  We analyze a stochastic optimal control problem, where the state process
follows a McKean-Vlasov dynamics and the diffusion coefficient can be
degenerate. We prove that its value function V admits a nonlinear Feynman-Kac
representation in terms of a class of forward-backward stochastic differential
equations, with an autonomous forward process. We exploit this probabilistic
representation to rigorously prove the dynamic programming principle (DPP) for
V. The Feynman-Kac representation we obtain has an important role beyond its
intermediary role in obtaining our main result: in fact it would be useful in
developing probabilistic numerical schemes for V. The DPP is important in
obtaining a characterization of the value function as a solution of a
non-linear partial differential equation (the so-called Hamilton-Jacobi-Belman
equation), in this case on the Wasserstein space of measures. We should note
that the usual way of solving these equations is through the Pontryagin maximum
principle, which requires some convexity assumptions. There were attempts in
using the dynamic programming approach before, but these works assumed a priori
that the controls were of Markovian feedback type, which helps write the
problem only in terms of the distribution of the state process (and the control
problem becomes a deterministic problem). In this paper, we will consider
open-loop controls and derive the dynamic programming principle in this most
general case. In order to obtain the Feynman-Kac representation and the
randomized dynamic programming principle, we implement the so-called
randomization method, which consists in formulating a new McKean-Vlasov control
problem, expressed in weak form taking the supremum over a family of equivalent
probability measures. One of the main results of the paper is the proof that
this latter control problem has the same value function V of the original
control problem.
