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

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.

💡 Research Summary

This paper investigates optimal control problems of McKean‑Vlasov type in which the cost functional and the dynamics depend on the joint law of the state and the control, and the system is driven by a common noise modeled by a Poisson random measure. The authors develop two complementary theoretical frameworks: a stochastic maximum principle (SMP) based on a strong relaxed‑control formulation, and a dynamic‑programming‑principle (DPP) based Hamilton‑Jacobi‑Bellman (HJB) equation on the Wasserstein space of probability measures.

1. Problem setting and challenges
Mean‑field control (MFC) problems traditionally involve Brownian common noise and dependence only on the marginal law of the state. Introducing a Poissonian common noise creates jumps in the conditional joint distribution of state and control, breaking the continuity assumptions that underlie most existing SMP and HJB results. Moreover, the control domain is allowed to be non‑convex, which prevents the direct use of first‑order variational methods.

2. Strong relaxed control and the extension transformation
To overcome non‑convexity, the authors adopt a relaxed‑control approach where a control is represented by a probability measure on the original control set (U). The key technical device is the extension transformation: a mapping (h:\mathcal P_2(\mathbb R^n\times U)\to\mathbb R) is lifted to (\tilde h:\mathcal P_2(\mathbb R^n\times\mathcal P(U))\to\mathbb R) by integrating (h) against the relaxed control measure. Lemmas 2.2 and 2.3 prove that Lipschitz continuity and Fréchet differentiability are preserved under this lift, allowing the authors to work with the joint law of ((X,\nu)) where (\nu) is a relaxed control.

3. Stochastic maximum principle
Within the relaxed framework, a first‑order variation of the cost functional yields a Hamiltonian (\mathcal H(t,x,\mu,u,\dots)) that depends on the state, the control, and the conditional joint law (\mu). Theorem 3.7 (relaxed SMP) states that any optimal relaxed control maximizes (\mathcal H) almost surely. To transfer this result back to strict (classical) controls, the authors prove an equivalence theorem (Lemmas 3.11, 3.16) using a chattering lemma: any relaxed optimal control can be approximated by a sequence of piecewise‑constant strict controls. Consequently, Theorems 3.18 and 3.19 provide a SMP for the original problem with non‑convex strict controls. The adjoint process is a backward stochastic differential equation (BSDE) whose driver contains the Wasserstein derivative of the Hamiltonian with respect to the joint law.

4. From SMP to HJB: a controlled Fokker‑Planck formulation
The presence of Poisson jumps makes the conditional law of the state discontinuous. To apply DPP, the authors introduce a controlled Fokker‑Planck (FP) problem in which the control enters as a kernel‑valued function adapted only to the Poisson common‑noise filtration. Lemma 4.2 characterizes the jump of the conditional law via a generalized measure‑shift operator, while Lemma 4.5 provides an Itô formula on the space of conditional measures. These tools lead to a HJB equation (52) on (\mathcal P_2(\mathbb R^n)) that contains a non‑local jump term reflecting the Poisson intensity and jump size distribution.

5. Equality of value functions
The authors prove that the value function of the original MFC problem coincides with that of the controlled FP problem. Lemma 4.9 shows (V_{\text{MFC}}\ge V_{\text{FP}}) by disintegrating the joint law using Bayes’ formula. Proposition 4.10 and 4.13 construct near‑optimal kernel‑valued feedback controls for the FP problem and then lift them to piecewise‑constant strict controls for the original problem, establishing the reverse inequality. Theorem 4.14 concludes that both value functions are equal and are given by the classical solution of the HJB equation.

6. Connection between SMP and HJB
When the HJB equation admits a smooth solution (V(t,\mu)), Theorem 4.17 shows that the adjoint BSDE solution can be expressed as
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