Abstract
We present a new notion of solution for mean field games master equations. This notion allows us to work with solutions which are merely continuous. We first prove results of uniqueness and stability for such solutions. It turns out that this notion is helpful to characterize the value function of mean field games of optimal stopping or impulse control and this is the topic of the second half of this paper. The notion of solution we introduce is only useful in the monotone case. In this article we focus on the finite state space case.
Highlights
This paper introduced a new notion of solution of the mean field games (MFG in short) master equation in the monotone setting and applies it to master equations of games involving optimal stopping or impulse control
The limit master equation. — We show how we can characterize the value function of a MFG of optimal stopping using the notion of monotone solutions
Results for particular mean field games of impulse control. — In the present section, we assume a particular form for the jump operator M and we prove a result of uniqueness for monotone solutions in the impulse control case
Summary
This paper introduced a new notion of solution of the mean field games (MFG in short) master equation in the monotone setting and applies it to master equations of games involving optimal stopping or impulse control. Several authors have worked to define weak solutions of the MFG master equation, in the non-monotone regime, namely [34, 25, 26] in the continuous state space case and [21] in the finite state space case Their approaches are quite different from the one we adopt here (because monotonicity assumptions are crucial to this paper), they provide interesting results and approaches. — An objective of this paper is to study the master equation, in finite state space, associated to optimal stopping or impulse controls In those situations, the players can, respectively, decide to exit the game or to change instantly their state to any other state. Acknowledgments. — We would like to thank Pierre-Louis Lions for pointing out to us the question of optimal stopping in MFG a few years ago and for the numerous discussions we had on this topic
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