully investigate the following cases: (1) State equation is a linear ordinary differential equation or stochastic differential equation and the cost functional is convex. For such a case, the quasi-Riccati equation technique will be applied/extended to find the time-consistent equilibrium control; (2) State equation is a stochastic differential equation with the cost functional containing some functions of conditional expectation of the state. For such a case, the problem will be transformed to a controlled forward-backward stochastic differential equation (FBSDE) parameterized by the initial time and the initial state. Proper techniques involving maximum principle and dynamic programming, including stochastic partial differential equations, will be developed to solve the problem; (3) State equation is a stochastic Volterra integral equation. For such a case, the theory of backward stochastic Volterra integral equations (BSVIE) recently developed by the investigator will play an essential role, and time-consistent solution will be expected by combining the theory of BSVIEs and multi-person differential games. In real world, as time goes by, it is common that people change their minds or objectives in what can be described as an inconsistent way (due to, for example, change of income and/or living standard, etc.). Similarly, various changes of the environment (advances of technology, new limits of resources, etc.) lead people to inconsistently modifying their ways of running business from time to time. In both cases, one faces time-inconsistent problems, yet time-consistent strategies are desirable. The above considerations served the main motivation of the research in this project.Mathematically, this project will substantially enrich the general theory of deterministic and stochastic optimal control theory from a new aspect. It will have impact on stochastic analysis, mathematical finance, optimal control theory, and differential games. From the point-of-view of applications, the theories developed in this project will provide useful insights for time-inconsistency, nonlinear preferences, and dynamic cumulative prospect theory, etc. Therefore, the results will provide principles for people who are handling problems of optimal investment, asset pricing, risk management, resource (such as oil, power, etc.) allocation, production planning, etc. The expected results will be of interest to relevant theoretic researchers, practitioners in various type industries, as well as a number of government agencies.