Stackelberg Stochastic Differential Games in Feedback Information Pattern with Applications

This paper is concerned with both nonzero-sum and zero-sum Stackelberg stochastic differential games on a finite horizon in feedback information pattern, where the control variables enter into the diffusion term. For the nonzero-sum case, a system of parabolic partial differential equations is obtained to give the verification theorem of the feedback Stackelberg equilibrium. As an example, a linear-quadratic nonzero-sum feedback Stackelberg stochastic differential game is investigated, where the diffusion term is also control-dependent. Coupled Riccati equations are introduced to express the feedback Stackelberg equilibrium, and sufficient conditions for the existence of their solutions are obtained. For the zero-sum case, a verification theorem that contains only one Hamilton–Jacobi–Bellman equation is given. In addition, the solvability of the Riccati equation, which is derived from the linear-quadratic zero-sum case with the leader’s control appearing in the diffusion term, is further discussed, and a sufficient and necessary condition for the existence and uniqueness of the solution is given.