Advertisement

Science China Information Sciences

, 60:092202 | Cite as

Linear-quadratic stochastic Stackelberg differential game with asymmetric information

Research Paper

Abstract

This paper is concerned with a linear-quadratic stochastic Stackelberg differential game, where players have asymmetric roles, with one leader and one follower in the context of two-person game. It is required that the information available to the follower is a sub-σ-algebra of the one of the leader. By maximum principle and optimal filtering, a feedback Stackelberg equilibrium of the game is given. A special example is used to elaborate the result.

Keywords

stochastic Stackelberg differential game linear-quadratic control asymmetric information conditional mean-field forward-backward stochastic differential equation optimal filtering 

Notes

Acknowledgements

SHI acknowledges the financial support from National Natural Science Fundation of China (Grant No. 11571205) and Natural Science Fund for Distinguished Young Scholars of Shandong Province of China (Grant No. JQ201401). WANG acknowledges the financial support from National Natural Science Fundation for Excellent Young Scholars of China (Grant No. 61422305), National Natural Science Fundation of China (Grant No. 11371228), Natural Science Fund for Distinguished Young Scholars of Shandong Province of China (Grant No. JQ201418), and Research Fund for the Taishan Scholar Project of Shandong Province of China. XIONG acknowledges the financial support from Macau Science and Technology Development Fund (Grant No. FDCT 025/2016/A1) and Multi-Year Research Grant of the University of Macau (Grant No. MYRG2014-00015-FST). The material in Section 3 of this work was partly presented at the 35th Chinese Control Conference, Chengdu, July 27–29, 2016. The authors thank three anonymous referees for their careful reading and valuable comments which have led to significant improvement on the previous version of the paper.

References

  1. 1.
    Baghery F, Øksendal B. A maximum principle for stochastic control with partial information. Stoch Anal Appl, 2007, 25: 705–717MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Shi J T, Wang G C, Xiong J. Leader-follower stochastic differential games with asymmetric information and applications. Automatica, 2016, 63: 60–73MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Meng Q X. A maximum principle for optimal control problem of fully coupled forward-backward stochastic systems with partial information. Sci China Ser A-Math, 2009, 52: 1579–1588MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Wu Z. A maximum principle for partially observed optimal control of forward-backward stochastic control systems. Sci China Inf Sci, 2010, 53: 2205–2214MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Wang G C, Wu Z, Xiong J. A linear-quadratic optimal control problem of forward-backward stochastic differential equations with partial information. IEEE Trans Automa Control, 2015, 60: 2904–2916MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hafayed M, Abbas S, Abba A. On mean-field partial information maximum principle of optimal control for stochastic systems with Lévy processes. J Optim Theory Appl, 2015, 167: 1051–1069MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Stackelberg H V. The Theory of the Market Economy. London: Oxford University Press, 1952Google Scholar
  8. 8.
    Bagchi A, Basar T. Stackelberg strategies in linear-quadratic stochastic differential games. J Optim Theory Appl, 1981, 35: 443–464MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Yong J M. A leader-follower stochastic linear quadratic differential game. SIAM J Control Optim, 2002, 41: 1015–1041MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Øksendal B, Sandal L, Ubøe J. Stochastic Stackelberg equilibria with applications to time dependent newsvendor models. J Econ Dyna Control, 2013, 37: 1284–1299MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bensoussan A, Chen S K, Sethi S P. The maximum principle for global solutions of stochastic Stackelberg differential games. SIAM J Control Optim, 2015, 53: 1956–1981MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Xu J J, Zhang H S. Sufficient and necessary open-loop Stackelberg strategy for two-player game with time delay. IEEE Trans Cyber, 2016, 46: 438–449MathSciNetCrossRefGoogle Scholar
  13. 13.
    Simaan M, Cruz J B. A Stackelberg solution for games with many players. IEEE Trans Autom Control, 1973, 18: 322–324MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Wang B C, Zhang J F. Hierarchical mean field games for multi-agent systems with tracking-type costs: distributed ϵ-Stackelberg equilibria. IEEE Trans Autom Control, 2014, 59: 2241–2247CrossRefGoogle Scholar
  15. 15.
    Xu J J, Zhang H S, Chai T Y. Necessary and sufficient condition for two-player stackelberg strategy. IEEE Trans Autom Control, 2015, 60: 1356–1361MathSciNetCrossRefGoogle Scholar
  16. 16.
    Xiong J. An Introduction to Stochastic Filtering Theory. London: Oxford University Press, 2008MATHGoogle Scholar
  17. 17.
    Yong J M, Zhou X Y. Stochastic Controls: Hamiltonian Systems and HJB Equations. New York: Springer-Verlag, 1999CrossRefMATHGoogle Scholar
  18. 18.
    Shi J T, Wang G C. A new kind of linear-quadratic leader-follower stochastic differential game. In: Proceeding of 10th IFAC Symposium on Nonlinear Control Systems, Monterey, 2016. 322–326Google Scholar

Copyright information

© Science China Press and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanChina
  2. 2.School of Control Science and EngineeringShandong UniversityJinanChina
  3. 3.Department of MathematicsUniversity of MacauMacauChina

Personalised recommendations