Zero-Sum Stackelberg Stochastic Linear-Quadratic Differential Games

Abstract

The paper is concerned with a zero-sum Stackelberg stochastic linear-quadratic (LQ) differential game over finite horizons. Under a fairly weak condition, the Stackelberg equilibrium is explicitly obtained by first solving a forward stochastic LQ optimal control problem (SLQ problem) and then a backward SLQ problem. Two Riccati equations are derived for constructing the Stackelberg equilibrium. An interesting finding is that the difference of these two Riccati equations coincides with the Riccati equation associated with the zero-sum Nash stochastic LQ differential game, which implies that under the uniform convexity-concavity condition, the Stackelberg equilibrium and the Nash equilibrium are actually identical. Consequently, the Stackelberg equilibrium admits a linear state feedback representation, and the Nash game can be solved in a leader-follower manner.