This paper considers the linear–quadratic (LQ) leader–follower Stackelberg differential game for Markov jump-diffusion stochastic differential equations (SDEs). We first obtain the open-loop type optimal solutions for the leader and the follower by establishing the general stochastic maximum principle for (indefinite) LQ control with Markovian jumps. Then we obtain the state-feedback representation of the open-loop type optimal solutions for the leader and the follower in terms of the coupled Riccati differential equations (CRDEs) by generalizing the classical Four-Step Scheme to the case with Markovian jumps. Unlike the existing literature, the Four-Step Scheme in our paper is not symmetric due to the presence of the nonsymmetric quadratic variation induced by the Markov chain. We develop the decoupling approach to find the simplified expression of the corresponding quadratic variation. Under the well-posedness of the CRDEs, the state-feedback type optimal solutions for the leader and the follower constitute the Stackelberg equilibrium. We also show the well-posedness of the CRDEs for the follower under suitable assumptions of the coefficients. Finally, we demonstrate the well-posedness of the CRDEs for the leader and the follower via numerical simulations for both indefinite and definite cost parameter cases.

本文考虑马尔可夫跳跃扩散随机微分方程 (SDE) 的线性二次 (LQ) 领导者–跟随者 Stackelberg 微分游戏。我们首先通过建立具有马尔可夫跳跃的（不定）LQ 控制的一般随机最大值原理，获得领导者和跟随者的开环型最优解。然后，我们通过将经典四步方案推广到马尔可夫跳跃的情况，根据耦合 Riccati 微分方程 (CRDE) 获得领导者和跟随者的开环型最优解的状态反馈表示。与现有文献不同，由于马尔可夫链引起的非对称二次方差的存在，我们论文中的四步方案不是对称的。我们开发了解耦方法来找到相应二次方差的简化表达式。在 CRDE 的适定性下，领导者和追随者的状态反馈型最优解构成 Stackelberg 均衡。我们还展示了在适当的系数假设下，跟随者的 CRDE 的适定性。最后，我们通过对不确定和确定成本参数情况的数值模拟，证明了领导者和追随者的 CRDE 适定性。