On The Mean Field Control And Stochastic Dynamic Games

In this thesis,using the mean field game theory,we consider three problems of stochastic differential game or optimal control as follows:(ⅰ)optimal consumption in stochastic growth model with jumps;(ⅱ)optimal production output with price stickiness in the market;(ⅲ)centralized systemic risk control in the interbank system.Approximating solutions are established through the optimal control of appropriate limiting control problem.In chapter 3,we study a class of dynamic games consisting of finite agents under a stochastic growth model with jumps.The jump process in the dynamics of the capital stock of each agent models announcements regarding each agent in the game occur at Poisson distributed random times.The aim of each agent is to maximize her objective functional with mean field interactions by choosing an optimal consumption strategy.We prove the existence of a unique fixed point related to the so-called consistence condition in the limit of the infinite number of agents.Building upon the fixed point,we establish an optimal feedback consumption strategy for all agents which is in fact an approximating Nash equilibrium which describes strategies for each agent such that no agent has any incentive to change her strategy.Chapter 4 considers a production control problem with price stickiness when the number of firms grows large in the market.Each firm aims to maximize its expectation of the total net profit relying on its production and the price process for all firms over an infinite horizon.The price process of each firm relies on the production rate of all firms via a mean field interaction.We establish a MFG(mean field game)of this production control problem.Moreover,a decentralized approximating Nash equilibrium is constructed as the number of firms goes to infinity.Chapter 5 studies a systemic risk control problem by the central bank,which dynamically plans monetary supply for the interbank system with borrowing and lending activities.Facing both heterogeneity among banks and the common noise,the central bank aims to find an optimal strategy to minimize the average distance between logmonetary reserves and some prescribed capital levels for all banks.A weak formulation is adopted,and an optimal randomized control can be obtained in the system with finite banks by applying Ekeland’s variational principle.As the number of banks grows large,we further prove the convergence of optimal strategies using the Gamma-convergence arguments,which yields an optimal weak control in the mean field model.It is shown that the limiting optimal control is linked to a solution of a stochastic Fokker-PlanckKolmogorov(FPK)equation.The uniqueness of the solution to the stochastic FPK equation is also established under some mild conditions.