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Extended Mean Field Type Control Theory: A Master Equation Approach with Some Applications

Author

Listed:
  • Alain Bensoussan

    (The University of Texas at Dallas)

  • Joohyun Kim

    (University of North Carolina at Charlotte)

  • Sheung Chi Phillip Yam

    (Chinese University of Hong Kong)

Abstract

The primary objective of this article is to present a general framework for users and applications of the master equation approach in extended mean field type control, formulated with a McKean-Vlasov stochastic differential equation that depends on the law of both the control and state variables. This control problem has recently gained significant attention and has been extensively studied at the level of the Bellman equation. Here, we extend the analysis to the master equation and derive the corresponding Hamilton-Jacobi-Bellman equation. A key novelty of our approach is that we do not directly rely on the Fokker-Planck equation, which surprisingly leads to a significant simplification. We provide a concise theoretical presentation with proofs, as the standard theory of stochastic control is not directly applicable. In the current work, the solution is constructed using an ansatz-based approach to dynamic programming via the master equation. We illustrate this method with a practical example. All proofs are presented in a self-contained manner. This paper offers a structured presentation of the extended mean field type control problem, serving as a valuable toolbox for users who are less focused on mathematical intricacies but seek a general framework for application.

Suggested Citation

  • Alain Bensoussan & Joohyun Kim & Sheung Chi Phillip Yam, 2025. "Extended Mean Field Type Control Theory: A Master Equation Approach with Some Applications," Journal of Optimization Theory and Applications, Springer, vol. 207(1), pages 1-49, October.
    • Handle: RePEc:spr:joptap:v:207:y:2025:i:1:d:10.1007_s10957-025-02776-4
    DOI: 10.1007/s10957-025-02776-4
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