ABSTRACT

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.

摘要

本文关注具有不对称噪声观察的 Stackelberg 随机微分博弈。在我们的模型中，follower 不能直接观察状态过程，但可以观察到有噪声的观察过程，而 leader 可以完全观察状态过程。考虑了开环 Stackelberg 平衡。跟随者首先求解一个局部观测的随机最优控制问题，得到极大值原理和验证定理。然后领导者转而求解条件平均场正反向随机微分方程的最优控制问题，并证明了极大原理和验证定理。为了说明本文的理论结果，讨论了具有不对称噪声观察的线性二次 Stackelberg 随机微分博弈。借助一些新的 Riccati 方程，开环 Stackelberg 均衡承认其状态估计反馈表示。最后，给出了资源分配的应用及其数值模拟，以证明所提结果的有效性。