Global solutions of stochastic Stackelberg differential games under convex control constraint

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 Γ of full space Rm. The state equations are represented by a class of fully coupled FBSDEs with projection operators on Γ. 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.