This was part of Mathematical Advances in Mean Field Games

## Mean Field Game of Mutual Holding

Fabrice Djete, Ecole Polytechnique
Wednesday, December 15, 2021

Abstract: In this presentation, we will talk about a new class of mean field game that we will call mean field game of mutual holding. We will begin by introducing this model which can be seen as a natural expected limit from a $N$-agents game where each agent can hold part of the assets of other agents. The induced mean field dynamics appear naturally in a form which is not covered by standard McKean-Vlasov stochastic differential equations.  In the absence of common noise, we will present results of the study of this mean field game model. Our main result provides existence of an explicit equilibrium of the mean field game of mutual holding, defined by a bang-bang control consisting in holding those competitors with non-negative drift coefficient of their dynamic value. Our analysis requires to prove an existence result for our new class of mean field SDE with the additional feature that the diffusion coefficient is irregular. If there is enough time, we will bring up the case with common noise which presents many additional difficulties.   This is a joint work with Nizar Touzi.