Discrete-time indefinite mean field linear quadratic games with multiplicative noise

This paper considers discrete-time mean field linear quadratic games with mul-tiplicative noises, where the weighting matrices in the cost functions are allowed to be indefinite. First, by expanding the dimension of the systems, we present the necessary and sufficient condition for the existence of Nash equilibrium. Second, by solving the limiting optimal control problem, we design a set of decentral-ized control strategies, which is shown to be an epsilon-Nash equilibrium. Finally, a numerical example is given to illustrate the effectiveness of the proposed control strategies.