Extended Mean Field Control Problems: stochastic maximum principle and transport perspective

TL;DR

This paper extends mean field stochastic control theory by developing a Pontryagin maximum principle for joint distribution-dependent dynamics and costs, and links it to optimal transport on path space for new discretization methods.

Contribution

It introduces a generalized Pontryagin maximum principle for mean field control problems with joint distribution dependence and connects it to optimal transport theory.

Findings

Extended maximum principle applicable to joint distribution dependence

Established connection between control problems and optimal transport on path space

Proposed a new discretization scheme inspired by this connection

Abstract

We study Mean Field stochastic control problems where the cost function and the state dynamics depend upon the joint distribution of the controlled state and the control process. We prove suitable versions of the Pontryagin stochastic maximum principle, both in necessary and in sufficient form, which extend the known conditions to this general framework. Furthermore, we suggest a variational approach to study a weak formulation of these control problems. We show a natural connection between this weak formulation and optimal transport on path space, which inspires a novel discretization scheme.