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

TL;DR

This work develops a rigorous framework for extended mean-field control with joint state-control dependence and Poissonian common noise, establishing both a stochastic maximum principle (SMP) and a Hamiltonian–Jacobi–Bellman (HJB) equation on Wasserstein space. It introduces an extension transformation and a strong relaxed-control formulation to handle nonconvex control domains and the joint law, and proves equivalence with strict controls via Chattering arguments. A novel controlled Fokker–Planck formulation is constructed to derive the HJB equation, and the authors prove the value functions of the original and FP problems coincide under suitable regularity, including a conditional-law invariance property. The paper also provides a linear-quadratic example that clarifies how Poissonian common noise affects the adjoint processes and the HJB structure, offering a tractable benchmark for these extended mean-field problems with jumps.

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.

Figures (2)

• Figure 1: Our methodology for SMP
• Figure 2: Our methodology for HJB equation

Theorems & Definitions (10)

• proof : Proof of \ref{['metriceq']}
• proof : Proof of \ref{['extensioneq']}
• proof : Proof of \ref{['lem:condexpect']}
• proof : Proof of \ref{['extensionLdif']}
• proof : Proof of \ref{['dif']}
• proof : Proof of \ref{['valuedif']}
• proof : Proof of \ref{['law_representation']}
• proof : Proof of \ref{['continuous_control']}
• proof : Proof of \ref{['L2con']}
• proof : Proof of \ref{['valueeq']}