A second-order Mean Field Games model with controlled diffusion

Vincenzo Ignazio; Michele Ricciardi

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arXiv · 2407.20826

A second-order Mean Field Games model with controlled diffusion

Vincenzo Ignazio·Michele Ricciardi

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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 $\mathcal C^3$ regularity for $u$ in the space variable. This allows us to prove a well-posedness result for the MFG system.

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Vincenzo Ignazio, Michele Ricciardi. 2024-07-30. A second-order Mean Field Games model with controlled diffusion. https://arxiv.org/abs/2407.20826

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