Mean field portfolio games

We study mean field portfolio games with random parameters, where each player is concerned with not only her own wealth, but also relative performance to her competitors. We use the martingale optimality principle approach to characterise the unique Nash equilibrium in terms of a mean field FBSDE with quadratic growth, which is solvable under a weak interaction assumption. Motivated by the latter, we establish an asymptotic expansion result in powers of the competition parameter. When the market parameters do not depend on the Brownian paths, we obtain the Nash equilibrium in closed form.