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arxiv: 2607.00799 · v1 · pith:NVLAYZQUnew · submitted 2026-07-01 · 🧮 math.OC · math.PR

Optimal control problem for reflected McKean--Vlasov stochastic differential equations with Poisson jumps

Wenrui Lu , Kai Wang This is my paper

Pith reviewed 2026-07-02 07:45 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords optimal relaxed controlMcKean-Vlasov SDEreflected processPoisson jumpsSkorokhod mapRoxin convexity conditionAldous tightness criterion
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The pith

An optimal relaxed control exists for reflected McKean-Vlasov stochastic differential equations with Poisson jumps under Lipschitz and growth conditions.

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

This paper proves existence of an optimal relaxed control for one-dimensional reflected McKean-Vlasov SDEs with Poisson jumps. Uniform moment estimates are first obtained for the state process and reflecting process. Tightness follows from Aldous' criterion, after which the continuity of the Skorokhod map and stability of jump integrals allow passage to a limit that yields the optimal relaxed control. When the Roxin convexity condition holds, a strict optimal control is shown to exist, and relaxed controls can always be approximated by sequences of strict controls.

Core claim

Under Lipschitz conditions and suitable growth conditions, uniform moment estimates for the state process and the reflecting process are established. By using Aldous' tightness criterion, the continuity of the Skorokhod map, and the stability results for stochastic integrals, the existence of an optimal relaxed control is proved. Furthermore, under the Roxin convexity condition, the existence of a strict optimal control is proved. In the general case, relaxed controls can be approximated by a sequence of strict controls.

What carries the argument

The continuity of the Skorokhod map together with stability of stochastic integrals with jumps, used to pass to the limit after establishing tightness of controlled processes under mean-field interaction and reflection.

If this is right

  • Uniform moment bounds hold for the state and reflecting processes.
  • An optimal relaxed control is guaranteed to exist.
  • A strict optimal control exists whenever the Roxin convexity condition is satisfied.
  • Any relaxed control admits approximation by a sequence of strict controls.

Where Pith is reading between the lines

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

  • The same tightness and limit arguments may apply to related mean-field control problems with other types of discontinuities if the Skorokhod map properties carry over.
  • Numerical schemes that first solve the relaxed problem and then project to strict controls could be validated using the approximation result.
  • The one-dimensional setting may allow explicit computation of the reflecting process in simple cases, providing testbeds for the general theory.

Load-bearing premise

The Skorokhod map must remain continuous and the stochastic integrals stable when passing to the limit under mean-field dependence and Poisson jumps.

What would settle it

A concrete set of Lipschitz coefficients and admissible controls for which the approximating sequence of state processes fails to converge in the Skorokhod space to a solution of the reflected equation would disprove the existence result.

read the original abstract

In this paper, we consider the optimal relaxed control problem for a class of one-dimensional reflected McKean--Vlasov stochastic differential equations with Poisson jumps. Due to the presence of the jump term, the state process generally belongs to the Skorokhod space $D([0,T],\Rp)$, which makes the proof of tightness and the passage to the limit more complicated. Under Lipschitz conditions and suitable growth conditions, we establish uniform moment estimates for the state process and the reflecting process. Then, by using Aldous' tightness criterion, the continuity of the Skorokhod map, and the stability results for stochastic integrals, we prove the existence of an optimal relaxed control. Furthermore, under the Roxin convexity condition, we prove the existence of a strict optimal control. In the general case, we show that relaxed controls can be approximated by a sequence of strict controls.

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

1 major / 2 minor

Summary. The paper studies the optimal relaxed control problem for one-dimensional reflected McKean-Vlasov SDEs driven by Brownian motion and Poisson jumps. Under Lipschitz and growth conditions it derives uniform moment bounds on the state and reflection processes, applies Aldous' criterion to obtain tightness in the Skorokhod space D([0,T],R+), invokes continuity of the Skorokhod map together with stability of stochastic integrals with jumps to pass to the limit, and concludes existence of an optimal relaxed control. Under the Roxin convexity condition it further obtains a strict optimal control, and shows that relaxed controls can be approximated by strict controls in the general case.

Significance. If the technical steps hold, the result provides an existence theory for optimal control of reflected mean-field jump processes, a setting relevant to constrained interacting particle systems. The combination of reflection, mean-field dependence, and jumps is technically demanding; the use of relaxed controls and the approximation statement are standard but necessary tools in this framework.

major comments (1)
  1. [Proof of existence of optimal relaxed control (limit passage step)] The existence proof for the optimal relaxed control rests on tightness via Aldous' criterion followed by passage to the limit using continuity of the Skorokhod map and stability of stochastic integrals with jumps. Because the driving Poisson random measure and the law-dependent coefficients both depend on the control sequence, the standard continuity results for the Skorokhod map (typically stated for fixed driving semimartingales) require a uniform-in-control version that accounts for the simultaneous convergence of the empirical measure and the jump measure; the manuscript invokes these properties but supplies no indication that the requisite uniform modulus or joint continuity lemma has been established for the reflected mean-field case.
minor comments (2)
  1. Clarify the precise statement of the Roxin convexity condition used for the strict-control result and confirm that it is compatible with the reflection and jump terms.
  2. The abstract states that the state process belongs to D([0,T],R+); confirm that all moment estimates and tightness arguments are carried out uniformly with respect to the control in this space.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comment on the limit passage step. We address the point below.

read point-by-point responses

  1. Referee: [Proof of existence of optimal relaxed control (limit passage step)] The existence proof for the optimal relaxed control rests on tightness via Aldous' criterion followed by passage to the limit using continuity of the Skorokhod map and stability of stochastic integrals with jumps. Because the driving Poisson random measure and the law-dependent coefficients both depend on the control sequence, the standard continuity results for the Skorokhod map (typically stated for fixed driving semimartingales) require a uniform-in-control version that accounts for the simultaneous convergence of the empirical measure and the jump measure; the manuscript invokes these properties but supplies no indication that the requisite uniform modulus or joint continuity lemma has been established for the reflected mean-field case.

    Authors: We agree that the dependence of both the Poisson measure and the coefficients on the control sequence requires care in the limit passage. The uniform moment bounds (established in Section 3 independently of the control) yield tightness via Aldous' criterion that is uniform over the sequence of relaxed controls. The subsequent identification of the limit then uses the pathwise continuity of the Skorokhod reflection map together with the stability of the stochastic integral with respect to the Poisson random measure under convergence in probability of the integrators. Nevertheless, the referee is correct that an explicit uniform-in-control joint continuity statement for the reflected mean-field setting is not spelled out. We will add a short lemma (or a clarifying remark in the proof of Theorem 4.1) that records the required uniform modulus of continuity, derived from the uniform integrability already obtained. revision: yes

Circularity Check

0 steps flagged

No circularity detected; proof chain relies on external classical results

full rationale

The derivation proceeds from Lipschitz/growth assumptions to uniform moment bounds on the state and reflection processes, followed by application of Aldous' tightness criterion, continuity of the Skorokhod map, and stability of stochastic integrals with jumps to obtain existence of an optimal relaxed control (and, under Roxin convexity, a strict control). These steps invoke standard external theorems rather than any self-definitional reduction, fitted parameter renamed as prediction, or load-bearing self-citation whose content is itself unverified. The argument is therefore self-contained against external benchmarks and receives score 0.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard domain assumptions from stochastic analysis; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Lipschitz conditions and suitable growth conditions on the coefficients
    Used to obtain uniform moment estimates and tightness.
  • domain assumption Roxin convexity condition
    Invoked to obtain existence of strict optimal controls.

pith-pipeline@v0.9.1-grok · 5679 in / 1425 out tokens · 22465 ms · 2026-07-02T07:45:30.754362+00:00 · methodology

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Reference graph

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