Stackelberg Stochastic Linear-Quadratic Differential Games: A Closed-Loop Equilibrium Approach

Abstract

This paper addresses a Stackelberg stochastic linear-quadratic (LQ) differential game under closed-loop information, a problem inherently time-inconsistent. Existing approaches rely on solving two coupled Hamilton-Jacobi-Bellman (HJB) equations derived via time discretization and a limiting argument, whose convergence remains an open problem. We propose an alternative framework based on closed-loop equilibrium strategies. We reformulate the leader's problem as a forward-backward optimal control problem involving a coupled system of forward SDEs and backward Riccati equations. Due to the presence of controlled Riccati equations, the leader's problem becomes essentially nonlinear. Using a variational method, we characterize the leader's closed-loop equilibrium strategy and derive the associated equilibrium Riccati equation (ERE). A key conceptual distinction is that the follower adopts a globally optimal strategy against any admissible control of the leader, whereas in previous literature the follower's strategy was only locally optimal along the leader's specific equilibrium path. This makes the follower's strategy more robust and the leader's commitment more credible. In our LQ setting, the resulting ERE coincides exactly with the coupled HJB system from the literature, showing the leader's strategy is equivalent to the feedback Stackelberg solution. Thus, our framework provides not only an alternative derivation but also a rigorous justification of the limiting argument. We establish a priori estimates for the ERE, covering 1D and high-dimensional cases, ensuring global well-posedness for any finite horizon. This significantly extends existing results which require a sufficiently short time horizon or control-independent diffusion. An application to an asset management problem with numerical simulations illustrates the theoretical results.