Linear-Quadratic Mean-Field Backward Stackelberg Game with Mixed Terminal Perturbation

Abstract

This paper investigates a linear-quadratic (LQ) Stackelberg game for mean-field type backward stochastic system, in which the cost functional is also mean-field type. In our model, the leader first announces the terminal goal satisfying pointwise and affine constraints and open-loop dynamic decisions at the initial time which takes into account the best response of the follower. Then two interrelated optimization problems are sequentially solved by the follower (a backward LQ problem) and the leader (a backward-forward LQ problem). The open-loop Stackelberg equilibrium is obtained by virtue of duality theory and represented by some fully coupled mean-field backward-forward stochastic differential equations with mixed initial-terminal conditions, whose global solvability is discussed in some nontrivial cases by Riccati decoupling method and discounting method. As an application, we address a product pricing problem.