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We discuss an open-loop backward Stackelberg differential game involvingsingle leader and single follower. Unlike most Stackelberg game literature, thestate to be controlled is characterized by a backward stochastic differentialequation (BSDE) for which the terminal- instead initial-condition is specifiedas a priori; the decisions of leader consist of a static terminal-perturbationand a dynamic linear-quadratic control. In addition, the terminal control issubject to (convex-closed) pointwise and (affine) expectation constraints. Bothconstraints are arising from real applications such as mathematical finance.For information pattern: the leader announces both terminal and open-loopdynamic decisions at the initial time while takes account the best response offollower. Then, two interrelated optimization problems are sequentially solvedby the follower (a backward linear-quadratic (BLQ) problem) and the leader (amixed terminal-perturbation and backward-forward LQ (BFLQ) problem). Ouropen-loop Stackelberg equilibrium is represented by some coupledbackward-forward stochastic differential equations (BFSDEs) with mixedinitial-terminal conditions. Our BFSDEs also involve nonlinear projectionoperator (due to pointwise constraint) combining with a Karush-Kuhn-Tucker(KKT) system (due to expectation constraint) via Lagrange multiplier. Theglobal solvability of such BFSDEs is also discussed in some nontrivial cases.Our results are applied to one financial example.

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