Mean-field quadratic BSDEs and related mean-field portfolio games of controls

We study a new class of mean-field quadratic backward stochastic differential equations (qBSDEs) arising from mean-field portfolio games with exponential utility. Typical examples of such games include a mean-field portfolio game with price impact, and a finite-contract pricing model with market clearing conditions. Generators of these mean-field qBSDEs contain quadratic terms $\mathbb{E}[Z{]}^{\mathrm{\top }}Z$ and $\mathrm{\mid }\mathbb{E}[Z]{\mathrm{\mid }}^2$, instead of the classical pathwise $Z^{\mathrm{\top }}Z$ term. We prove local well-posedness under $L^q$-integrability assumptions on terminals and their Malliavin derivatives, and global well-posedness under an extra exponential integrability condition on the Malliavin derivatives. Then we show the existence and uniqueness of global equilibria of the above two mean-field games via our qBSDE theory.