Mean-field linear-quadratic stochastic differential games

The paper is concerned with two-person zero-sum mean-field linear-quadratic stochastic differential games over finite horizons. By a Hilbert space method, necessary condition and sufficient are derived for the existence of an open-loop saddle point. It shown that under condition, associated two Riccati equations admit unique strongly regular solutions, in terms which point can be represented as linear feedback current state. When game only satisfies approximate sequence constructed by solving family closed-loop systems. convergence turns out to equivalent solvability game, limit exactly point, provided solvable.