Dynamic programming for mean-field type control

For mean-field type control problems, stochastic dynamic programming requires adaptation. We propose to reformulate the problem as a distributed control problem by assuming that the PDF ρ of the stochastic process exists. Then we show that Bellman's principle applies to the dynamic programming value function V ( τ , ρ τ ) , where the dependency on ρ τ is functional as in P.-L. Lions' analysis of mean-field games (2007) [10]. We derive HJB equations and apply them to two examples, a portfolio optimization and a systemic risk model.