Incomplete Information Mean-Field Games and Related Riccati Equations

We study a class of mean-field games with incomplete information in this paper. For each agent, the state is given by a linear forward stochastic differential equation with common noise. Moreover, both the state and control variables can enter the diffusion coefficients of the state equation. We deduce the open-loop adapted decentralized strategies and feedback decentralized strategies by a mean-field forward–backward stochastic differential equation and Riccati equations, respectively. The well-posedness of the corresponding consistency condition system is obtained and the limiting state-average turns out to be the solution of a mean-field stochastic differential equation driven by common noise. We also verify the \(\varepsilon \) $\epsilon$ -Nash equilibrium property of the decentralized strategies. Finally, a network security problem is studied to illustrate our results as an application.