Equilibrium in Functional Stochastic Games with Mean-Field Interaction

We consider a general class of nite-player stochastic games with mean-eld interaction, in which the linear-quadratic cost functional includes linear operators acting on controls in L2. We propose a novel approach for deriving the Nash equilibrium of the game explicitly in terms of operator resolvents, by reducing the associated rst order conditions to a system of stochastic Fredholm equations of the second kind and deriving their closed form solution. Furthermore, by proving stability results for the system of stochastic Fredholm equations we derive the convergence of the equilibrium of the N-player game to the corresponding mean-eld equilibrium. As a by-product we also derive an ε-Nash equilibrium for the mean-eld game, which is valuable in this setting as we show that the conditions for existence of an equilibrium in the meaneld limit are less restrictive than in the nite-player game. Finally we apply our general framework to solve various examples, such as stochastic Volterra linear-quadratic games, models of systemic risk and advertising with delay, and optimal liquidation games with transient price impact.