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A Stackelberg Game of Backward Stochastic Differential Equations with Applications

Abstract

This paper is concerned with a Stackelberg game of backward stochastic differential equations (BSDEs), where the coefficients of the backward system and the cost functionals are deterministic, and the control domain is convex. Necessary and sufficient conditions of the optimality for the follower and the leader are first given for the general problem, by the stochastic maximum principles of BSDEs and forward–backward stochastic differential equations (FBSDEs), respectively. Then, a linear quadratic (LQ) Stackelberg game of BSDEs is investigated under standard assumptions. The state feedback representation for the optimal control of the follower is first given via two Riccati equations. Then, the leader’s problem is formulated as an optimal control problem of FBSDE with the control-independent diffusion term. Two high-dimensional Riccati equations are introduced to represent the state feedback for the optimal control of the leader. The solvability of the four Riccati equations are discussed. Theoretical results are applied to a pension fund problem of two players in the financial market.

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Acknowledgements

The authors would like to thank the editor and the two anonymous referees for their constructive and insightful comments for improving the quality of this work. The main content of this paper is presented by the first author in the 16th Annual Conference of Chinese Society of Industrial and Applied Mathematics, Foshan, P.R. China, September 2019. Many thanks for discussions and suggestions with Professor Guangchen Wang and Professor Jie Xiong. The second author would like to thank Department of Mathematics and SUSTech International Center for Mathematics, Southern University of Science and Technology for their hospitality during their visit to Shenzhen.

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Correspondence to Jingtao Shi.

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This work is financially supported by the National Key R&D Program of China with Grant Number 2018YFB1305400, and the National Natural Science Foundations of China with Grant Numbers 11971266, 11831010, 11571205.

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Cite this article

Zheng, Y., Shi, J. A Stackelberg Game of Backward Stochastic Differential Equations with Applications. Dyn Games Appl 10, 968–992 (2020). https://doi.org/10.1007/s13235-019-00341-z

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  • DOI: https://doi.org/10.1007/s13235-019-00341-z

Keywords

  • Stackelberg differential game
  • Backward stochastic differential equation
  • Maximum principle
  • Linear quadratic control
  • Pension fund

Mathematics Subject Classification

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