Zero-Sum Stochastic Linear-Quadratic Stackelberg Differential Games with Jumps

This paper investigates a zero-sum Stackelberg stochastic linear-quadratic differential game with jumps. The coefficients of the state equation and the weighting matrices in the performance functional are allowed to be random. We first derive the optimality system of the follower's problem and give the unique solvability of the optimality system by a Hilbert space method. The state feedback representation of the follower's rational reaction is obtained under the assumption that the corresponding integro-stochastic Riccati differential equation admits a unique solution. We then explore the leader's problem. The optimality system and its unique solvability are obtained using a similar method to the follower's problem. To get the explicit optimal control of the leader, we consider two special cases and derive the explicit equilibrium control in terms of the stochastic Riccati differential equation.