  We study a class of linear-quadratic mean-field games with incomplete
information. 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 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 $\varepsilon$-Nash equilibrium property of
the decentralized control strategies. Finally, a network security problem is
studied to illustrate our results as an application.
