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Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

Author

Listed:
  • Alain Bensoussan

    (International Center for Decision and Risk Analysis, Naveen Jindal School of Management, University of Texas at Dallas, USA)

  • Ziyu Huang

    (School of Mathematical Sciences, Fudan University, P. R. China)

  • Sheung Chi Phillip Yam

    (Department of Statistics, The Chinese University of Hong Kong, Hong Kong, P. R. China)

Abstract

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations.

Suggested Citation

  • Alain Bensoussan & Ziyu Huang & Sheung Chi Phillip Yam, 2024. "Maximum Principle for Mean Field Type Control Problems with General Volatility Functions," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 26(02), pages 1-31, June.
    • Handle: RePEc:wsi:igtrxx:v:26:y:2024:i:02:n:s0219198924400036
    DOI: 10.1142/S0219198924400036
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