Mean Field Theory, Stochastic Control and Systems of Partial Differential Equations

Description

Since the introduction of the concept of mean field games, an intensive research has been conducted in many countries. The interest stems from both the new mathematical problems which have emerged and the diversity of applications. Mean field type control, which has similarities with mean field games, is a different direction. It has led to new systems of forward-backward stochastic differential equations with applications to non-standard stochastic control problems. Our first objective is to provide a unified approach to these two research directions and to compare the corresponding solutions. A very interesting comparison can be done in the Linear-Quadratic case.Our comparison relates primarily to methodology; indeed, optimal control theory has developed on two major approaches: Dynamic Programming and Maximum Principle. The first one leads to a partial differential equation (PDE) called the Hamilton-Jacobi-Bellman (HJB) and the second one to a system of forward-backward stochastic differential equations. When one considers mean field games, the HJB approach has been first developed, the HJB equation becomes a coupled HJB-FP (Fokker-Planck) equation. On the other hand, for the mean field type control problem, it is mainly the stochastic maximum principle approach. We believe it is important to show that, like in traditional control theory, the two approaches are possible. Besides, to our best knowledge, the HJB-FP approach to mean field type control leads to interesting systems of PDE which have never appeared in the literature. What can be said in general remains an open question. The remarkable success of mean field games is that, although not directly a game but a control problem, it provides a very simple approximation to differential games of Nash type, when the number of players is very large each of whom is identical with others. The limitation is, however, serious: all players must be identical. Our second objective is to waive this limitation, and explore the situation of competition between large coalitions. We encounter here new systems of partial differential equations, which are mathematically challenging. Indeed, systems of coupled HJB-FP equations are much more complicated than a single coupled HJB-FP equation. In addition to large coalitions, we also plan to consider the situation of major players which is similar to Stackelberg games. In this case, because the representative agent of the large group takes into account the major player, his/her decisions become conditional on the major player and hence random. The system of HJB-FP coupled equations becomes random, and one has to solve a system of forward-backward stochastic partial differential equations; so far only partial solutions exist in the literature. Among the applications, we are interested in risk management. In this context, the objective functionals to be optimized are not just expected values, but take into account other moments, primarily the variance of the random objective. These problems are not standard stochastic control problems, but interestingly, they lead to HJB-FP coupled equations or systems of equations. That opens a very fruitful direction of research in the blossoming area of risk management.