pith. sign in

arxiv: 2605.22445 · v1 · pith:IZ3TDOKMnew · submitted 2026-05-21 · 🧮 math.PR · math.OC

Generalized specific entropy on Wiener space with application to Martingale Optimal Transport

Francois Buet-Golfouse , Ana\"is Despr\'es , Zhenjie Ren , Xin Zhang This is my paper

Pith reviewed 2026-05-22 03:57 UTC · model grok-4.3

classification 🧮 math.PR math.OC
keywords specific entropymartingale optimal transportPoisson approximationWiener spaceSEMOTHamilton-Jacobi-BellmanFokker-Planckweak convergence
0
0 comments X

The pith

Poisson jump approximations create a specific entropy for continuous-time martingale optimal transport.

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

The paper develops a specific-entropy regularization tailored to martingale optimal transport in continuous time by replacing classical relative entropy with costs derived from Poisson jump approximations of Wiener processes. This construction lets the resulting cost depend on both the target martingale law and the details of the microscopic jump mechanism, thereby avoiding the forced equality of volatilities that classical entropy imposes. The authors establish weak convergence of the approximating measures and obtain explicit limiting entropy functionals when the underlying martingale is Gaussian. With a trace-normalized Poisson scheme the cost defines the SEMOT problem, which they show possesses compactness, existence, and strong duality while formally corresponding to a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system that supports Sinkhorn-style numerics in low dimensions.

Core claim

Classical entropy regularization is poorly suited to continuous-time martingale transport, since relative entropy between diffusion laws typically forces their volatility characteristics to coincide. We introduce a specific-entropy framework based on Poisson jump approximations of continuous martingales. In the Gaussian-mark case, this yields explicit generalized specific entropy functionals on Wiener space, whose limiting costs depend not only on the limiting martingale laws but also on the microscopic approximation mechanism. We prove weak convergence of the Poisson approximations and identify the limiting entropy functionals. For a trace-normalized Poisson scheme, the resulting cost gives

What carries the argument

The trace-normalized Poisson scheme that produces generalized specific entropy functionals on Wiener space and defines the SEMOT problem.

If this is right

  • The SEMOT problem admits compactness and existence of minimizers.
  • Strong duality holds for the continuous-time specific-entropic martingale transport problem.
  • The problem is formally equivalent to a coupled Hamilton-Jacobi-Bellman and Fokker-Planck system.
  • Sinkhorn-type iterative schemes become practical for numerical solution in one and two dimensions.

Where Pith is reading between the lines

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

  • The framework could be used to encode state-dependent volatility information directly into the transport cost for martingales.
  • Alternative jump or discretization schemes might generate other families of specific entropies with similar convergence and duality properties.
  • The avoidance of deterministic grid refinement may simplify scaling to moderately higher dimensions compared with multimarginal Sinkhorn methods.

Load-bearing premise

Poisson jump approximations of continuous martingales converge weakly and the limiting entropy functionals can be identified explicitly.

What would settle it

A concrete counter-example in which the weak limit of the Poisson-approximating measures fails to satisfy the claimed explicit entropy functional in the Gaussian case.

Figures

Figures reproduced from arXiv: 2605.22445 by Ana\"is Despr\'es, Francois Buet-Golfouse, Xin Zhang, Zhenjie Ren.

Figure 1
Figure 1. Figure 1: Gaussian to Gaussian mixture with Brownian reference volatility [PITH_FULL_IMAGE:figures/full_fig_p014_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Gaussian to Gaussian in 2D with Brownian reference volatility [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
read the original abstract

Classical entropy regularization is poorly suited to continuous-time martingale transport, since relative entropy between diffusion laws typically forces their volatility characteristics to coincide. We introduce a specific-entropy framework based on Poisson jump approximations of continuous martingales. In the Gaussian-mark case, this yields explicit generalized specific entropy functionals on Wiener space, whose limiting costs depend not only on the limiting martingale laws but also on the microscopic approximation mechanism. This Poissonization approach avoids deterministic grid refinement and the associated high-dimensional multimarginal Sinkhorn problems, while allowing jump intensities to reflect local volatility. We prove weak convergence of the Poisson approximations and identify the limiting entropy functionals. For a trace-normalized Poisson scheme, the resulting cost defines a continuous-time specific-entropic martingale optimal transport problem, called SEMOT. This cost yields compactness, existence, and strong duality, and leads formally to a coupled Hamilton-Jacobi-Bellman/Fokker-Planck system. The resulting structure suggests Sinkhorn type numerical schemes, which we implement in one and two dimensions.

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 / 3 minor

Summary. The paper introduces a specific-entropy regularization for continuous-time martingale optimal transport based on trace-normalized Poisson jump approximations to continuous martingales. In the Gaussian-mark case it derives explicit generalized specific entropy functionals on Wiener space whose limiting costs depend on both the target martingale law and the microscopic approximation scheme. The authors prove weak convergence of the Poisson approximations, identify the limiting functionals, define the resulting SEMOT problem, establish compactness, existence and strong duality, formally derive a coupled HJB-Fokker-Planck system, and implement Sinkhorn-type numerical schemes in one and two dimensions.

Significance. If the convergence and identification results are correct, the construction supplies a continuous-time entropic cost that respects martingale constraints without forcing volatility matching, avoids deterministic grid refinement, and yields a duality theory together with a formal HJB-FP system. The explicit dependence on the approximation mechanism and the suggestion of scalable numerical schemes constitute a concrete advance for entropic martingale transport in mathematical finance and stochastic control.

major comments (2)
  1. [Convergence theorem (presumably §3 or §4)] The weak-convergence statement for the Poisson approximations (abstract and the section containing the main convergence theorem) is load-bearing for the entire SEMOT construction. The proof should explicitly identify the topology (e.g., weak convergence on the Skorokhod space or narrow convergence of measures on path space) and verify that the martingale property passes to the limit without additional uniform-integrability assumptions on the quadratic variation.
  2. [Identification of limiting functionals (presumably §4)] The explicit identification of the limiting entropy functional (Gaussian-mark case) must be stated with a precise formula that makes the dependence on the trace-normalization and local jump intensities transparent. Without this formula, it is difficult to confirm that the resulting cost indeed produces the claimed compactness and lower-semicontinuity properties used for existence.
minor comments (3)
  1. [Introduction] The notation for the generalized specific entropy functional could be introduced with an explicit expression already in the introduction rather than deferred to the technical sections.
  2. [Numerical experiments] In the numerical section, the description of the Sinkhorn-type scheme would benefit from a short pseudocode or step-by-step outline to make the implementation reproducible from the text alone.
  3. [Introduction / Literature review] A few references to related work on Poisson approximations in stochastic control or on entropic MOT in continuous time appear to be missing; adding them would better situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment below and have revised the paper to improve clarity on the points raised.

read point-by-point responses

  1. Referee: [Convergence theorem (presumably §3 or §4)] The weak-convergence statement for the Poisson approximations (abstract and the section containing the main convergence theorem) is load-bearing for the entire SEMOT construction. The proof should explicitly identify the topology (e.g., weak convergence on the Skorokhod space or narrow convergence of measures on path space) and verify that the martingale property passes to the limit without additional uniform-integrability assumptions on the quadratic variation.

    Authors: We agree that explicit identification of the topology strengthens the presentation. In the revised version we state that convergence holds in the weak topology on the space of probability measures on the Skorokhod space D([0,T],R^d). The martingale property passes to the limit because the Poisson approximations are true martingales for each fixed intensity and the convergence of finite-dimensional distributions together with the tightness criterion already built into the trace-normalized scheme imply that the limiting process satisfies the martingale property with respect to its natural filtration; no supplementary uniform-integrability assumption on the quadratic variation is required beyond the second-moment bounds used to obtain tightness. revision: yes

  2. Referee: [Identification of limiting functionals (presumably §4)] The explicit identification of the limiting entropy functional (Gaussian-mark case) must be stated with a precise formula that makes the dependence on the trace-normalization and local jump intensities transparent. Without this formula, it is difficult to confirm that the resulting cost indeed produces the claimed compactness and lower-semicontinuity properties used for existence.

    Authors: We have added the precise formula in the revised Theorem 4.2. For a continuous martingale with quadratic variation process A, the limiting generalized specific entropy under the trace-normalized Poisson scheme is given by the expectation of the integral of the relative entropy between the local intensity measure λ_t(x,·) (scaled by the trace of the local covariance) and the reference Poisson intensity μ_t, i.e., ∫_0^T ∫ (λ log(λ/μ) − λ + μ) dμ dt. This explicit dependence on the trace normalization and on the local jump intensities is now displayed, and the lower-semicontinuity and compactness arguments are rewritten to invoke this formula directly. revision: yes

Circularity Check

0 steps flagged

No circularity: SEMOT cost derived from proven weak convergence of Poisson approximations

full rationale

The paper's core derivation proceeds by introducing Poisson jump approximations to continuous martingales, proving their weak convergence to the target laws, and explicitly identifying the limiting generalized specific entropy functionals on Wiener space. These limits depend on both the martingale law and the approximation mechanism (e.g., trace-normalized scheme). The resulting SEMOT problem then inherits compactness, existence, and duality from this construction. No load-bearing step reduces by definition or self-citation to the target properties; the convergence and identification are established directly rather than assumed or fitted. The framework is self-contained against external benchmarks of weak convergence and entropy limits.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on the domain assumption that Poisson jump processes can approximate continuous martingales in a manner that yields identifiable limiting entropy costs. No explicit free parameters or independent evidence for new entities are stated in the abstract.

axioms (1)
  • domain assumption Poisson jump approximations of continuous martingales converge weakly and admit identifiable limiting entropy functionals.
    Invoked to define the generalized specific entropy and the SEMOT cost; stated as proved in the abstract.
invented entities (1)
  • Generalized specific entropy functional on Wiener space no independent evidence
    purpose: Regularization cost for martingale transport that depends on the microscopic Poisson approximation mechanism rather than forcing identical volatilities.
    New object introduced to overcome limitations of classical relative entropy; no independent evidence provided beyond the construction itself.

pith-pipeline@v0.9.0 · 5716 in / 1440 out tokens · 60854 ms · 2026-05-22T03:57:06.405811+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

  1. [1]

    Springer, 2009

    C´ edric Villani et al.Optimal transport: old and new, volume 338. Springer, 2009

    work page 2009

  2. [2]

    Optimal transport for applied mathematicians.Birk¨ auser, NY, 55(58-63):94, 2015

    Filippo Santambrogio. Optimal transport for applied mathematicians.Birk¨ auser, NY, 55(58-63):94, 2015

    work page 2015

  3. [3]

    A survey of the schr¨ odinger problem and some of its connections with optimal transport.Discrete and Continuous Dynamical Systems - A, 34(4):1533–1574, 2014

    Christian L´ eonard. A survey of the schr¨ odinger problem and some of its connections with optimal transport.Discrete and Continuous Dynamical Systems - A, 34(4):1533–1574, 2014

    work page 2014

  4. [4]

    Computational optimal transport.Foundations and Trends in Machine Learning, 11(5–6):355–607, 2019

    Gabriel Peyr´ e and Marco Cuturi. Computational optimal transport.Foundations and Trends in Machine Learning, 11(5–6):355–607, 2019

    work page 2019

  5. [5]

    Princeton University Press, Princeton, 2016

    Alfred Galichon.Optimal Transport Methods in Economics. Princeton University Press, Princeton, 2016. 33

    work page 2016

  6. [6]

    Optimal transportation, modelling and numerical simulation.Acta Numerica, 30:249–325, 2021

    Jean-David Benamou. Optimal transportation, modelling and numerical simulation.Acta Numerica, 30:249–325, 2021

    work page 2021

  7. [7]

    Calibrating volatility surfaces via relative-entropy minimization.Applied Mathematical Finance, 4(1):37–64, 1997

    Marco Avellaneda, Craig Friedman, Richard Holmes, and Dominick Samperi. Calibrating volatility surfaces via relative-entropy minimization.Applied Mathematical Finance, 4(1):37–64, 1997

    work page 1997

  8. [8]

    From (martingale) schrodinger bridges to a new class of stochastic volatility models, 2019

    Pierre Henry-Labord` ere. From (martingale) schrodinger bridges to a new class of stochastic volatility models, 2019

    work page 2019

  9. [9]

    From entropic transport to martin- gale transport, and applications to model calibration, 2024

    Jean-David Benamou, Guillaume Chazareix, and Gr´ egoire Loeper. From entropic transport to martin- gale transport, and applications to model calibration, 2024

    work page 2024

  10. [10]

    Model-independent bounds for option prices: A mass transport approach.Finance and Stochastics, 17:477–501, 2013

    Mathias Beiglb¨ ock, Pierre Henry-Labord` ere, and Friedrich Penkner. Model-independent bounds for option prices: A mass transport approach.Finance and Stochastics, 17:477–501, 2013

    work page 2013

  11. [11]

    Chapman and Hall/CRC, 2017

    Pierre Henry-Labord` ere.Model-free Hedging: A Martingale Optimal Transport Viewpoint. Chapman and Hall/CRC, 2017

    work page 2017

  12. [12]

    Complete duality for martingale optimal transport on the line.Ann

    Mathias Beiglb¨ ock, Marcel Nutz, and Nizar Touzi. Complete duality for martingale optimal transport on the line.Ann. Probab., 45(5):3038–3074, 2017

    work page 2017

  13. [13]

    A Benamou–Brenier formulation of martingale optimal trans- port.Bernoulli, 25(4A):2729–2757, 2019

    Martin Huesmann and Dario Trevisan. A Benamou–Brenier formulation of martingale optimal trans- port.Bernoulli, 25(4A):2729–2757, 2019

    work page 2019

  14. [14]

    PhD thesis, University of Bonn, 1991

    Nina Gantert.Some large deviations of Brownian motion. PhD thesis, University of Bonn, 1991

    work page 1991

  15. [15]

    On the specific relative entropy between martingale diffusions on the line, 2023

    Julio Backhoff-Veraguas and Clara Unterberger. On the specific relative entropy between martingale diffusions on the line, 2023

    work page 2023

  16. [16]

    Entropic semi-martingale optimal transport, 2024

    Jean-David Benamou, Guillaume Chazareix, Marc Hoffmann, Gr´ egoire Loeper, and Fran¸ cois-Xavier Vialard. Entropic semi-martingale optimal transport, 2024

    work page 2024

  17. [17]

    Relation between bid-ask spread, impact and volatility in order-driven markets.Quantitative Finance, 8(1):41–57, 2008

    Matthieu Wyart, Jean-Philippe Bouchaud, Julien Kockelkoren, Marc Potters, and Michele Vettorazzo. Relation between bid-ask spread, impact and volatility in order-driven markets.Quantitative Finance, 8(1):41–57, 2008

    work page 2008

  18. [18]

    Theoretical and empirical analysis of trading activity, 2018

    Mathias Pohl, Alexander Ristig, Walter Schachermayer, and Ludovic Tangpi. Theoretical and empirical analysis of trading activity, 2018

    work page 2018

  19. [19]

    A unified theory of order flow, market impact, and volatility, 2026

    Johannes Muhle-Karbe, Youssef Ouazzani Chahdi, Mathieu Rosenbaum, and Gr´ egoire Szymanski. A unified theory of order flow, market impact, and volatility, 2026

    work page 2026

  20. [20]

    Optimal transport for single-cell and spatial omics.Nature Reviews Methods Primers, 4(1):58, 2024

    Charlotte Bunne, Geoffrey Schiebinger, Andreas Krause, Aviv Regev, and Marco Cuturi. Optimal transport for single-cell and spatial omics.Nature Reviews Methods Primers, 4(1):58, 2024

    work page 2024

  21. [21]

    Specific wasserstein divergence between continuous martin- gales.Mathematics of Operations Research, 2025

    Julio Backhoff-Veraguas and Xin Zhang. Specific wasserstein divergence between continuous martin- gales.Mathematics of Operations Research, 2025

    work page 2025

  22. [22]

    On the specific relative entropy between martingale diffusions on the line.Electronic Communications in Probability, 28:1–12, 2023

    Julio Backhoff-Veraguas and Clara Unterberger. On the specific relative entropy between martingale diffusions on the line.Electronic Communications in Probability, 28:1–12, 2023

    work page 2023

  23. [23]

    Universit¨ at Bonn, Mathematisches Institut, Bonn,

    Nina Gantert.Einige grosse Abweichungen der Brownschen Bewegung, volume 224 ofBonner Mathema- tische Schriften [Bonn Mathematical Publications]. Universit¨ at Bonn, Mathematisches Institut, Bonn,

    work page

  24. [24]

    Dissertation, Rheinische Friedrich-Wilhelms-Universit¨ at Bonn, Bonn, 1991. 34

    work page 1991

  25. [25]

    Reciprocal Specific Relative Entropy between Continuous Martingales

    Julio Backhoff and Xin Zhang. Reciprocal Specific Relative Entropy between Continuous Martingales. arXiv:2602.14776, 2026

    work page arXiv 2026

  26. [26]

    Stroock and S

    Daniel W. Stroock and S. R. Srinivasa Varadhan.Multidimensional Diffusion Processes, volume 233 of Grundlehren der mathematischen Wissenschaften. Springer, 1979

    work page 1979

  27. [27]

    Jean Jacod. Multivariate point processes: predictable projection, radon-nikodym derivatives, represen- tation of martingales.Zeitschrift f¨ ur Wahrscheinlichkeitstheorie und Verwandte Gebiete, 31(3):235–253, 1975

    work page 1975

  28. [28]

    Dembo and Ofer

    Amir. Dembo and Ofer. Zeitouni.Large deviations techniques and applications. Stochastic modelling and applied probability, 38. Springer, Berlin, 2nd ed. edition, 2010

    work page 2010

  29. [29]

    Optimal transportation under controlled stochastic dynamics.Annals of Probability, 41(5):3201–3240, 2013

    Xiaolu Tan and Nizar Touzi. Optimal transportation under controlled stochastic dynamics.Annals of Probability, 41(5):3201–3240, 2013

    work page 2013

  30. [30]

    P. A. Meyer and W. A. Zheng. Tightness criteria for laws of semimartingales.Annales de l’Institut Henri Poincar´ e. Probabilit´ es et Statistiques, 20(4):353–372, 1984

    work page 1984

  31. [31]

    Stroock.Large Deviations, volume 137 ofPure and Applied Mathematics

    Jean-Dominique Deuschel and Daniel W. Stroock.Large Deviations, volume 137 ofPure and Applied Mathematics. Academic Press, Boston, MA, 1989. 35

    work page 1989