Degenerate Mean Field Type Control with Linear and Unbounded Diffusion, and their Associated Equations

We study the well-posedness of a system of forward-backward stochastic differential equations (FBSDEs) corresponding to a degenerate mean field type control problem, when the diffusion coefficient depends on the state together with its measure and also the control. Degenerate mean field type control problems are rarely studied in the literature. Our method is based on a lifting approach which embeds the control problem and the associated FBSDEs in Wasserstein spaces into certain Hilbert spaces. We use a continuation method to establish the solvability of the FBSDEs and that of the G\^ateaux derivatives of this FBSDEs. We then explore the regularity of the value function in time and in measure argument, and we also show that it is the unique classical solution of the associated Bellman equation. We also study the higher regularity of the linear functional derivative of the value function, by then, we obtain the classical solution of the mean field type master equation.