  We investigate the linear quadratic Gaussian Stackelberg game under a class
of nested observation information pattern. Two decision makers implement
control strategies relying on different information sets: The follower uses its
observation data to design its strategy, whereas the leader implements its
strategy using global observation data. We show that the solution requires
solving a new type of forward-backward stochastic differential equations whose
drift terms contain two types of conditional expectation terms associated to
the adjoint variables. We then propose a method to find the functional
relations between each adjoint pair, i.e., each pair formed by an adjoint
variable and the conditional expectation of its associated state. The proposed
method follows a layered pattern. More precisely, in the inner layer, we seek
the functional relation for the adjoint pair under the sigma-sub-algebra
generated by follower's observation information; and in the outer layer, we
look for the functional relation for the adjoint pair under the
sigma-sub-algebra generated by leader's observation information. Our result
shows that the optimal open-loop solution admits an explicit feedback type
representation. More precisely, the feedback coefficient matrices satisfy
tuples of coupled forward-backward differential Riccati equations, and feedback
variables are computed by Kalman-Bucy filtering.
