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

We analyze the regularity of solutions and discrete-time approximations of extended mean-field control (extended MFC) problems, which seek optimal control of McKean-Vlasov dynamics with coefficients involving mean-field interactions both on the  state and actions, and where objectives are optimized over
open-loop strategies.

We show for a large class of extended MFC problems that the unique optimal open-loop control is 1/2-Hölder continuous in time. Based on the regularity of the solution, 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. Further, we show that any epsilon-optimal control 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.

This is based on join work joint work with C. Reisinger and Y. Zhang.