A linear-quadratic partially observed Stackelberg stochastic differential game with application

Abstract

This paper is concerned with a linear-quadratic partially observed Stackelberg stochastic differential game with correlated state and observation noises, where the control set is not necessarily convex. Both the leader and the follower have their own observation equations, and the information filtration available to the leader is contained in that available to the follower. Necessary and sufficient conditions of the Stackelberg equilibrium points are derived. In the follower’s problem, the state estimation feedback of optimal control can be represented by a forward-backward stochastic differential filtering equation and some Riccati equation. In the leader’s problem, via the innovation process, the state estimation feedback of optimal control is represented by a stochastic differential filtering equation, a semi-martingale process and three high-dimensional Riccati equations. As an application, a dynamic advertising problem with asymmetric information is studied, and the effectiveness and reasonability of the theoretical result are illustrated by numerical simulations.