随机微分对策问题中开环与闭环纳什均衡的研究

The Nash equilibrium, named after American mathematician John Forbes Nash Jr., who was also a winner of Nobel Memorial Prize in Economic Sciences, is one of the central research objects in differential games. It could be classified into two types according to the dependence on the state of controlled system: open-loop and closed-loop Nash equilibria, both of which are significantly important in theoretical research and practical applications. The purpose of this project is to study these two kinds of equilibria in the following frameworks: games of pure controls, in which both players control the state process; and games of control and stopping, in which the controller can affect the state process while the stopper decides the duration of the game. We shall deeply investigate the operator structure of the game from a functional analysis point of view, use dynamic programming principle and forward-backward stochastic differential equations to establish verification theorems, and combine the theories of optimal control and optimal stopping to develop characterizations and construction methods for both equilibria. Further, we shall deepen the results on controllability of stochastic systems and then explore differential games with state constraints by means of Lagrangian duality. For linear-quadratic stochastic differential games which have nice mathematical structures, we are expecting to obtain some more delicate results.

以美国数学家、诺贝尔经济学奖得主 John Forbes Nash Jr. 命名的纳什均衡是微分对策中的核心研究对象之一。根据对受控系统状态的依赖性，纳什均衡可分为开环纳什均衡和闭环纳什均衡。两者在理论研究与实际应用中均有着重要的研究价值。本项目旨在研究两类对策中的开环纳什均衡和闭环纳什均衡：博弈双方均对状态系统施加控制的纯控制型随机微分对策和博弈双方分别控制系统状态及博弈时长的控制-停时混合型随机微分对策。我们将从泛函分析角度出发深入研究博弈的算子结构，利用动态规划原理、正倒向随机微分方程等工具建立验证定理，结合随机最优控制和随机最优停时理论给出两种纳什均衡的刻画及构造方法。进一步，我们将深化随机系统能控性方面的结果，利用拉格朗日共轭理论研究状态受约束的对策问题。对于数学结构较好的线性二次随机微分对策问题，我们将给出更为精细的结果。

随机微分对策在诸多领域具有广阔的应用背景，特别是在金融经济和工业工程中。在庞特里亚金最大值原理建立之后，人们清楚地认识到微分对策和最优控制有着紧密的联系。本项目深入研究了随机微分对策的相关理论及其衍生出的随机最优控制问题，在多个方面取得了一批国内外领先的理论成果。这些成果包括：随机微分对策的纳什均衡理论，平均场随机微分对策理论，Stackelberg随机微分对策理论，不定型正向随机线性二次最优控制理论，倒向随机线性二次最优控制理论，能控条件下的终端约束最优控制问题，随机最优控制的大道理论等。本项目在国家自然科学基金的资助下，已在本领域国际权威期刊发表了8篇高水平的学术论文以及2部英文学术专著，并另有2篇论文即将发表。所发表的期刊包括SIAM Journal on Control and Optimization、Annals of Applied Probability、Journal of Differential Equations、ESAIM: Control, Optimisation and Calculus of Variations、Chinese Annals of Mathematics-Series B、Mathematical Control and Related Fields等。2部英文学术专著于2020年由Springer出版社出版。此外，在项目执行期间，项目负责人成功组织了3次大型学术会议，多次参加国内外学术会议，并培养了2名硕士研究生。