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Backward Stackelberg Games with Delay and Related Forward–Backward Stochastic Differential Equations

Author

Listed:
  • Li Chen

    (School of Science, China University of Mining and Technology, Beijing 100083, China)

  • Peipei Zhou

    (School of Science, China University of Mining and Technology, Beijing 100083, China)

  • Hua Xiao

    (School of Mathematics and Statistics, Shandong University, Weihai 264209, China)

Abstract

In this paper, we study a kind of Stackelberg game where the controlled systems are described by backward stochastic differential delayed equations (BSDDEs). By introducing a new kind of adjoint equation, we establish the sufficient verification theorem for the optimal strategies of the leader and the follower in a general case. Then, we focus on the linear–quadratic (LQ) backward Stackelberg game with delay. The backward Stackelberg equilibrium is presented by the generalized fully coupled anticipated forward–backward stochastic differential delayed Equation (AFBSDDE), which is composed of anticipated stochastic differential equations (ASDEs) and BSDDEs. Moreover, we obtain the unique solvability of the AFBSDDE using the continuation method. As an application of the theoretical results, the pension fund problem with delay effect is considered.

Suggested Citation

  • Li Chen & Peipei Zhou & Hua Xiao, 2023. "Backward Stackelberg Games with Delay and Related Forward–Backward Stochastic Differential Equations," Mathematics, MDPI, vol. 11(13), pages 1-18, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:13:p:2898-:d:1181913
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    References listed on IDEAS

    as
    1. Zheng, Yueyang & Shi, Jingtao, 2022. "A linear-quadratic partially observed Stackelberg stochastic differential game with application," Applied Mathematics and Computation, Elsevier, vol. 420(C).
    2. Yueyang Zheng & Jingtao Shi, 2020. "A Stackelberg Game of Backward Stochastic Differential Equations with Applications," Dynamic Games and Applications, Springer, vol. 10(4), pages 968-992, December.
    3. Øksendal, Bernt & Sandal, Leif & Ubøe, Jan, 2013. "Stochastic Stackelberg equilibria with applications to time-dependent newsvendor models," Journal of Economic Dynamics and Control, Elsevier, vol. 37(7), pages 1284-1299.
    4. Na Li & Yuan Wang & Zhen Wu, 2018. "An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations," Journal of Optimization Theory and Applications, Springer, vol. 179(2), pages 722-744, November.
    5. Feng Zhang, 2022. "Sufficient Maximum Principle for Stochastic Optimal Control Problems with General Delays," Journal of Optimization Theory and Applications, Springer, vol. 192(2), pages 678-701, February.
    6. Li Chen & Jianhui Huang, 2015. "Stochastic Maximum Principle for Controlled Backward Delayed System via Advanced Stochastic Differential Equation," Journal of Optimization Theory and Applications, Springer, vol. 167(3), pages 1112-1135, December.
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