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Mixed Linear Quadratic Stochastic Differential Leader-Follower Game with Input Constraint

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Abstract

This paper investigates a mixed leader-follower differential games problem, where the model involves two players with the same hierarchy in decision making and each player has two controls which act as a leader and a follower, respectively. Specifically, we solve a follower problem with unconstrained controls and obtain the corresponding Nash equilibrium. Then a leader problem with constrained controls is tackled and a pair of optimal constrained controls are presented by a projection mapping. Furthermore, the control weights are allowed to be singular. In this case, we first investigate the uniform convexity of the cost functional whose corresponding states are fully-coupled forward-backward stochastic differential equation. After that, the minimizing sequence of solutions with non-degenerate control weights are constructed to study the weak convergence of the corresponding cost functionals. Finally, two examples are addressed for non-singular and singular cases, respectively.

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Acknowledgements

The authors would like to thank the editors and two anonymous referees for their very extensive and constructive suggestions that helped to improve this paper considerably. Xinwei Feng’s work is supported by National Natural Science Foundation of China (No. 12001317), Shandong Provincial Natural Science Foundation (No. ZR2020QA019) and QILU Young Scholars Program of Shandong University; Tinghan Xie and Jianhui Huang’s work are supported by RGC 153005/14P, 153275/16P, P0030808, P0008686, P0031044; the authors also acknowledge the support from The PolyU-SDU Joint Research Centre on Financial Mathematics.

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Authors and Affiliations

  1. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Tinghan Xie & Jianhui Huang

  2. Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan, 250100, Shandong, China

    Xinwei Feng

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  1. Tinghan Xie
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  2. Xinwei Feng
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Correspondence to Xinwei Feng.

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Xie, T., Feng, X. & Huang, J. Mixed Linear Quadratic Stochastic Differential Leader-Follower Game with Input Constraint. Appl Math Optim 84 (Suppl 1), 215–251 (2021). https://doi.org/10.1007/s00245-021-09767-7

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