A Kind Of Linear-Quadratic Mean-Field Game Of Stochastic System With Incomplete Information

Large-population system exists in finance,management and other fields,and it has wide applications.Recently,with the development of the theory of stochastic control,the largepopulation game of stochastic system has attracted more attentions,which has important theoretical significance and practical value.In most of the existing works,it assumed that the participants can observe the full information.In general,due to existence of external interference factors,participants can not observe the full information.Therefore,in this dissertation,a kind of linear-quadratic mean-field game of stochastic system with incomplete information is studied.Incomplete information includes two cases:partial information and partially observable information.Partial information represents a subfiltration of the full information,which does not depend on control;partially observable information is a filtration generated by the observation process,where the observation process is related with the state,and it relies on control.The research contents of this dissertation include the following four aspects:1.Optimal control of backward stochastic system with partial informationThe partial information linear-quadratic optimal control problems driven by BSDE and MF-BSDE are considered.Using stochastic maximum principle,the corresponding Hamiltonian system is derived,which is a conditional MF-FBSDE.By backward separation approach and filtering technique,decoupling the Hamiltonian system two times,a feedback form of the optimal control is obtained.The results obtained provides a foundation for mean-field game of backward stochastic system with partial information.In fact,in order to study the linear-quadratic mean-field game for backward stochastic system with partial information,one needs to introduce an auxiliary control problem,which is a kind of linear-quadratic control problem of backward stochastic system with partial information.By virtue of the theoretical results obtained in this section,the optimal control of auxiliary control problem can be obtained.2.Mean-field game of backward stochastic system with partial informationA linear-quadratic mean-field game of backward stochastic system with partial information and common noise is investigated.By virtue of stochastic maximum principle and optimal filter technique,a Hamiltonian system is obtained first,which is a fully-coupled FBSDE.By virtue of three ODEs,a forward and a backward optimal filtering equations,an optimal control(decentralized control strategy)of the auxiliary limiting control problem is obtained.Furthermore,it verifies that a decentralized control strategy obtained is an ε-Nash equilibrium of the original mean-field game.The theoretical results can be used to solve a product pricing problem and a network security control problem.However,the backward stochastic system is used to describe the scenario with a prescribed terminal condition,and it is difficult to describe the recursive utility optimization problem in the large-population setting.Forward-backward stochastic system provides an effective tool to solve recursive utility optimization problem.Therefore,it has important significance to study the mean-field game driven by forward-backward stochastic system.In addition,partial information is a special case of incomplete information,and partially observable information is a more general case.Thus,one needs to study the mean-field game of forward-backward stochastic system with partial observation information.3.Mean-field game of forward-backward stochastic system with partial observation informationA linear-quadratic mean-field game driven by FBSDE with partial observation and common noise is investigated.Applying variational technique and optimal filter technique,a decentralized control strategy and a consistency condition system are derived.Furthermore,by virtue of dimensional-expansion technique and ODEs,the solvability of the consistency condition system is established.Employing the estimate techniques of FBSDE,the asymptotic equilibrium property of the decentralized control strategy obtained is given.Moreover,the theoretical results can be used to solve a kind of wealth investment problem.In some practical problems,in order to achieve a more satisfactory result,the state of the system may be constrained.Therefore,it is necessary to study the mean-field game with terminal state constraint.4.Mean-field game of stochastic system with state constraintA linear-quadratic mean-field game of stochastic differential system with terminal state constraint and common noise is studied.By virtue of mean-field method,an auxiliary problem of the original game is introduced,which is a constrained optimal control problem.Then,applying Lagrangian multiplier technique,duality theory and stochastic maximum principle,a constrained control problem is transformed to an unconstrained control problem,and an explicit form of the optimal Lagrangian multiplier and a decentralized control strategy depending on the optimal Lagrangian multiplier are derived.Furthermore,one can prove that the decentralized control strategy obtained is an ε-Nash equilibrium of the linear-quadratic mean-field game.The theoretical results obtained can be used to solve a kind of production problem.