Pith. sign in

REVIEW 3 minor 45 references

Semimartingale Optimal Transport with Jumps: A General Framework and Equivalent Formulations

T0 review · 0 major / 3 minor · reviewed 2026-06-26 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Semimartingale optimal transport admits an optimal plan and Kantorovich duality under minimal cost assumptions.

desk verdict Claims a unification of OT variants into jump semimartingales with four equivalent formulations and duality under minimal assumptions, but the abstract alone leaves the proofs unverified. read the letter →

arxiv 2606.27078 v1 pith:GZL4P4MF submitted 2026-06-25 math.PR math.DS

Boyi Hou , Zhenxin Liu This is my paper · ORCID
classification math.PRmath.DS
keywords semimartingaleoptimaltransportKantorovichdualityHamilton-Jacobi-BellmanequationFokker-Planck-KolmogorovmartingaleSchrödingerbridgeequivalentformulationsweak
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper develops a framework for semimartingale optimal transport where the cost depends on the full differential characteristics of the process and the minimization runs over semimartingale laws with fixed marginals whose absolutely continuous characteristics lie inside a prescribed closed convex set. It shows that existence of an optimal plan and a Kantorovich-type duality hold when the cost is merely measurable, convex, and lower semicontinuous, without any time-regularity requirements. The same problem is proved to be equivalent to four different formulations: a Kantorovich duality statement, a viscosity solution of a Hamilton-Jacobi-Bellman equation, a martingale solution obtained via Markovian projection, and a weak solution of a non-local Fokker-Planck-Kolmogorov equation. This simultaneously recovers classical optimal transport, martingale optimal transport, Schrödinger bridges, and barycentric weak optimal transport as special cases. A reader cares because the equivalences let one switch between analytic, probabilistic, and PDE viewpoints inside a single setting.

What carries the argument

The semimartingale optimal transport problem over laws with prescribed marginals whose absolutely continuous characteristics belong to a given closed convex set, with cost depending on the full differential characteristics.

What would settle it

A concrete measurable convex lower-semicontinuous cost function together with a closed convex set of characteristics for which no optimal semimartingale plan exists with the given marginals.

Watch

Extended reading notes

Core claim

Under only minimal assumptions of measurability, convexity, and lower semicontinuity on the cost function, we prove the existence of an optimal plan for SOT and establish a Kantorovich-type duality without time-regularity conditions. We further prove that SOT admits four equivalent formulations: (i) a Kantorovich duality formulation, (ii) a viscosity solution formulation of the Hamilton--Jacobi--Bellman equation, (iii) a martingale solution formulation via Markovian projection, (iv) a PDE formulation via weak solutions of a non-local Fokker--Planck--Kolmogorov equation. The framework simultaneously generalises classical optimal transport, martingale optimal transport, Schrödinger bridge prob

Load-bearing premise

The cost function is measurable, convex and lower semicontinuous while the admissible set of absolutely continuous characteristics is a prescribed closed convex set.

Editorial extensions

If this is right

  • An optimal plan exists for the semimartingale optimal transport problem.
  • Kantorovich duality holds without time-regularity conditions on the processes.
  • The problem is equivalent to a viscosity solution of the associated Hamilton-Jacobi-Bellman equation.
  • The problem is equivalent to a martingale solution obtained via Markovian projection.
  • The problem is equivalent to a weak solution of the non-local Fokker-Planck-Kolmogorov equation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The four equivalent formulations may allow selection of the computationally or analytically simplest representation for a given application.
  • The absence of time-regularity assumptions could extend the framework to processes with irregular time dependence in the characteristics.
  • Equivalence results may link the transport problem directly to stochastic control problems whose value functions satisfy the same Hamilton-Jacobi-Bellman equation.
  • Numerical approximation schemes could be built by discretizing whichever of the four formulations is most convenient for the chosen marginals.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper develops a semimartingale optimal transport (SOT) framework in which the cost depends on the full differential characteristics of the semimartingale and the minimization is performed over laws with prescribed marginals whose absolutely continuous characteristics belong to a given closed convex set. Under the sole assumptions of measurability, convexity and lower semicontinuity of the cost, the authors establish existence of an optimal plan and a Kantorovich duality without any time-regularity requirement. They further prove that the SOT problem admits four equivalent formulations: a Kantorovich duality formulation, a viscosity solution formulation of the associated Hamilton–Jacobi–Bellman equation, a martingale solution formulation obtained via Markovian projection, and a PDE formulation via weak solutions of a non-local Fokker–Planck–Kolmogorov equation. The setting simultaneously generalizes classical optimal transport, martingale optimal transport, Schrödinger bridges and barycentric weak optimal transport.

Significance. If the stated existence, duality and equivalence results hold under the announced minimal hypotheses, the work supplies a unified, regularity-free theory that connects several previously separate strands of optimal transport and stochastic analysis. The explicit equivalence between probabilistic, viscosity and PDE formulations is a notable strength that may enable transfer of techniques across communities. The absence of time-regularity assumptions broadens the scope relative to earlier semimartingale transport papers.

minor comments (3)
  1. [Abstract] The abstract asserts equivalence of four formulations but does not indicate whether the equivalences are proved under exactly the same hypotheses as existence and duality; a single sentence clarifying the common assumption set would improve readability.
  2. [§1] Notation for the set of admissible absolutely continuous characteristics is introduced only in the abstract; an early dedicated paragraph or displayed definition in §1 would help readers track the constraint throughout the subsequent statements.
  3. [Introduction] The manuscript would benefit from a short table or diagram in the introduction that explicitly maps the four equivalent formulations to the corresponding mathematical objects (e.g., plans, measures, functions, PDEs).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the accurate summary of our contributions, and the recommendation for minor revision. The report correctly identifies the key strengths of the work, including the minimal-assumption existence and duality results and the four equivalent formulations. No major comments are listed in the provided referee report, so we have no specific points requiring point-by-point rebuttal. We remain available to address any minor suggestions or clarifications that may arise during the revision process.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper establishes existence of an optimal plan, Kantorovich duality, and four equivalent formulations (Kantorovich, viscosity HJB, martingale projection, and non-local FPK) under explicit minimal assumptions of measurability, convexity, and lower semicontinuity on the cost together with a closed convex constraint on absolutely continuous characteristics. These results rest on standard convex-analysis and stochastic-process arguments that do not reduce any claimed output to a definition or fitted input by construction. No load-bearing self-citation chains, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation are present in the stated claims; the derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The framework rests on standard domain assumptions from optimal transport and semimartingale theory; no free parameters or invented entities are introduced.

assumptions (2)
  • domain assumption Cost function is measurable, convex, and lower semicontinuous.
    Invoked as the only conditions needed for existence and duality.
  • domain assumption The admissible set of absolutely continuous characteristics is closed and convex.
    Prescribed constraint set in the problem definition.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Semimartingale Optimal Transport with Jumps: A General Framework and Equivalent Formulations." pith.science (2026). https://pith.science/paper/GZL4P4MF

@misc{pith2026260627078,
  author       = {Pith},
  title        = {Pith review of: Semimartingale Optimal Transport with Jumps: A General Framework and Equivalent Formulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZL4P4MF}},
  note         = {Machine review of arXiv:2606.27078}
}
read the original abstract

We study a semimartingale optimal transport (SOT) problem where the cost depends on the full differential characteristics, and the minimisation is over all semimartingale laws with given marginals whose absolutely continuous characteristics lie in a prescribed closed convex set. Under only minimal assumptions of measurability, convexity, and lower semicontinuity on the cost function, we prove the existence of an optimal plan for SOT and establish a Kantorovich-type duality without time-regularity conditions. We further prove that SOT admits four equivalent formulations: (i) a Kantorovich duality formulation, (ii) a viscosity solution formulation of the Hamilton--Jacobi--Bellman equation, (iii) a martingale solution formulation via Markovian projection, (iv) a PDE formulation via weak solutions of a non-local Fokker--Planck--Kolmogorov equation. The framework simultaneously generalises classical optimal transport, martingale optimal transport, Schr\"odinger bridge problems, and barycentric weak optimal transport.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

45 extracted references · 2 canonical work pages

  1. [1]

    Alfonsi and G

    A. Alfonsi and G. Szulda, On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients.Bernoulli,31(2025), 2890–2915

    2025

  2. [2]

    Backhoff-Veraguas, M

    J. Backhoff-Veraguas, M. Beiglb¨ ock, M. Huesmann and S. K¨ allblad, Martingale Benamou–Brenier: a probabilistic perspective.Ann. Probab.,48(2020), 2258–2289

    2020

  3. [3]

    Backhoff-Veraguas and G

    J. Backhoff-Veraguas and G. Pammer, Applications of weak transport theory.Bernoulli,28(2022), 370–394

    2022

  4. [4]

    E. J. Balder, A general approach to lower semicontinuity and lower closure in optimal control theory.SIAM J. Control Optim.,22(1984), 570–598

    1984

  5. [5]

    E. J. Balder, On seminormality of integral functionals and their integrands.SIAM J. Control Optim.,24(1986), 95–121

    1986

  6. [6]

    D. S. Bates, Empirical option pricing models.Annu. Rev. Financ. Econ.,14(2022), 369–389

    2022

  7. [7]

    Bayraktar and E

    E. Bayraktar and E. Cl´ ement, Volatility and jump activity estimation in a stable Cox-Ingersoll-Ross model. Bernoulli,32(2026), 926–951

    2026

  8. [8]

    Beiglb¨ ock, A

    M. Beiglb¨ ock, A. M. G. Cox and M. Huesmann, Optimal transport and Skorokhod embedding.Invent. Math.,208 (2017), 327–400

    2017

Show all 45 references
  1. [9]

    Beiglb¨ ock, P

    M. Beiglb¨ ock, P. Henry-Labord` ere and F. Penkner, Model-independent bounds for option prices—a mass transport approach.Finance Stoch.,17(2013), 477–501

    2013

  2. [10]

    Beiglb¨ ock and N

    M. Beiglb¨ ock and N. Juillet, On a problem of optimal transport under marginal martingale constraints.Ann. Probab., 44(2016), 42–106

    2016

  3. [11]

    Beiglb¨ ock, M

    M. Beiglb¨ ock, M. Nutz and F. Stebegg, Fine properties of the optimal Skorokhod embedding problem.J. Eur. Math. Soc. (JEMS),24(2022), 1389–1429

    2022

  4. [12]

    Benamou and Y

    J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge–Kantorovich mass transfer problem.Numer. Math.,84(2000), 375–393

    2000

  5. [13]

    Benamou, G

    J.-D. Benamou, G. Chazareix, M. Hoffmann, G. Loeper and F.-X. Vialard, Entropic semi-martingale optimal trans- port, arXiv: 2408.09361

  6. [14]

    Cheridito, M

    P. Cheridito, M. Kiiski, D. J. Pr¨ omel and H. M. Soner, Martingale optimal transport duality.Math. Ann.,379 (2021), 1685–1712

    2021

  7. [15]

    Criens and L

    D. Criens and L. Niemann, Nonlinear continuous semimartingales.Electron. J. Probab.,28(2023), Paper No. 146, 40 pp. 26 BOYI HOU AND ZHENXIN LIU

    2023

  8. [16]

    Criens and L

    D. Criens and L. Niemann, Markov selections and Feller properties of nonlinear diffusions.Stochastic Process. Appl., 173(2024), Paper No. 104354, 30 pp

    2024

  9. [17]

    R. Denk, M. Kupper and M. Nendel, A semigroup approach to nonlinear L´ evy processes.Stochastic Process. Appl., 130(2020), 1616–1642

    2020

  10. [18]

    Deuschel and D

    J.-D. Deuschel and D. W. Stroock,Large Deviations. Pure and Applied Mathematics, 137, Academic Press, Inc., Boston, MA, 1989, xiv+307 pp

    1989

  11. [19]

    Diestel, Uniform integrability: an introduction.Rend

    J. Diestel, Uniform integrability: an introduction.Rend. Istit. Mat. Univ. Trieste,23(1991), 41–80. (School on Measure Theory and Real Analysis, Grado, 1991)

    1991

  12. [20]

    Dolinsky and H

    Y. Dolinsky and H. M. Soner, Martingale optimal transport in the Skorokhod space.Stochastic Process. Appl.,125 (2015), 3893–3931

    2015

  13. [21]

    Dweik, N

    S. Dweik, N. Ghoussoub, Y.-H. Kim and A. Z. Palmer, Stochastic optimal transport with free end time.Ann. Inst. Henri Poincar´ e Probab. Stat.,57(2021), 700–725

    2021

  14. [22]

    Eraker, M

    B. Eraker, M. Johannes and N. Polson, The impact of jumps in volatility and returns.J. Finance,58(2003), 1269–1300

    2003

  15. [23]

    G. Guo, X. Tan and N. Touzi, Optimal Skorokhod embedding under finitely many marginal constraints.SIAM J. Control Optim.,54(2016), 2174–2201

    2016

  16. [24]

    G. Guo, X. Tan and N. Touzi, Tightness and duality of martingale transport on the Skorokhod space.Stochastic Process. Appl.,127(2017), 927–956

    2017

  17. [25]

    I. Guo, G. Loeper and S. Wang, Local volatility calibration by optimal transport. In2017 MATRIX Annals, MATRIX Book Ser., 2, Springer, Cham, 2019, pp. 51–64

    2019

  18. [26]

    I. Guo, G. Loeper and S. Wang, Calibration of local-stochastic volatility models by optimal transport.Math. Finance, 32(2022), 46–77

    2022

  19. [27]

    I. Guo, S. Nilsson and J. Wiesel, Dynamic characterization of barycentric optimal transport problems and their martingale relaxation, arXiv: 2511.21287

  20. [28]

    Hu and S

    M. Hu and S. Peng,G-L´ evy processes under sublinear expectations.Probab. Uncertain. Quant. Risk,6(2021), 1–22

    2021

  21. [29]

    Huesmann and F

    M. Huesmann and F. Stebegg, Monotonicity preserving transformations of MOT and SEP.Stochastic Process. Appl., 128(2018), 1114–1134

    2018

  22. [30]

    Huesmann and D

    M. Huesmann and D. Trevisan, A Benamou–Brenier formulation of martingale optimal transport.Bernoulli,25 (2019), 2729–2757

    2019

  23. [31]

    A. D. Ioffe, On lower semicontinuity of integral functionals. I.SIAM J. Control Optim.,15(1977), 521–538

    1977

  24. [32]

    Jacod and A

    J. Jacod and A. N. Shiryaev,Limit Theorems for Stochastic Processes. Second edition, Grundlehren Math. Wiss., 288, Springer-Verlag, Berlin, 2003, xx+661 pp

    2003

  25. [33]

    Joseph, G

    B. Joseph, G. Loeper and J. Ob l´ oj, Joint calibration of local volatility models with stochastic interest rates using semimartingale optimal transport.Quant. Finance,24(2024), 1597–1620

    2024

  26. [34]

    K¨ uhn, Viscosity solutions to Hamilton–Jacobi–Bellman equations associated with sublinear L´ evy(-type) processes

    F. K¨ uhn, Viscosity solutions to Hamilton–Jacobi–Bellman equations associated with sublinear L´ evy(-type) processes. ALEA Lat. Am. J. Probab. Math. Stat.,16(2019), 531–559

    2019

  27. [35]

    Larsson and S

    M. Larsson and S. Long, Markovian projections for Itˆ o semimartingales with jumps.Electron. Commun. Probab., 29(2024), Paper No. 65, 13 pp

    2024

  28. [36]

    Liu and A

    C. Liu and A. Neufeld, Compactness criterion for semimartingale laws and semimartingale optimal transport.Trans. Amer. Math. Soc.,372(2019), 187–231

    2019

  29. [37]

    Mikami and M

    T. Mikami and M. Thieullen, Duality theorem for the stochastic optimal control problem.Stochastic Process. Appl., 116(2006), 1815–1835

    2006

  30. [38]

    Neufeld and M

    A. Neufeld and M. Nutz, Measurability of semimartingale characteristics with respect to the probability law.Sto- chastic Process. Appl.,124(2014), 3819–3845

    2014

  31. [39]

    Neufeld and M

    A. Neufeld and M. Nutz, Nonlinear L´ evy processes and their characteristics.Trans. Amer. Math. Soc.,369(2017), 69–95

    2017

  32. [40]

    Peng,Nonlinear Expectations and Stochastic Calculus under Uncertainty: With Robust CLT and G-Brownian Motion

    S. Peng,Nonlinear Expectations and Stochastic Calculus under Uncertainty: With Robust CLT and G-Brownian Motion. Probab. Theory Stoch. Model., 95, Springer, Berlin, 2019, xiii+212 pp

    2019

  33. [41]

    Richard, X

    A. Richard, X. Tan and N. Touzi, On the Root solution to the Skorokhod embedding problem given full marginals. SIAM J. Control Optim.,58(2020), 1874–1892

    2020

  34. [42]

    R¨ ockner, L

    M. R¨ ockner, L. Xie and X. Zhang, Superposition principle for non-local Fokker–Planck–Kolmogorov operators. Probab. Theory Related Fields,178(2020), 699–733

    2020

  35. [43]

    Tan and N

    X. Tan and N. Touzi, Optimal transportation under controlled stochastic dynamics.Ann. Probab.,41(2013), 3201–3240

    2013

  36. [44]

    Tr` eves,Topological Vector Spaces, Distributions and Kernels

    F. Tr` eves,Topological Vector Spaces, Distributions and Kernels. Academic Press, New York–London, 1967, xvi+624 pp

    1967

  37. [45]

    Villani,Optimal Transport: Old and New

    C. Villani,Optimal Transport: Old and New. Grundlehren Math. Wiss., 338, Springer-Verlag, Berlin, 2009, xxii+973 pp. SEMIMARTINGALE OPTIMAL TRANSPORT WITH JUMPS 27 B. Hou: School of Mathematics, Dalian University of Technology, Dalian 116024, P. R. China Email address:boyi.hou...

    2009

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.