Stochastic maximum principle for optimal control problem of non exchangeable mean field systems

TL;DR

This paper develops a stochastic maximum principle for non exchangeable mean field control systems with heterogeneous agent interactions, deriving conditions via FBSDEs and illustrating with a linear-quadratic example.

Contribution

It introduces necessary and sufficient optimality conditions for non exchangeable mean field systems, extending classical results to asymmetric agent interactions.

Findings

Derived solvability conditions for the FBSDE system

Characterized optimal control via infinite-dimensional Riccati equations

Applied framework to linear-quadratic case

Abstract

We study the Pontryagin maximum principle by deriving necessary and sufficient conditions for a class of optimal control problems arising in non exchangeable mean field systems, where agents interact through heterogeneous and asymmetric couplings. Our analysis leads to a collection of forward-backward stochastic differential equations (FBSDE) of non exchangeable mean field type. Under suitable assumptions, we establish the solvability of this system. As an illustration, we consider the linear-quadratic case, where the optimal control is characterized by an infinite dimensional system of Riccati equations.