  This paper is concerned with a Stackelberg stochastic differential game,
where the systems are driven by stochastic differential equation (SDE for
short), in which the control enters the randomly disturbed coefficients (drift
and diffusion). The control region is postulated to be convex. By making use of
the first-order adjoint equation (backward stochastic differential equation,
BSDE for short), we are able to establish the Pontryagin's maximum principle
for the leader's global Stackelberg solution, within adapted open-loop
structure and closed-loop memoryless information one, respectively, where the
term global indicates that the leader's domination over the entire game
duration. Since the follower's adjoint equation turns out to be a BSDE, the
leader will be confronted with a control problem where the state equation is a
kind of \emph{fully} coupled forward-backward stochastic differential equation
(FBSDE for short).
  As an application, we study a class of linear-quadratic (LQ for short)
Stackelberg games in which the control process is constrained in a closed
convex subset $\Gamma $ of full space $\mathbb{R}^{m}$. The state equations are
represented by a class of fully coupled FBSDEs with projection operators on
$\Gamma $. By means of monotonicity condition method, the existence and
uniqueness of such FBSDEs are obtained. When the control domain is full space,
we derive the resulting backward stochastic Riccati equations.
