Extended mean-field control problems with Poissonian common noise: Stochastic maximum principle and Hamiltonian-Jacobi-Bellman equation (2407.05356v2)

Abstract: This paper studies mean-field control problems with state-control joint law dependence and Poissonian common noise. We develop the stochastic maximum principle (SMP) and establish its connection to the Hamiltonian-Jacobi-BeLLMan (HJB) equation on the Wasserstein space. The presence of the conditional joint law in the McKean-Vlasov dynamics and its discontinuity caused by the Poissonian common noise bring new technical challenges. To develop the SMP when the control domain is not necessarily convex, we first consider a strong relaxed control formulation that allows us to perform the first-order variation. We propose the technique of extension transformation to overcome the compatibility issues arising from the joint law in the relaxed control formulation. By further establishing the equivalence between the relaxed control and the strict control formulations, we obtain the SMP for the original problem with strict controls. In the part to investigate the HJB equation, we formulate an auxiliary control problem subjecting to a controlled measure-valued dynamics with Poisson jumps, which allows us to derive the HJB equation of the original problem under the open-loop strict control by some newly established equivalence formulations. We also establish the connection between the SMP and the HJB equation.