  The maximum principle for optimal control problems of fully coupled
forward-backward doubly stochastic differential equations (FBDSDEs in short) in
the global form is obtained, under the assumptions that the diffusion
coefficients do not contain the control variable, but the control domain need
not to be convex. We apply our stochastic maximum principle (SMP in short) to
investigate the optimal control problems of a class of stochastic partial
differential equations (SPDEs in short). And as an example of the SMP, we solve
a kind of forward-backward doubly stochastic linear quadratic optimal control
problems as well. In the last section, we use the solution of FBDSDEs to get
the explicit form of the optimal control for linear quadratic stochastic
optimal control problem and open-loop Nash equilibrium point for nonzero sum
differential games problem.
