Abstract

We introduce a mean field model for optimal holding of a representative agent of her peers as a natural expected scaling limit from the corresponding N-agent model. The induced mean field dynamics appear naturally in a form which is not covered by standard McKean-Vlasov stochastic differential equations. We study the corresponding mean field game of mutual holding in the absence of common noise. Our first main result provides an explicit equilibrium of this mean field game, defined by a bang–bang control consisting in holding those competitors with positive drift coefficient of their dynamic value. We next use this mean field game equilibrium to construct (approximate) Nash equilibria for the corresponding N–player game. Finally, we extend the model to allow for default upon hitting the origin. Then, the equilibrium distribution exhibits an atom at the origin whose mass evolution is characterized by a ordinary differential equation.