2024/07/07 by Lijun Bo, Jingfei Wang, Bo, Lijun +5 · 2 citations
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Optimization and Control (math.OC) #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2407.05356
openalex publication_date 2024/07/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.