We study two kinds of stochastic recursive optimal control problems. The cost function in the first kind is defined by the solution of the backward stochastic differential equation (BSDE). The control domain does not need to be convex, and the generator of the BSDE can contain z. We obtain the variational equations for BSDE, and then obtain the global maximum principle which solves completely Peng’s open problem. The cost function in the second kind is defined by the solution of the backward stochastic differential equation driven by G-Brownian motion (G-BSDE). The control domain is convex. Unlike the classical variational approach, we establish the local maximum principle by the linearization and weak convergence methods.

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