Mean field optimal stopping

Abstract

We are interested in the study of the mean field optimal stopping problem, that is the optimal stopping of a McKean-Vlasov diffusion, when the criterion to optimize is a function of the distribution of the stopped process. This problem models the situation where a central planner controls a continuous infinity of interacting agents by assigning a stopping time to each of them, in order to maximize some criterion which depends on the distribution of the system. We study this problem via a dynamic programming approach, which allows to characterize its value function by a partial differential equation on the space of probability measures, that we call obstacle problem (or equation) on Wasserstein space by analogy with the classical obstacle problem, which arises in particular in standard optimal stopping. We also study the corresponding finite population problem and prove its convergence to the mean field problem, and provide an application of our theory to several problems arising in economics and in finance.