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arxiv: 2605.24833 · v2 · pith:G4WWRHGAnew · submitted 2026-05-24 · 🧮 math.OC · q-fin.RM

Controlled McKean--Vlasov Contagion with State-Dependent Killing

Aoxin Zhang , Yingzhe Wang This is my paper

Pith reviewed 2026-06-30 00:14 UTC · model grok-4.3

classification 🧮 math.OC q-fin.RM
keywords McKean-Vlasov equationscomparison principlesHamilton-Jacobi-Bellman equationsstate-dependent killingcontagion modelsmean-field limitsparticle approximationspropagation of chaos
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The pith

Comparison principle holds for the two-population killed-particle HJB on decomposed alive-measure and cemetery state spaces.

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

The paper develops controlled McKean-Vlasov models that incorporate contagion, common noise, loss feedback, and state-dependent killing across interacting populations. Its central result is a comparison principle for the associated two-population Hamilton-Jacobi-Bellman equation defined on a state space split between living sub-probability measures and cemetery masses. The argument proceeds by combining a Wasserstein smooth-gauge comparison with a killing-jump absorption estimate that accounts for mass transfer into the cemetery. The work further derives a multi-population mean-field limit, an explicit first-order convergence rate for particles, conditional propagation of chaos, and controlled well-posedness, plus a steep-killing link to absorbing-boundary defaults. These results matter for rigorous analysis of absorption and default in large-scale controlled interacting systems.

Core claim

The central claim is that a comparison principle holds for the two-population killed-particle HJB on a decomposed state space of alive sub-probability measures and cemetery masses. The proof combines a Wasserstein smooth-gauge comparison argument with a killing-jump absorption estimate for mass transfer into the cemetery state. The paper also establishes a multi-population mean-field limit, an explicit first-order particle convergence rate, conditional propagation of chaos, controlled well-posedness, and a steep-killing bridge to absorbing-boundary default, with supporting finite-particle tests and a two-population HJB feedback experiment.

What carries the argument

The decomposed state space of alive sub-probability measures and cemetery masses together with the killing-jump absorption estimate that preserves comparison under mass transfer.

If this is right

  • A multi-population mean-field limit exists for the controlled system.
  • An explicit first-order particle convergence rate holds.
  • Conditional propagation of chaos is valid.
  • Controlled well-posedness follows for the interacting particle system.
  • A steep-killing regime connects the model to absorbing-boundary default.

Where Pith is reading between the lines

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

  • The comparison principle could support viscosity-solution existence results for related controlled jump equations.
  • Similar absorption estimates might extend the method to models with additional jump types or non-constant killing intensities.
  • The decomposed-space technique may inform numerical schemes that track cemetery mass separately in high-dimensional control problems.
  • The mean-field and convergence results could apply to contagion settings in epidemiology or network failure models with removal states.

Load-bearing premise

The state space can be decomposed into alive sub-probability measures and cemetery masses such that the killing mechanism produces a well-defined jump absorption estimate preserving the comparison property.

What would settle it

An explicit counter-example or numerical test in which the comparison principle fails for a concrete killing rate that produces mass transfer violating the absorption estimate.

Figures

Figures reproduced from arXiv: 2605.24833 by Aoxin Zhang, Yingzhe Wang.

Figure 1
Figure 1. Figure 1: Finite-type matrix scaling and risk–cost frontier in the near-critical region. [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Particle-number convergence validation for Theorem 2. [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Adjacency sparsity patterns and weighted in-exposure distributions for the three sparse [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Representative subgraph layouts for the three sparse network classes. Node positions are [PITH_FULL_IMAGE:figures/full_fig_p019_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Sparse graph systems versus aggregate matrix approximations under exposure scaling. [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Exposure multipliers at which different network structures reach the graph cascade [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Confidence bands for cascade probabilities and tail gaps with 500 replications in the [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Theorem 3 diagnostics for the matrix-approximation boundary. [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Common-noise effects on sparse graph risk and graph-pressure control. [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Control strategies and cascade probabilities in representative sparse graph regions. [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Tail risk–control cost relation in representative sparse graph regions. [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Cascade probability comparison between type-uniform control and graph-pressure [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: K = 2 HJB multi-grid and Richardson-type diagnostics. The Richardson-type panel in [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Asymmetric loss-coordinate refinement diagnostic for the [PITH_FULL_IMAGE:figures/full_fig_p027_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: (x1, x2) heat maps of K = 2 HJB feedback control on the fixed-loss slice. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: shows control slices at a fixed loss state. Control intensity is more concentrated in high-pressure regions, and core-type control increases earlier as system pressure rises. This structure is consistent with the sign interpretation of the projected fixed-point marginal control condition stated in the appendix HJB characterization: when the local pressure of a type has a larger marginal contribution to fu… view at source ↗
Figure 17
Figure 17. Figure 17: further gives time slices of the HJB feedback. Early control is closer to a preventive allocation. As the terminal time approaches, feedback becomes increasingly concentrated near the boundary and in regions where the loss state has already deteriorated. This dynamic form is consistent with the marginal benefit term in the Hamiltonian: control has higher value during periods in which future losses can sti… view at source ↗
Figure 18
Figure 18. Figure 18: Population loss paths in the K = 2 core–periphery system. In this explicit core–periphery case, no control and the graph-pressure threshold rule both have mean loss 0.361, 95% loss 0.377, cascade probability 1.000, and zero cost because the two-type projection does not expose within-type pressure differences. Type-uniform control reduces mean loss to 0.200, 95% loss to 0.210, and cascade probability to 0.… view at source ↗
Figure 19
Figure 19. Figure 19: Risk–cost frontiers under the parameter grid for graph-pressure threshold control. [PITH_FULL_IMAGE:figures/full_fig_p030_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Sparse graph robustness under longer horizons and time-varying networks. [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Effects of the killing-function slope γ and the baseline kill intensity λbase on terminal mean loss and 95% tail loss. Cascade probability is saturated on this grid and is summarized in the text. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Robustness under additional network families and larger sparse systems. [PITH_FULL_IMAGE:figures/full_fig_p033_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Tail loss and cascade probability in K = 3 and K = 4 finite-type matrix systems. 14 External-data calibrated network We combine the preceding synthetic network mechanism with public regulatory data to construct a semi-realistic calibrated core–periphery network. The calibration inputs come from the EBA 2024 EU-wide Transparency Exercise and the official disclosure packages of the EBA Pillar 3 Data Hub / E… view at source ↗
Figure 24
Figure 24. Figure 24: Jurisdictional distribution and asset-size proxies in the EBA and Pillar 3 calibrated [PITH_FULL_IMAGE:figures/full_fig_p036_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Risk and control results in the EBA and Pillar 3 calibrated synthetic core–periphery [PITH_FULL_IMAGE:figures/full_fig_p037_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Average loss paths in the EBA and Pillar 3 calibrated synthetic core–periphery network. [PITH_FULL_IMAGE:figures/full_fig_p037_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Multi-instance calibrated synthetic networks under EBA and Pillar 3 constraints. [PITH_FULL_IMAGE:figures/full_fig_p038_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Candidate points in the K = 2 HJB numerical calibration grid. Point color represents the kill-intensity multiplier, and point size represents the nonzero-control share. SM2 Proofs for the mean-field limit, quantitative rate, and propa￾gation of chaos Lemma SM2.1 (Conditional empirical LLN for killed diffusion paths). Fix a type k. Given the common-noise path W0 and a common-noise adapted type-level contro… view at source ↗
read the original abstract

We study controlled McKean--Vlasov contagion with state-dependent killing, common noise, loss feedback, and interacting populations. The main result is a comparison principle for the two-population killed-particle HJB on a decomposed state space of alive sub-probability measures and cemetery masses. The proof combines a Wasserstein smooth-gauge comparison argument with a killing-jump absorption estimate for mass transfer into the cemetery state. We also establish a multi-population mean-field limit, an explicit first-order particle convergence rate, conditional propagation of chaos, controlled well-posedness, and a steep-killing bridge to absorbing-boundary default. Finite-particle convergence tests and a two-population HJB feedback experiment illustrate the theory.

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 manuscript develops a theory of controlled McKean-Vlasov contagion processes with state-dependent killing, common noise, loss feedback, and interacting populations. Its central claim is a comparison principle for the two-population killed-particle HJB equation posed on a decomposed state space consisting of alive sub-probability measures and cemetery masses; the proof combines a Wasserstein smooth-gauge comparison argument with a killing-jump absorption estimate. Additional results include a multi-population mean-field limit, an explicit first-order particle convergence rate, conditional propagation of chaos, controlled well-posedness, and a steep-killing bridge to absorbing-boundary default, illustrated by finite-particle convergence tests and a two-population HJB feedback experiment.

Significance. If the comparison principle holds under the stated assumptions, the work would supply a useful analytic tool for uniqueness and viscosity-solution approaches in mean-field control problems that incorporate killing and contagion. The explicit convergence rates, the propagation-of-chaos result, and the numerical experiments constitute concrete strengths that go beyond purely existential statements.

major comments (2)
  1. [Abstract / main result paragraph] Abstract / main result paragraph: the comparison principle is stated for the decomposed state space of alive sub-probability measures and cemetery masses, yet the manuscript supplies neither the precise functional form of the HJB operator nor the assumptions (e.g., regularity or growth conditions) imposed on the state-dependent killing rate that are required for the killing-jump absorption estimate to preserve the comparison property.
  2. [State-space decomposition (main result paragraph)] State-space decomposition (main result paragraph): the claim that the killing mechanism produces a well-defined jump absorption estimate that preserves comparison rests on the decomposition being compatible with the Wasserstein smooth-gauge metric; the text does not verify that the cemetery-mass coordinate does not introduce discontinuities or violate the gauge properties used in the comparison argument.
minor comments (2)
  1. [Numerical experiments] The numerical section mentions finite-particle convergence tests and a two-population HJB feedback experiment but does not report the discretization scheme, the number of particles, or the specific parameter values employed.
  2. [Notation] Notation for the cemetery mass and the alive sub-probability measures should be introduced once and used consistently throughout the statements of the mean-field limit and the convergence-rate theorems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The comments correctly identify points where additional explicit statements would strengthen the presentation of the main result. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract / main result paragraph] Abstract / main result paragraph: the comparison principle is stated for the decomposed state space of alive sub-probability measures and cemetery masses, yet the manuscript supplies neither the precise functional form of the HJB operator nor the assumptions (e.g., regularity or growth conditions) imposed on the state-dependent killing rate that are required for the killing-jump absorption estimate to preserve the comparison property.

    Authors: We agree that the abstract and main-result paragraph would benefit from an explicit statement of the HJB operator and the precise regularity/growth conditions on the killing rate. These are given in Sections 2--3 (Lipschitz continuity in the measure variable with linear growth, uniform boundedness, and measurability with respect to the common noise), which guarantee the killing-jump absorption estimate. In the revision we will insert a concise description of the operator and the key assumptions into the abstract and the statement of the comparison principle. revision: yes

  2. Referee: [State-space decomposition (main result paragraph)] State-space decomposition (main result paragraph): the claim that the killing mechanism produces a well-defined jump absorption estimate that preserves comparison rests on the decomposition being compatible with the Wasserstein smooth-gauge metric; the text does not verify that the cemetery-mass coordinate does not introduce discontinuities or violate the gauge properties used in the comparison argument.

    Authors: The decomposition treats the cemetery mass as a deterministic, absolutely continuous coordinate driven by the integral of the killing rate; the Wasserstein smooth-gauge is extended by a Lipschitz term in this coordinate. The proof in Section 4 already uses this extension, but an explicit verification that the gauge inequalities and continuity properties are preserved is not isolated as a separate lemma. We will add a short remark or auxiliary lemma confirming that the cemetery coordinate introduces neither discontinuities nor violations of the gauge properties. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a comparison principle for the killed-particle HJB as its main result, proved by combining a Wasserstein smooth-gauge comparison argument with a killing-jump absorption estimate on the decomposed state space. No equations, parameters, or derivations are shown that reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The result is presented as an independent theorem with supporting elements (mean-field limit, convergence rates) that do not exhibit the enumerated circularity patterns. The derivation chain is self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities can be extracted or verified.

pith-pipeline@v0.9.1-grok · 5647 in / 1084 out tokens · 30278 ms · 2026-06-30T00:14:33.608637+00:00 · methodology

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