Mean field control with stopping

TL;DR

This paper develops a mean field control framework incorporating stopping, where a central planner manages a large population with the ability to remove particles, leading to a novel QVI characterization and convergence results.

Contribution

It introduces a new mean field control model with stopping, characterizes the value function via viscosity solutions of a QVI, and proves convergence of finite particle systems.

Findings

Characterization of the mean field value function as a unique viscosity solution.

Establishment of a comparison principle for the QVI.

Proof of convergence of N-particle value functions to the mean field limit.

Abstract

We study a high-dimensional stochastic optimization problem which features both control and stopping. In particular, a central planner steers a large population of particles, and can also remove particles at any time by paying a penalty. In the limit, we obtain a mean field control problem with discontinuous dynamics, in the sense that the controlled Fokker-Planck equation can have jumps. The value of the N-particle problem is characterized by a hierarchy of non-linear obstacle problems. The value of the limiting problem, meanwhile, solves an infinite-dimensional quasi-variational inequality (QVI). We introduce a notion of viscosity solution for this QVI, and obtain a comparison principle. Together with various regularity estimates, this comparison principle allows us to characterize the mean field value function as the unique viscosity solution of the QVI, and to establish the convergence of the N-particle value functions to the mean field value function.