Mean field stochastic linear quadratic games for continuum-parameterized multi-agent systems

In this paper, we consider a stochastic linear quadratic mean field game for the continuum-parameterized multi-agent systems with multiplicative noise. Based on the Nash certain equivalence principle, we obtain a series of decentralized control laws. Then, Dynkin's formula and comparison principle are employed to prove the boundedness of the state of the closed loop system in the mean square sense. Finally, we show that the set of decentralized controls has an ϵ-Nash equilibrium property.