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We analyze a stochastic optimal control problem, where the state processfollows a McKean-Vlasov dynamics and the diffusion coefficient can bedegenerate. We prove that its value function V admits a nonlinear Feynman-Kacrepresentation in terms of a class of forward-backward stochastic differentialequations, with an autonomous forward process. We exploit this probabilisticrepresentation to rigorously prove the dynamic programming principle (DPP) forV. The Feynman-Kac representation we obtain has an important role beyond itsintermediary role in obtaining our main result: in fact it would be useful indeveloping probabilistic numerical schemes for V. The DPP is important inobtaining a characterization of the value function as a solution of anon-linear partial differential equation (the so-called Hamilton-Jacobi-Belmanequation), in this case on the Wasserstein space of measures. We should notethat the usual way of solving these equations is through the Pontryagin maximumprinciple, which requires some convexity assumptions. There were attempts inusing the dynamic programming approach before, but these works assumed a priorithat the controls were of Markovian feedback type, which helps write theproblem only in terms of the distribution of the state process (and the controlproblem becomes a deterministic problem). In this paper, we will consideropen-loop controls and derive the dynamic programming principle in this mostgeneral case. In order to obtain the Feynman-Kac representation and therandomized dynamic programming principle, we implement the so-calledrandomization method, which consists in formulating a new McKean-Vlasov controlproblem, expressed in weak form taking the supremum over a family of equivalentprobability measures. One of the main results of the paper is the proof thatthis latter control problem has the same value function V of the originalcontrol problem.

Randomized dynamic programming principle and Feynman-Kac representation for optimal control of McKean-Vlasov dynamics