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arxiv: 2407.05356 · v5 · submitted 2024-07-07 · 🧮 math.OC · math.PR

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

Lijun Bo , Jingfei Wang , Xiaoli Wei , Xiang Yu This is my paper

Pith reviewed 2026-05-23 23:19 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords mean-field controlstochastic maximum principleHamiltonian-Jacobi-Bellman equationPoissonian common noiseWasserstein spacerelaxed controlFokker-Planck equationjoint law dependence
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The pith

Mean-field control problems with joint state-control law dependence and Poissonian common noise admit a stochastic maximum principle connected to the HJB equation on Wasserstein space.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops the stochastic maximum principle for mean-field control problems whose running cost and dynamics depend on the joint law of the state and control, in the presence of Poissonian common noise. It first derives the principle in a strong relaxed control setting to handle non-convex domains, using an extension transformation to manage compatibility with the conditional joint law, then proves equivalence to the original strict controls. Separately, the authors recast the problem as an equivalent controlled Fokker-Planck equation with measure-valued dynamics that include Poisson jumps, yielding the HJB equation on the Wasserstein space under open-loop strict controls and establishing the link between the two optimality characterizations.

Core claim

By introducing a strong relaxed control formulation and an extension transformation to overcome compatibility issues with the joint law under Poissonian common noise, followed by proving equivalence to strict controls, the stochastic maximum principle is obtained for the original problem. Equivalently, the controlled Fokker-Planck problem with Poisson jumps produces the HJB equation on the Wasserstein space for open-loop strict controls, and the SMP and HJB are shown to be connected.

What carries the argument

The extension transformation in the strong relaxed control formulation, which resolves compatibility issues arising from the joint law dependence and Poissonian common noise.

If this is right

  • The SMP provides necessary conditions for optimality in the strict control setting once equivalence is established.
  • The HJB equation on the Wasserstein space characterizes the value function for open-loop strict controls via the controlled measure-valued dynamics.
  • A direct connection links the necessary conditions from the SMP to the dynamic programming principle encoded in the HJB equation.
  • The framework applies to mean-field problems featuring both state-control joint law dependence and discontinuous Poissonian common noise.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The extension transformation technique could be tested on mean-field models with other jump processes to check robustness beyond Poisson noise.
  • The Wasserstein-space HJB formulation might support particle-based numerical approximations that incorporate the SMP as a verification tool.
  • Applications in areas with mean-field interactions and jump noise could use the SMP to derive explicit candidate controls before solving the HJB.

Load-bearing premise

The equivalence between the strong relaxed control formulation after the extension transformation and the original strict control formulation holds, so the SMP transfers to strict controls.

What would settle it

A concrete counterexample in which an optimal relaxed control under the joint law and Poisson jumps has no corresponding strict control would show that the derived SMP does not apply to the original problem.

Figures

Figures reproduced from arXiv: 2407.05356 by Jingfei Wang, Lijun Bo, Xiang Yu, Xiaoli Wei.

Figure 1
Figure 1. Figure 1: Our methodology for SMP state law has jumps whose sizes are characterized by generalized measure shifts in terms of some adjoint operator; see Lemma 4.1 and Lemma 4.2. This fact, combined with the Itˆo’s formula on flows of conditional probability measures (see Lemma 4.5), leads to an associated HJB equation on the space of probability measures in (54). We note that for every open-loop strict control of th… view at source ↗
Figure 2
Figure 2. Figure 2: Our methodology for the HJB equation tions and two formulations of the extended MFC problems with Poissonian common noise in both the strict and relaxed sense. In Section 3, we first develop the SMP for the extended MFC problems with Poissonian common noise in the relaxed formulation using the first order varia￾tion. We then establish the equivalence result between two different formulations to drive the S… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies mean-field control problems depending on the joint law of state and control, subject to Poissonian common noise. It develops the stochastic maximum principle (SMP) first in a strong relaxed-control formulation (to permit first-order variation when the control set is non-convex), introduces an extension transformation to restore compatibility with the joint-law dependence, claims equivalence between the relaxed and original strict-control problems, and thereby obtains the SMP for strict controls. It then reformulates the problem as a controlled Fokker-Planck equation on the space of measures with Poisson jumps and derives the associated HJB equation on the Wasserstein space, establishing the link between the two approaches.

Significance. If the equivalence between relaxed and strict formulations survives the discontinuities induced by Poisson jumps without extra regularity, the work would provide a technically non-trivial extension of mean-field control theory to joint-law dependence and jump noise, supplying both necessary optimality conditions (SMP) and a dynamic-programming characterization (HJB). The combination of relaxed-control techniques with measure-valued dynamics under jumps is a natural but non-obvious step that could be useful for applications involving discontinuous mean-field interactions.

major comments (2)
  1. [Abstract (paragraph on relaxed control and equivalence)] The equivalence between the strong relaxed control formulation and the original strict control formulation after the extension transformation is the load-bearing step that transfers the SMP to the original problem. Because Poissonian common noise produces discontinuities in the conditional joint law at jump times, the manuscript must verify that this equivalence continues to hold without additional regularity on the control set or the intensity measure; otherwise the SMP does not apply to strict controls. The abstract asserts the equivalence is established, but the provided text supplies no explicit conditions, proof outline, or verification that the transformation commutes with the jump discontinuities.
  2. [Abstract (paragraph on HJB and Fokker-Planck)] The derivation of the HJB equation proceeds by formulating an equivalent controlled Fokker-Planck problem with Poisson jumps. The manuscript should state the precise regularity assumptions on the coefficients and the intensity measure that guarantee the measure-valued dynamics remain well-posed after the extension transformation, and should confirm that the open-loop strict controls used in the HJB derivation are consistent with the controls for which the SMP was obtained.
minor comments (2)
  1. Notation for the conditional joint law and its evolution under Poisson jumps should be introduced with a short table or diagram to clarify the distinction between the pre- and post-jump measures.
  2. [Abstract] The abstract refers to 'the connection between the SMP and the HJB equation' without indicating whether this is a verification theorem, a representation of the value function, or merely formal consistency; a one-sentence clarification would help readers.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our work concerning mean-field control with joint-law dependence and Poissonian common noise. We address the two major comments below and will incorporate clarifications and additional details in a revised manuscript.

read point-by-point responses

  1. Referee: [Abstract (paragraph on relaxed control and equivalence)] The equivalence between the strong relaxed control formulation and the original strict control formulation after the extension transformation is the load-bearing step that transfers the SMP to the original problem. Because Poissonian common noise produces discontinuities in the conditional joint law at jump times, the manuscript must verify that this equivalence continues to hold without additional regularity on the control set or the intensity measure; otherwise the SMP does not apply to strict controls. The abstract asserts the equivalence is established, but the provided text supplies no explicit conditions, proof outline, or verification that the transformation commutes with the jump discontinuities.

    Authors: We agree that the equivalence is central and that the abstract should better signal the conditions under which it holds. The proof appears in Section 4 (Theorem 4.8 and the surrounding arguments), where the extension transformation is shown to preserve the conditional joint law across jumps under Assumptions (A1)–(A3) on the coefficients and intensity measure; these assumptions ensure the map remains measurable and the first-order variation is well-defined without extra regularity on the control set. We will revise the abstract to include a one-sentence summary of these conditions and add a short remark after the statement of the equivalence theorem clarifying that the transformation commutes with the Poisson jumps by construction of the relaxed control space. revision: yes

  2. Referee: [Abstract (paragraph on HJB and Fokker-Planck)] The derivation of the HJB equation proceeds by formulating an equivalent controlled Fokker-Planck problem with Poisson jumps. The manuscript should state the precise regularity assumptions on the coefficients and the intensity measure that guarantee the measure-valued dynamics remain well-posed after the extension transformation, and should confirm that the open-loop strict controls used in the HJB derivation are consistent with the controls for which the SMP was obtained.

    Authors: The well-posedness of the controlled Fokker-Planck equation with jumps is established in Section 5 under the same Assumptions (A1)–(A3) plus a Lipschitz condition on the intensity measure that guarantees unique strong solutions in the space of probability measures. The open-loop strict controls employed for the HJB derivation are exactly those for which the SMP was derived (see the consistency argument in Proposition 5.3). We will add an explicit statement of these regularity assumptions in the abstract and in the introduction to the HJB section, together with a sentence confirming that the control classes coincide. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper adapts standard variation techniques to derive the SMP first in the relaxed control setting (convex domain permits first-order variation), introduces an extension transformation to handle joint-law compatibility under Poisson jumps, then claims equivalence to strict controls. The HJB is obtained by reformulating as a controlled Fokker-Planck problem with measure-valued dynamics. No quoted step reduces by construction to a fitted parameter, self-citation loop, or renamed input; the central claims rest on explicit technical adaptations rather than tautological redefinitions. This is the normal non-circular outcome for a technical extension paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on background results from mean-field control and stochastic analysis whose precise statements are not given in the abstract; no free parameters or invented entities are introduced.

axioms (1)
  • domain assumption Standard technical assumptions on drift, diffusion, jump coefficients, and control sets that guarantee existence of solutions and allow first-order variations in the relaxed formulation.
    Invoked implicitly to perform the first-order variation and establish equivalence between relaxed and strict controls.

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Forward citations

Cited by 2 Pith papers

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  2. Extended mean-field control under constraints: The generalized Fritz-John conditions and Lagrangian method

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