Optimal Control Problems with Time-Inconsistency and Related Topics

本项目旨在建立连续时间最优控制问题（包括确定性和随机性）的一般理论，其中状态方程和成本泛函由初始时间和/或初始状态参数化。 对于这种类型的问题，最优控制/策略（如果存在）也将取决于初始时间和/或初始状态。 因此，一般来说，在初始时间之后立即将不再是最佳的。 这个特征被称为时间不一致性。 为了找到时间不一致问题的时间一致解决方案，研究者将介绍一种多人分层微分对策的方法。 预计当参与者的数量达到无穷大时，限制博弈应该导致对原始问题的时间一致的均衡控制/策略。 利用这样的通用方法，本项目将仔细研究以下情况：（1）状态方程是线性常微分方程或随机微分方程，并且成本泛函是凸的。 对于这种情况，拟Riccati方程技术将被应用/推广到寻找时间一致的平衡控制;（2）状态方程是一个随机微分方程，其代价泛函包含一些状态的条件期望函数。在这种情况下，问题将转化为一个由初始时间和初始状态参数化的受控正倒向随机微分方程。 运用最大值原理和动态规划的方法，包括随机偏微分方程的求解方法;（3）状态方程是一个随机的沃尔泰拉积分方程。 对于这种情况，研究者最近发展的倒向随机沃尔泰拉积分方程（BSVIE）理论将发挥重要作用，并将BSVIE理论与多人微分对策相结合，有望得到时间一致的解. 在真实的世界中，随着时间的推移，人们通常会以一种不一致的方式改变他们的想法或目标（例如，由于收入和/或生活水平的变化等）。 同样，环境的各种变化（技术的进步、资源的新限制等）导致人们不时地改变他们的经营方式。 在这两种情况下，人们都面临着时间不一致的问题，但时间一致的策略是可取的。 以上考虑是本项目研究的主要动机，在数学上，本项目将从一个新的角度丰富确定性和随机最优控制理论的一般理论。 它将对随机分析、数学金融、最优控制理论和微分对策产生影响。 从应用的角度来看，本研究所发展的理论将为时间不一致性、非线性偏好、动态累积前景理论等提供有益的启示，从而为人们处理最优投资、资产定价、风险管理、资源（如石油、电力等）分配，生产计划等。预期的结果将感兴趣的相关理论研究人员，从业人员在各种类型的行业，以及一些政府机构。

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.