Probabilistic Analysis of Graphon Mean Field Control

TL;DR

This paper offers a comprehensive probabilistic framework for graphon mean field control problems, establishing existence, uniqueness, and optimality conditions for systems with heterogeneous interactions.

Contribution

It introduces a novel probabilistic analysis of GMFC, including existence, uniqueness, Pontryagin maximum principle, and stability results for complex stochastic systems.

Findings

Proved existence and uniqueness of graphon mean field FBSDEs.

Derived a Pontryagin stochastic maximum principle for GMFC.

Showed approximate optimality for large heterogeneous systems.

Abstract

Motivated by recent interest in graphon mean field games and their applications, this paper provides a comprehensive probabilistic analysis of graphon mean field control (GMFC) problems, where the controlled dynamics are governed by a graphon mean field stochastic differential equation with heterogeneous mean field interactions. We formulate the GMFC problem with general graphon mean field dependence and establish the existence and uniqueness of the associated graphon mean field forward-backward stochastic differential equations (FBSDEs). We then derive a version of the Pontryagin stochastic maximum principle tailored to GMFC problems. Furthermore, we analyze the solvability of the GMFC problem for linear dynamics and study the continuity and stability of the graphon mean field FBSDEs under the optimal control profile. Finally, we show that the solution to the GMFC problem provides an approximately optimal solution for large systems with heterogeneous mean field interactions, based on a propagation of chaos result.