Closed-Loop Solvability of Linear Quadratic Mean-Field Type Stackelberg Stochastic Differential Games

This paper is devoted to a Stackelberg stochastic differential game for a linear mean-field type stochastic differential system with a mean-field type quadratic cost functional in finite horizon. The coefficients in the state equation and weighting matrices in the cost functional are all deterministic. Closed-loop Stackelberg equilibrium strategies are introduced which require to be independent of initial states. Follower's problem is solved firstly, which is a stochastic linear quadratic optimal control problem. By converting the original problem into a new one whose optimal control is known, the closed-loop optimal strategy of the follower is characterized by two coupled Riccati equations as well as a linear mean-field type backward stochastic differential equation. Then the leader turns to solve a stochastic linear quadratic optimal control problem for a mean-field type forward-backward stochastic differential equation. Necessary conditions for the existence of closed-loop optimal strategies for the leader is given by the existence of two coupled Riccati equations with a linear mean-field type backward stochastic differential equation. The solvability of Riccati equations of leader's optimization problem is discussed in the case where the diffusion term of the state equation does not contain the control process of the follower. Moreover, leader's value function is expressed via two backward stochastic differential equations and two Lyapunov equations.Comment: 44 page

Backward Stackelberg Differential Game with Constraints: a Mixed Terminal-Perturbation and Linear-Quadratic Approach

We discuss an open-loop backward Stackelberg differential game involving single leader and single follower. Unlike most Stackelberg game literature, the state to be controlled is characterized by a backward stochastic differential equation (BSDE) for which the terminal- instead initial-condition is specified as a priori; the decisions of leader consist of a static terminal-perturbation and a dynamic linear-quadratic control. In addition, the terminal control is subject to (convex-closed) pointwise and (affine) expectation constraints. Both constraints are arising from real applications such as mathematical finance. For information pattern: the leader announces both terminal and open-loop dynamic decisions at the initial time while takes account the best response of follower. Then, two interrelated optimization problems are sequentially solved by the follower (a backward linear-quadratic (BLQ) problem) and the leader (a mixed terminal-perturbation and backward-forward LQ (BFLQ) problem). Our open-loop Stackelberg equilibrium is represented by some coupled backward-forward stochastic differential equations (BFSDEs) with mixed initial-terminal conditions. Our BFSDEs also involve nonlinear projection operator (due to pointwise constraint) combining with a Karush-Kuhn-Tucker (KKT) system (due to expectation constraint) via Lagrange multiplier. The global solvability of such BFSDEs is also discussed in some nontrivial cases. Our results are applied to one financial example.Comment: 38 page

Mean field games with imperfect information

In this thesis, three topics in Mean Field Games in the absence of complete information have been studied. The first part of the thesis focus on Mean Field Stackelberg Games between a large group of followers and a leader, in such a way that each follower is subject to a delay effect inherited from the leader. The case with delays being identical among the followers in the population is first considered. Under mild assumptions of regular enough coefficients, the whole Stackelberg game problem is solved via stochastic maximum principle. The solution could be represented by a system of six coupled forward backward stochastic differential equations. A comprehensive study on the particular Linear Quadratic case has been provided. By considering the corresponding linear functional, the time-independent sufficient condition which warrants the unique existence of the solution of the whole Stackelberg game is obtained. Several numerical examples are also demonstrated. The second work studies another class of Stackelberg games, under a Linear Quadratic setting, in the presence with an additional leader. Given the trajectories of the mean field term and two leaders, the follower's optimal control problem is first solved. Depending on whether or not the leaders cooperate, the solutions of the respective Pareto and Nash games between the leaders are obtained, which can be represented by systems of forward backward stochastic functional differential equations. To numerically implement the obtained results, explicit expression of solutions of the whole problem: Mean Field Game among the followers and Nash (and Pareto) Game between the leaders, are provided. Finally, several examples are given to study the impact of different games on the cost functionals of the followers. An interesting example shows that the population are worse off as the leaders cooperate. The last part of the thesis studies discrete time partially observable mean field systems in the presence of a common noise. Each player makes decision solely based on the observable processes but not the common noise. Both the mean field game and the associated mean field type stochastic control problem are formulated. The mean field type control problem is solved by adopting the classical discrete time Kalman filter with notable modifications; indeed, the unique existence of the resulting forward-backward stochastic difference system is then established by Separation Principle. The mean field game problem is also solved via an application of stochastic maximum principle, while the existence of the mean field equilibrium is shown by the Schauder's fixed point theorem.Open Acces