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arxiv: 2606.27078 · v1 · pith:GZL4P4MFnew · submitted 2026-06-25 · 🧮 math.PR · math.DS

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

Boyi Hou , Zhenxin Liu This is my paper

Pith reviewed 2026-06-26 03:33 UTC · model grok-4.3

classification 🧮 math.PR math.DS
keywords semimartingale optimal transportKantorovich dualityHamilton-Jacobi-Bellman equationFokker-Planck-Kolmogorov equationmartingale optimal transportSchrödinger bridgeequivalent formulationsweak optimal transport
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The pith

Semimartingale optimal transport admits an optimal plan and Kantorovich duality under minimal cost assumptions.

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

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.

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

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.

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.

Where Pith is reading between the lines

These are 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.

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.

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.

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.

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

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

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.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

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

axioms (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.

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discussion (0)

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