A second-order Mean Field Games model with controlled diffusion

TL;DR

This paper extends Mean Field Games theory by allowing agents to control both drift and diffusion, resulting in a fully nonlinear system with new existence and regularity results for solutions.

Contribution

It introduces a novel second-order MFG model with controlled diffusion and establishes existence and regularity of solutions using viscosity and Krylov's methods.

Findings

Proves existence of solutions for the fully nonlinear MFG system.

Establishes $ ext{C}^3$ regularity for the value function.

Develops a well-posedness framework for the new model.

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.