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A second-order Mean Field Games model with controlled diffusion

Author

Listed:
  • Vincenzo Ignazio

    (ETH Zürich)

  • Michele Ricciardi

    (Università LUISS Guido Carli)

Abstract

Mean Field Games (MFG) theory describes strategic interactions in differential games with a large number of small and indistinguishable players. Traditionally, the players’ control impacts only the drift term in the system’s dynamics, leaving the diffusion term uncontrolled. This paper explores a novel scenario where agents control both drift and diffusion. This leads to a fully non-linear MFG system with a fully non-linear Hamilton–Jacobi–Bellman equation. We use viscosity arguments to prove existence of solutions for the HJB equation, and then we adapt and extend a result from Krylov to prove a $${\mathcal {C}}^3$$ C 3 regularity for u in the space variable. This allows us to prove a well-posedness result for the MFG system.

Suggested Citation

  • Vincenzo Ignazio & Michele Ricciardi, 2025. "A second-order Mean Field Games model with controlled diffusion," Partial Differential Equations and Applications, Springer, vol. 6(2), pages 1-34, April.
    • Handle: RePEc:spr:pardea:v:6:y:2025:i:2:d:10.1007_s42985-025-00323-4
    DOI: 10.1007/s42985-025-00323-4
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    References listed on IDEAS

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    1. Fausto Gozzi & Federica Masiero & Mauro Rosestolato, 2024. "An optimal advertising model with carryover effect and mean field terms," Mathematics and Financial Economics, Springer, volume 18, number 9, September.
    2. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    3. Marco Avellaneda & Antonio ParAS, 1996. "Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(1), pages 21-52.
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    More about this item

    Keywords

    35D40; 35Q84; 35Q89; 49L25; 49N80;
    All these keywords.

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