部分信息下状态受约束平均场随机控制理论研究

Lots of phenomena arising from the areas of quantum chemistry, statistical mechanics and mathematical finance can be described by mean-field type stochastic systems. Optimal control problems of mean-field type stochastic systems are of important theoretical and practical significance, and then, it is worth making a further research about the control problems. The existing corresponding works are mainly based on the assumptions of full information, non-constraint states and convex control domains. This program, however, is interested in studying optimal control problems of mean-field type stochastic systems with partial information, especially with the case of state constraints and non-convex control domains. By virtue of some approaches and techniques from stochastic system, filtering and control, non-linear programming, particle system and mathematical finance, the existence and uniqueness of mean-field type stochastic systems with square-intergrable parameters are proved, and the corresponding numerical algorithm is proposed. General maximum principle (necessary condition) and verification theorem (sufficient condition) for optimality are obtained. Optimal feedback solutions for linear-quadratic control problems of mean-field type stochastic systems with state constraints are studied, and are used to solve several important problems from mathematical finance. We hope the theoretical results obtained in this program can provide help for the financial analysis.

平均场随机系统能有效描述量子化学、统计力学、数理经济领域的许多现象，其最优控制问题具有重要的理论意义与实际价值，值得深入研究。现有相关文献大多基于完全信息、自由状态和凸控制域的假定，而本项目则侧重研究部分信息下的平均场随机优化控制问题，特别是状态受约束、控制域非凸情形下的控制理论和数值算法。运用随机系统、滤波控制、非线规划、粒子系统、数理经济等领域的理论工具，证明具有平方可积系数平均场随机系统解的存在唯一性，提出其数值解法，推出最优控制满足的一般极大值原理（必要条件）和验证定理（充分条件），探讨状态受约束平均场随机系统线性二次控制的最优反馈解，并用于求解数理经济领域的几个热点问题，以期为金融实务分析提供理论指导。

平均场随机系统能有效描述量子化学、统计力学、数理经济领域的许多现象，其最优控制问题具有重要的理论意义与实际价值，值得深入研究。现有相关文献大多基于完全信息、自由状态等假定，而本项目则侧重研究部分信息下的平均场随机优化控制问题，特别是状态受约束情形下的控制理论。例如：运用随机系统、滤波控制、数理经济等领域的理论工具，推出了最优控制满足的极大值原理，得到了状态受约束平均场随机系统线性二次控制的最优反馈控制，求解了一类资产-债务管理问题，为主从博弈研究和金融实务分析提供了理论指导。