A second-order Mean Field Games model with controlled diffusion

Vincenzo Ignazio () and Michele Ricciardi ()
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Vincenzo Ignazio: ETH Zürich
Michele Ricciardi: Università LUISS Guido Carli

Partial Differential Equations and Applications, 2025, vol. 6, issue 2, 1-34

Abstract: 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.

Keywords: 35D40; 35Q84; 35Q89; 49L25; 49N80 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s42985-025-00323-4

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