  This paper considers linear-quadratic (LQ) stochastic leader-follower
Stackelberg differential games for jump-diffusion systems with random
coefficients. We first solve the LQ problem of the follower using the
stochastic maximum principle and obtain the state-feedback representation of
the open-loop optimal solution in terms of the integro-stochastic Riccati
differential equation (ISRDE), where the state-feedback type control is shown
to be optimal via the completion of squares method. Next, we establish the
stochastic maximum principle for the indefinite LQ stochastic optimal control
problem of the leader using the variational method. However, to obtain the
state-feedback representation of the open-loop solution for the leader, there
is a technical challenge due to the jump process. To overcome this limitation,
we consider two different cases, in which the state-feedback type optimal
control for the leader in terms of the ISRDE can be characterized by
generalizing the Four-Step Scheme. Finally, in these two cases, we show that
the state-feedback representation of the open-loop optimal solutions for the
leader and the follower constitutes the Stackelberg equilibrium. Note that the
(indefinite) LQ control problem of the leader is new and nontrivial due to the
coupled FBSDE constraint induced by the rational behavior of the follower.
