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

Alain Bensoussan (), Ziyu Huang () and Sheung Chi Phillip Yam ()
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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

International Game Theory Review (IGTR), 2024, vol. 26, issue 02, 1-31

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.

Keywords: Mean field type control problem; optimality condition; Wasserstein space; linear functional derivative; subspace of L2-random variables (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1142/S0219198924400036

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