|Title:||Linear quadratic mean field games of forward-backward stochastic systems|
Mean field theory.
Hong Kong Polytechnic University -- Dissertations
|Department:||Department of Applied Mathematics|
|Pages:||xvi, 154 pages : illustrations|
|Abstract:||The thesis is concerned with the linear quadratic (LQ) mean field games (MFGs) involving forward-backward stochastic differential equations (FBSDEs). Five topics are under consideration: 1. The large-population dynamic optimization in forward-backward setting. 2. The backward LQ games of stochastic large-population systems. 3. The large-population systems in major-minor framework. 4. The combination problems of leader-follower and major-minor large-population systems. 5. The dynamic optimization of large-population systems with partial information. For the first topic, a class of dynamic optimization problems of large-population 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 forward-backward sense, is also formulated to seek the decentralized strategies. Unlike the forward case, the consistency conditions of the forward-backward MFGs involve six Riccati and force rate equations. Moreover, their initial and terminal conditions are mixed which requires some special decoupling technique. The ε-Nash 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 forward-backward system, all arguments here are not parallel to those in its forward case. For the second topic, the backward LQMFGs of weakly coupled stochastic large-population system are studied. In contrast to the well-studied forward LQMFGs, the individual state in this large-population system follows the backward stochastic differential equation (BSDE) whose terminal instead of initial condition should be prescribed. The individual agents of large-population 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 ε-Nash equilibrium of the decentralized control strategy are investigated. To this end, some estimates to BSDE, are presented in the large-population setting.|
For the third topic, the backward-forward 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 backward-forward 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 fixed-point analysis. The decentralized strategies are derived, which are also shown to satisfy the ε-Nash equilibrium property. For the fourth topic, the combination problems of leader-follower and major-minor large-population systems are proposed. In the entire system, the major and minor agents are together regarded as the leaders, which are called major-leader and minor-leaders, respectively. The major-leader tracks a convex combination of the centroid of the minor-leaders and the followers; the minor-leaders track a convex combination of their own centroid and the major-leader's dynamics; and the followers track a convex combination of their own centroid and the centroid of the minor-leaders or a convex combination of the centroid of the minor-leaders and the major-leader's dynamics. As the applications of leader-follower and major-minor 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 large-population 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 state-average 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 ε-Nash equilibrium properties are presented.
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