  In this paper we study stochastic optimal control problems of fully coupled
forward-backward stochastic differential equations (FBSDEs). The recursive cost
functionals are defined by controlled fully coupled FBSDEs. We study two cases
of diffusion coefficients $\sigma$ of FSDEs. We use a new method to prove that
the value functions are deterministic, satisfy the dynamic programming
principle (DPP), and are viscosity solutions to the associated generalized
Hamilton-Jacobi-Bellman (HJB) equations. The associated generalized HJB
equations are related with algebraic equations when $\sigma$ depends on the
second component of the solution $(Y, Z)$ of the BSDE and doesn't depend on the
control. For this we adopt Peng's BSDE method, and so in particular, the notion
of stochastic backward semigroup in [16]. We emphasize that the fact that
$\sigma$ also depends on $Z$ makes the stochastic control much more complicate
and has as consequence that the associated HJB equation is combined with an
algebraic equation, which is inspired by Wu and Yu [19]. We use the
continuation method combined with the fixed point theorem to prove that the
algebraic equation has a unique solution, and moreover, we also give the
representation for this solution. On the other hand, we prove some new basic
estimates for fully coupled FBSDEs under the monotonic assumptions. In
particular, we prove under the Lipschitz and linear growth conditions that
fully coupled FBSDEs have a unique solution on the small time interval, if the
Lipschitz constant of $\sigma$\ with respect to $z$ is sufficiently small. We
also establish a generalized comparison theorem for such fully coupled FBSDEs.
