Mean field games with unbounded controlled common noise in portfolio management with relative performance criteria

Motivated by optimal allocation models with relative performance criteria, we introduce a mean field game in which the terminal expected utility of the representative agent depends on her own state as well as the average of her peers. We derive the master equation, which, in view of the presence of controls in the volatility, needs to be coupled with a compatibility condition for the mean field optimal feedback control. We concentrate on the class of separable payoffs under both general utilities and couplings. We derive a solution to the master equation and find the associated optimal feedback control expressed via the value function in the absence of competition and a dynamic coupling function solving a non-local quasilinear equation. In turn, we construct the related optimal state and control processes, and give representative examples. Projecting the mean field solutions on finite dimensions, we recover the solution of the N-game for linear couplings and arbitrary utilities, and we study the proximity of these approximations to their N-player game counterparts.