【作者】 李敏；

【导师】 吴臻； 李娜； 周超；

【作者基本信息】 山东大学 ， 概率论与数理统计， 2022， 博士

【摘要】 在金融、经济、工程、管理、生物和社会学等实际背景下,随机大种群系统的动态优化问题一直是复杂系统研究领域的热门课题.事实上,由于外部噪声干扰、账户信息离散、技术手段受限以及潜在过程存在等因素,大种群问题的信息结构往往不能被个体全部获取.基于上述观察,完全信息和部分信息随机线性二次平均场博弈问题在理论研究与实际应用中具有重要的意义.本文围绕完全信息和部分信息随机大种群系统的动态优化问题,以平均场博弈理论、随机控制理论以及滤波技术为指导,以正倒向随机微分方程、Riccati方程以及偏微分方程为工具,对控制平均与状态平均多种耦合关系的随机大种群系统在不定情形带平均场、带跳扩散、部分观测、控制受约束部分信息以及非单调系数下的动态优化相关问题展开了全面深入地研究.(1)在不定系数情形下研究带平均场的随机大种群系统的动态优化问题(对应本文第二章).相应的分散化策略由代数方程和带平均场正倒向随机微分方程组成的随机Hamiltonian系统表示.进一步,利用解耦方法和两个Riccati方程,分散化策略可表示为反馈形式.同时,我们证明了不定情形下随机Hamiltonian系统和Riccati方程的适定性.另外,本章首次提出分离技术来确定控制平均极限的显式结构和相应的带平均场的纳什确定性等价系统.可以验证得到的分散化策略满足渐近均衡性质.最后,通过研究工程领域中的实例说明了理论结果的良好性能.(2)将带跳扩散的随机线性二次平均场博弈问题扩展到一般模型(对应本文第三章).本章解决了一类带泊松跳随机大种群系统的线性二次平均场博弈问题,其中状态过程和控制过程允许进入个体系统方程的跳扩散项.通过极限控制问题的随机Hamiltonian系统和Riccati方程,相应的分散化策略分别表示为开环形式和反馈形式.此外,利用第二章提出的分离技术,控制平均极限可显式确定出来.与此同时,可以验证分散化策略是原问题的ε-纳什均衡.最后,利用工程和经济领域两个实际控制问题说明了理论结果的良好性能.(3)针对部分观测的随机系统探索相应的线性二次平均场博弈问题(对应本文第四章).本章研究了一类部分观测的线性二次随机大种群问题,其中每个个体只能获得与状态相关的噪声观测过程.个体的系统状态满足由个人噪声和公共噪声联合驱动的线性随机微分方程且所有个体的相互作用通过控制平均实现.基于对平均场博弈问题的研究,本章利用带分解技术的倒向分离原理并借助带条件期望的正倒向随机微分方程,得到了开环形式的分散化策略.同时,我们也建立了相应的最优滤波方程.根据解耦方法和Riccati方程,分散化策略可以表示为状态滤波的反馈形式.另外,本章给出了控制平均极限的显式结构并讨论了相应的一致相容性条件系统.对应的渐近均衡性质也得到验证.最后,本章用一个金融中的实例说明了理论结果的有效性.(4)在部分信息框架下解决了控制受约束和控制无约束随机线性二次大种群动态优化问题(对应本文第五章).在控制受约束情形,借助Hamiltonian方法和凸分析的结果,分散化策略可以通过投影算子显式表达出来.相应的Hamiltonian型一致相容性条件系统满足一类带投影算子的非线性平均场正倒向随机微分方程.利用discounting方法构造压缩映射,本章证明了此类正倒向随机微分方程的全局适定性.相应的渐近均衡性质也得到验证.在控制无约束的情形,通过Riccati方法,分散化策略可进一步表示为状态滤波的反馈形式-与此同时,我们讨论了一类全新Riccati型一致相容性条件系统的解的存在唯一性结果.作为应用,本章考虑了一般银行间的借贷问题,并由此说明了部分信息的影响是不可忽视的.(5)实现了在非单调性条件假设下随机线性二次平均场控制博弈的均衡控制(对应本文第六章).本章个体之间相互作用通过状态和控制的联合分布实现.根据随机最大值原理,我们首先分析了代表个体的极限行为,并在给定的状态和控制分布流下得到其反馈形式最优控制.相应的平均场均衡由纳什确定性等价系统决定.由于公共噪声的存在,本章在不需要任何单调性条件下建立了纳什确定性等价系统和主方程的全局适定性结果.进一步假设异质噪声的非退化性,我们直接求解了相应的N-玩家博弈问题.利用矩阵理论的精确分析以及高维隐函数定理,反馈形式的开环纳什均衡可通过两个可解N-耦合正倒向随机微分方程刻画.与此同时,我们给出了对应于上述两个正倒向随机微分方程系统的N-耦合抛物偏微分方程适定性结果.基于方程的适定性,本章的最后证明了从N-玩家博弈到平均场博弈问题的定量收敛性结果以及相应最优轨迹的混沌传播性质.更多还原

【Abstract】 Under the practical background of finance,economics,engineering,management,biology and sociology,the dynamic optimization problems of stochastic large population systems have always been a hot topic in the field of complex systems.In fact,due to the external noise interference,the discrete account information,the restriction of technology and potential processes,individuals usually cannot obtain the full information of large population systems.Based on the above observations,the stochastic linearquadratic mean field game problems with full information and partial information are of great significance in theory and practical applications.This dissertation focuses on the dynamic optimization problems of stochastic large population systems with full information and partial information.Based on the mean field game theory,stochastic control theory and filtering techniques,using forwardbackward stochastic differential equations,Riccati equations and partial differential equations as tools,we study a series of dynamic optimization problems for the control average and state average stochastic large population systems,including indefinite mean field case,jump diffusion model,partial observation case,partial information control constrained and non-monotone data case.(1)We research on dynamic optimization problems of mean field stochastic large population systems with indefinite matrix coefficients(Corresponding to Chapter 2 of this dissertation).The related decentralized strategies are represented through a stochastic Hamiltonian system,which consists of an algebra equation and a mean field forward-backward stochastic differential equation.Moreover,using decoupling methods and two Riccati equations,the decentralized strategies are further presented in the feedback form.Meanwhile,we also prove the solvability of the stochastic Hamiltonian system and Riccati equation under the indefinite condition.This chapter first proposes separation techniques to determine the explicit structure of control average limit and the corresponding mean field Nash certainty equivalence system.It is verified that the decentralized strategies satisfy the approximate Nash equilibrium property.In the end,a practical example from the engineering field is discussed to demonstrate the good performance of theoretical results.(2)We extend the stochastic linear-quadratic mean field game problem with jump diffusion to a more general model(Corresponding to Chapter 3 of this dissertation).This chapter studies a class of linear-quadratic mean field game problems for stochastic large population systems with Poisson jumps.The state process and the control process are allowed into the jump diffusion term of system dynamics.By stochastic Hamiltonian system and Riccati equation of the limiting control problem,the corresponding decentralized strategies are represented in the open-loop form and the feedback form,respectively.Furthermore,by using the separation techniques introduced in Chapter 2,we can determine the control average limit explicitly.The decentralized strategies turn out to be the e-Nash equilibrium of the original problem.For illustrations,two control problems in engineering and economy are well discussed.(3)As for the partially observed stochastic system,we explore the corresponding linear-quadratic mean field game problem(Corresponding to Chapter 4 of this dissertation).This part considers a class of linear-quadratic stochastic large population problems with partial observation,where each agent can only observe a noisy process related to the state.The system dynamic of each agent satisfies a stochastic differential equation driven by individual noise and common noise,and all the agents are coupled through control average.Based on the mean field game theory,this chapter uses the backward separation principle with state decomposition technique and the forwardbackward stochastic differential equation with conditional expectation to give the openloop decentralized strategies.At the same time,we establish the corresponding optimal filtering equation.According to the decoupling method and Riccati equation,the decentralized strategies are further derived in the feedback form of the filtered state.Moreover,this chapter gives the explicit control average limit and discusses the consistency condition system.The approximate Nash equilibrium property is also verified.Finally,this chapter illustrates the validity of theoretical results with an example in finance.(4)We solve the dynamic optimization problems for control constrained and control unconstrained stochastic linear-quadratic large population systems with partial information(Corresponding to Chapter 5 of this dissertation).In control constrained case,by using Hamiltonian approach and convex analysis,the explicit decentralized strategies can be obtained through projection operator.The corresponding Hamiltonian type consistency condition system is derived,which turns out to be a nonlinear mean field forward-backward stochastic differential equation with projection operator.The well-posedness of such kind of equations is proved by using discounting method.In control unconstrained case,the decentralized strategies can be further represented explicitly as the feedback of filtered state through Riccati approach.The existence and uniqueness of solution to a new Riccati type consistency condition system is also discussed.As an application,a general inter-bank borrowing and lending problem is studied to illustrate the effect of partial information cannot be ignored.(5)We realize equilibrium for stochastic linear-quadratic mean field games of controls with non-monotone data(Corresponding to Chapter 6 of this dissertation).In this chapter,we study a class of linear-quadratic mean field game of controls with common noise and their corresponding N-player game.The theory of mean field game of controls considers a class of mean field games where the interaction is via the joint law of both the state and control.By the stochastic maximum principle,we first analyze the limiting behavior of the representative player and obtain the optimal control in a feedback form with the given distributional flow of the population and its control.The mean field equilibrium is determined by the Nash certainty equivalence system.Thanks to the common noise,we do not require any monotonicity conditions for the solvability of the Nash certainty equivalence system.We also study the master equation arising from the linear-quadratic mean field game of controls,which is a finitedimensional second-order parabolic equation.It can be shown that the master equation admits a unique classical solution over an arbitrary time horizon without any monotonicity conditions.Beyond that,we can solve the N-player game directly by further assuming the non-degeneracy of the idiosyncratic noises.As by products,we prove the quantitative convergence results from the N-player game to the mean field game and the propagation of chaos property for the related optimal trajectories.更多还原