Graphon Mean Field Game of mutual holding

Abstract

This paper studies the mean field game of mutual holding proposed by Djete and Touzi(AAP, 2024), and consider the case where the interactions among agents are described by a graphon. We adopt the formulation on the enlarged space which is modeled using the joint law of the value process and the graphon label, as in Lacker and Soret(MOR, 2023). Under suitable conditions on the graphon function, we are able to provide the explicit characterization of the optimal strategy, prove the wellposedness of associated Mckean-Vlasov SDE and establish the convergence results of the Nash equilibria. The key technique consists in a detailed analysis of the continuity property under the $\mathcal{WOP}_2$ metric, and tailor-made arguments for different graphon equilibria under different regularities of the model.