Stochastic Linear-Quadratic Mean Field Game Problems With Full Information And Partial Information

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.