Stackelberg stochastic differential game with asymmetric noisy observations

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.

KEYWORDS:

• Stackelgerg stochastic differential game
• asymmetric noisy observation
• open-loop Stackelberg equilibrium
• maximum principle
• verification theorem
• conditional mean-field forward–backward stochastic differential equation

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 93E20
• 49K45
• 49N10
• 49N70
• 60H10

Acknowledgments

The authors would like to thank the associate editor and the anonymous referee for his/her constructive and insightful comments for improving the quality of this paper. Many thanks for discussion and suggestions with Professor Guangchen Wang at Shandong University, and Professor Jie Xiong at Southern University of Science and Technology.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Key R&D Program of China [grant number 2018YFB1305400], National Natural Science Foundations of China [grant numbers 11971266, 11831010, 11571205], and Shandong Provincial Natural Science Foundations [grant numbers ZR2020ZD24, ZR2019ZD42].