  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.
