Stackelberg stochastic differential game with asymmetric noisy observations

This paper is concerned with a Stackelberg stochastic differential game with asymmetric noisy observation. In our model, the follower cannot observe the state process directly, but could observe a noisy observation process, while the leader can completely observe the state process. Open-loop Stackelberg equilibrium is considered. The follower first solve a stochastic optimal control problem with partial observation, the maximum principle and verification theorem are obtained. Then the leader turns to solve an optimal control problem for a conditional mean-field forward-backward stochastic differential equation, and both maximum principle and verification theorem are proved. A linear-quadratic Stackelberg stochastic differential game with asymmetric noisy observation is discussed to illustrate the theoretical results in this paper. With the aid of some new Riccati equations, the open-loop Stackelberg equilibrium admits its state estimate feedback representation. Finally, an application to the resource allocation and its numerical simulation are given to show the effectiveness of the proposed results.