Regularity and time discretization of extended mean field control problems: a McKean-Vlasov FBSDE approach

We analyze the solution regularity and discrete-time approximations of extended mean field control (extended MFC) problems, which seek optimal control of McKean-Vlasov dynamics whose coefficients involve mean field interactions both on the state and actions, and where objectives are optimized over open-loop strategies. We show that for a large class of extended MFC problems, the unique optimal open-loop control is 1/2-Holder continuous in time. Based on the solution regularity, we prove that the value functions of such extended MFC problems can be approximated by those with piecewise constant controls and discrete-time state processes arising from Euler-Maruyama time stepping up to an order 1/2 error, which is optimal in our setting. We further show that any $\epsilon$-optimal controls of these discrete-time problems converge to the optimal control of the original problems. To establish the time regularity of optimal controls and the convergence of time discretizations, we extend the canonical path regularity results to general coupled McKean-Vlasov forward-backward stochastic differential equations, which are of independent interest.