  This paper is concerned with two-person mean-field linear-quadratic non-zero
sum stochastic differential games in an infinite horizon. Both open-loop and
closed-loop Nash equilibria are introduced. Existence of an open-loop Nash
equilibrium is characterized by the solvability of a system of mean-field
forward-backward stochastic differential equations in an infinite horizon and
the convexity of the cost functionals, and the closed-loop representation of an
open-loop Nash equilibrium is given through the solution to a system of two
coupled non-symmetric algebraic Riccati equations. The existence of a
closed-loop Nash equilibrium is characterized by the solvability of a system of
two coupled symmetric algebraic Riccati equations. Two-person mean-field
linear-quadratic zero-sum stochastic differential games in an infinite time
horizon are also considered. Both the existence of open-loop and closed-loop
saddle points are characterized by the solvability of a system of two coupled
generalized algebraic Riccati equations with static stabilizing solutions.
Mean-field linear-quadratic stochastic optimal control problems in an infinite
horizon are discussed as well, for which it is proved that the open-loop
solvability and closed-loop solvability are equivalent.
