Linear Quadratic Stackelberg Stochastic Differential Games: Closed-Loop Solvability

This paper is concerned with the closed-loop solvability of one kind of linear-quadratic Stackelberg stochastic differential game, where the coefficients are deterministic. The notion of the closed-loop solvability is introduced, which require to be independent of the initial state. The follower's problem is solved first, and the closed-loop optimal strategy is characterized by a Riccati equation, together with an adapted solution to a linear backward stochastic differential equation. Then the necessary conditions of the existence of the leader's nonanticipating closed-loop optimal strategy is obtained via a system of cross-coupled Riccati equations. The sufficiency is open since the completion-of-square method is invalid.