The maximum principle for optimal control problems of fully coupledforward-backward doubly stochastic differential equations (FBDSDEs in short)in the global form is obtained, under the assumptions that the diffusioncoefficients do not contain the control variable, but the control domainneed not to be convex. We apply our stochastic maximum principle (SMP inshort) to investigate the optimal control problems of a class of stochasticpartial differential equations (SPDEs in short). And as an example of theSMP, we solve a kind of forward-backward doubly stochastic linear quadraticoptimal control problems as well. In the last section, we use the solutionof FBDSDEs to get the explicit form of the optimal control for linearquadratic stochastic optimal control problem and open-loop Nash equilibriumpoint for nonzero sum stochastic differential games problem.
