Backward Stackelberg Differential Game with Constraints: A Mixed Terminal-Perturbation and Linear-Quadratic Approach

We discuss an open-loop backward Stackelberg differential game involving a single leader and single follower. Unlike most Stackelberg game literature, the state to be controlled is characterized by a backward stochastic differential equation for which the terminal- instead of the initial-condition is specified a priori; the decisions of the leader consist of a static terminal-perturbation and a dynamic linear-quadratic control. In addition, the terminal control is subject to (convex-closed) pointwise and (affine) expectation constraints. Both constraints arise from real applications such as mathematical finance. For the information pattern, the leader announces both terminal and open-loop dynamic decisions at the initial time while taking into account the best response of the follower. Then, two interrelated optimization problems are sequentially solved by the follower (a backward linear-quadratic problem) and the leader (a mixed terminal-perturbation and backward-forward LQ problem). Our open-loop Stackelberg equilibrium is represented by some coupled backward-forward stochastic differential equations (BFSDEs) with mixed initial-terminal conditions. Our BFSDEs also involve a nonlinear projection operator (due to pointwise constraint) combining with a Karush--Kuhn--Tucker system (due to expectation constraint) via Lagrange multiplier. The global solvability of such BFSDEs is also discussed in some nontrivial cases. Our results are applied to one financial example.