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

TL;DR

This paper advances mean-field control theory by deriving stochastic maximum principles and HJB equations for problems with Poissonian noise and joint law dependence, addressing technical challenges with new formulations.

Contribution

It introduces a relaxed control framework and extension techniques to handle non-convex control domains and discontinuities in the joint law under Poisson noise.

Findings

Established SMP for non-convex control domains with Poisson noise

Derived HJB equation via controlled measure-valued dynamics with jumps

Connected SMP and HJB in the context of mean-field control with common noise

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 and its discontinuity under 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 equivalent controlled Fokker-Planck problem subjecting to a controlled measure-valued dynamics with Poisson jumps, which allows us to derive the HJB equation of the original problem under open-loop strict controls. We also establish the connection between the SMP and the HJB equation.