A Weak Martingale Approach to Linear-Quadratic McKean-Vlasov Stochastic Control Problems

TL;DR

This paper introduces a novel martingale-based method for solving linear-quadratic mean-field stochastic control problems, including stochastic coefficients and common noise, with applications demonstrated in resource production.

Contribution

It develops a new approach using martingale formulation for verification in mean-field control, extending to stochastic coefficients and common noise scenarios.

Findings

Solution involves Riccati ODEs and linear mean-field BSDEs

Existence and uniqueness conditions established for the system

Application to exhaustible resource production demonstrated

Abstract

We propose a simple and original approach for solving linear-quadratic mean-field stochastic control problems. We study both finite-horizon and infinite-horizon pro\-blems, and allow notably some coefficients to be stochastic. Extension to the common noise case is also addressed. Our method is based on a suitable version of the martingale formulation for verification theorems in control theory. The optimal control involves the solution to a system of Riccati ordinary differential equations and to a linear mean-field backward stochastic differential equation; existence and uniqueness conditions are provided for such a system. Finally, we illustrate our results through an application to the production of an exhaustible resource. MSC Classification: 49N10, 49L20, 93E20.