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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.
Bensoussan et al. (Tue,) studied this question.