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 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.

本文关注的是 Stackelberg 随机微分博弈，其中系统由随机微分方程（简称 SDE）驱动，其中控制进入随机扰动系数（漂移和扩散）。假设控制区域是凸的。通过使用一阶伴随方程（反向随机微分方程，简称BSDE），我们能够在适应的开环结构和闭环无记忆信息的情况下建立领导者全局Stackelberg解的庞特里亚金极大值原理，分别，其中术语 global 表示领导者在整个游戏持续时间内的统治。由于跟随者的伴随方程结果是一个 BSDE，领导者将面临一个控制问题，其中状态方程是一种完全耦合的前向后向随机微分方程（简称 FBSDE）。

作为一个应用，我们研究了一类线性二次（简称 LQ）Stackelberg 博弈，其中控制过程被约束在一个封闭的凸子集中 $\Gamma$ 全空间 ${\mathbb{电阻}}^{米}$. 状态方程由一类具有投影算子的全耦合 FBSDE 表示$\Gamma$. 通过单调性条件方法，得到了这种FBSDEs的存在性和唯一性。当控制域为全空间时，我们推导出所得的后向随机 Riccati 方程。