Extended mean field control problem: a propagation of chaos result

TL;DR

This paper introduces an extended mean field control framework involving joint distributions of states and controls, formulates it via a controlled Fokker-Planck equation, and proves its limit relation to large population control problems.

Contribution

It develops a new optimization formulation for extended mean field control problems using measure-valued processes and establishes their propagation of chaos property.

Findings

Formulation of an optimization problem over measure-valued processes.

Proof of equivalence between the optimization problem and the extended mean field control.

Establishment of the limit of large population control problems to the extended mean field control.

Abstract

In this paper, we study the $extended$ mean field control problem, which is a class of McKean-Vlasov stochastic control problem where the state dynamics and the reward functions depend upon the joint (conditional) distribution of the controlled state and the control process. By considering an appropriate controlled Fokker-Planck equation, we can formulate an optimization problem over a space of measure-valued processes and, under suitable assumptions, prove the equivalence between this optimization problem and the $extended$ mean-field control problem. Moreover, with the help of this new optimization problem, we establish the associated limit theory i.e. the $extended$ mean field control problem is the limit of a large population control problem where the interactions are achieved via the empirical distribution of state and control processes.