Author:  Wang, Shujun 
Title:  Linear quadratic mean field games of forwardbackward stochastic systems 
Degree:  Ph.D. 
Year:  2016 
Subject:  Mathematical optimization. Mean field theory. Game theory. Hong Kong Polytechnic University  Dissertations 
Department:  Dept. of Applied Mathematics 
Pages:  xvi, 154 pages : illustrations 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b2890621 
URI:  http://theses.lib.polyu.edu.hk/handle/200/8482 
Abstract:  The thesis is concerned with the linear quadratic (LQ) mean field games (MFGs) involving forwardbackward stochastic differential equations (FBSDEs). Five topics are under consideration: 1. The largepopulation dynamic optimization in forwardbackward setting. 2. The backward LQ games of stochastic largepopulation systems. 3. The largepopulation systems in majorminor framework. 4. The combination problems of leaderfollower and majorminor largepopulation systems. 5. The dynamic optimization of largepopulation systems with partial information. For the first topic, a class of dynamic optimization problems of largepopulation are formulated. The most significant feature in this setup is the dynamics of individual agents follow the FBSDEs in which the forward and backward states are coupled at the terminal time. The related LQMFG, in its forwardbackward sense, is also formulated to seek the decentralized strategies. Unlike the forward case, the consistency conditions of the forwardbackward MFGs involve six Riccati and force rate equations. Moreover, their initial and terminal conditions are mixed which requires some special decoupling technique. The εNash equilibrium property of the derived decentralized strategies is also verified. To this end, some estimates to backward stochastic system are employed. In addition, due to the adaptiveness requirement to forwardbackward system, all arguments here are not parallel to those in its forward case. For the second topic, the backward LQMFGs of weakly coupled stochastic largepopulation system are studied. In contrast to the wellstudied forward LQMFGs, the individual state in this largepopulation system follows the backward stochastic differential equation (BSDE) whose terminal instead of initial condition should be prescribed. The individual agents of largepopulation system are weakly coupled in their state dynamics and the full information is accessible to all agents. The explicit form of the limiting process and εNash equilibrium of the decentralized control strategy are investigated. To this end, some estimates to BSDE, are presented in the largepopulation setting. For the third topic, the backwardforward LQ games with major and minor players are investigated. In this topic, the dynamics of major player is given by a BSDE; while dynamics of minor players are described by (forward) SDEs. A backwardforward stochastic differential equation (BFSDE) system is established in which a large number of negligible agents are coupled in their dynamics via state average. The problem when major player takes into account the relative performance by comparison to minor players is under consideration. Some auxiliary mean field (MF) SDEs and a 3 x 2 mixed FBSDE system are considered and analyzed instead of involving the fixedpoint analysis. The decentralized strategies are derived, which are also shown to satisfy the εNash equilibrium property. For the fourth topic, the combination problems of leaderfollower and majorminor largepopulation systems are proposed. In the entire system, the major and minor agents are together regarded as the leaders, which are called majorleader and minorleaders, respectively. The majorleader tracks a convex combination of the centroid of the minorleaders and the followers; the minorleaders track a convex combination of their own centroid and the majorleader's dynamics; and the followers track a convex combination of their own centroid and the centroid of the minorleaders or a convex combination of the centroid of the minorleaders and the majorleader's dynamics. As the applications of leaderfollower and majorminor theory, the analysis of this problem is only presented as a framework and three consistency condition systems are obtained. For the fifth topic, the dynamic optimization of largepopulation systems with partial information is considered. In this topic, the individual agents can only access the filtration generated by one observable component of the underlying Brownian motion. The stateaverage limit in this setup turns out to be some stochastic process driven by the common Brownian motion. Two classes of MFGs are proposed in this framework: one is governed by forward dynamics, and the other involves the backward one. In the forward case, the associated MFG is formulated and its consistency condition is equivalent to the wellposedness of some Riccati equation system. In the backward case, the explicit forms of the decentralized strategies and some BSDE (satisfied by the limiting process) are obtained. In both cases, the εNash equilibrium properties are presented. 
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