{"id":"b9b21efa-0de3-4a61-b3bd-12a86a738846","arxiv_id":"2606.27078","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves existence of optimal plans and Kantorovich duality for semimartingale optimal transport with jumps, plus four equivalent formulations (Kantorovich, viscosity HJB, martingale projection, non-local FPK PDE) under minimal measurability-convexity-lsc assumptions on the cost.","lead":"The paper develops a general semimartingale optimal transport framework where the cost depends on full differential characteristics of processes that may include jumps. It proves existence of optimal plans, Kantorovich duality, and shows equivalence to four other formulations under minimal assumptions, unifying several prior optimal transport variants.","discovery_kind":"unification","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flags that full proofs are unavailable, precluding any technical diagnosis of gaps in the equivalence arguments or duality construction. No load-bearing concern can be located without the manuscript.","tokens_in":1688,"tokens_out":208,"duration_ms":33635,"concrete_test":"Re-derive the equivalence (iii) martingale solution to (iv) non-local FPK from the Markovian projection step using only the closed convex set property; confirm the projected characteristics remain inside the set for a jump process example.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract states existence, duality, and four equivalent formulations under measurability/convexity/lsc of the cost and a closed convex constraint on absolutely continuous characteristics. No internal inconsistency or hidden regularity assumption is visible from the stated claims; the weakest_assumption identified by the reader (the cost and constraint set) is explicitly the setting in which the results are claimed to hold.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1736,"tokens_out":475,"duration_ms":31765,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"§1"},{"comment":"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).","section":"Introduction"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[],"tokens_in":1310,"tokens_out":114,"duration_ms":18646,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper sets up semimartingale optimal transport where the cost depends on the full differential characteristics (including jumps) and admissible laws are constrained to a closed convex set of absolutely continuous characteristics. It asserts existence of an optimal plan plus Kantorovich duality under only measurability, convexity and lower semicontinuity of the cost, with no time-regularity required. It further claims four equivalent formulations: the Kantorovich problem, a viscosity HJB equation, a Markovian-projection martingale problem, and a weak non-local Fokker-Planck-Kolmogorov equation. The framework is said to cover classical OT, martingale OT, Schrödinger bridges and barycentric weak OT at once.\n\nThe removal of time-regularity and the explicit jump setting are the clearest novelties relative to the cited literature. Stating the assumptions so plainly is also useful. The argument is described as resting on standard convex-analysis and stochastic-process tools rather than circular definitions.\n\nThe main limitation is that only the abstract is available here. Existence, duality and the four equivalences are asserted, but there are no proof sketches or counter-example checks, so it is impossible to judge whether the jump case introduces gaps or whether the equivalences really hold at the stated level of generality. The reader's soundness score of 6 tracks that uncertainty.\n\nThis is for people working on stochastic optimal transport and related PDEs who need a broad reference framework. A reader looking for a single setting that drops time-regularity would find it relevant. It deserves peer review because the scope is wide and the minimal-assumption claim is strong enough to be worth checking in detail.","headline":"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.","tokens_in":2206,"tokens_out":407,"would_cite":false,"duration_ms":29744,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Semimartingale optimal transport admits an optimal plan and Kantorovich duality under minimal cost assumptions.","keywords":["semimartingale optimal transport","Kantorovich duality","Hamilton-Jacobi-Bellman equation","Fokker-Planck-Kolmogorov equation","martingale optimal transport","Schrödinger bridge","equivalent formulations","weak optimal transport"],"falsifier":"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.","tokens_in":2587,"feed_emoji":"","tokens_out":855,"duration_ms":40289,"temperature":0.7,"pith_summary":"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.","feed_headline":"Semimartingale optimal transport has four equivalent formulations","feed_subtitle":"Existence and Kantorovich duality hold under only measurability, convexity and lower semicontinuity of the cost, without time regularity.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["SOT existence and duality hold with minimal cost assumptions","Four SOT formulations: Kantorovich HJB martingale and non-local PDE","Semimartingale optimal transport equals viscosity solution and PDE forms","Minimal conditions prove optimal semimartingale transport plans","SOT unifies classical optimal transport and Schrodinger bridges"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The cost function is measurable, convex and lower semicontinuous while the admissible set of absolutely continuous characteristics is a prescribed closed convex set.","fun_headline_variants_meta":{"raw":{"variants":["SOT existence and duality hold with minimal cost assumptions","Four SOT formulations: Kantorovich HJB martingale and non-local PDE","Semimartingale optimal transport equals viscosity solution and PDE forms","Minimal conditions prove optimal semimartingale transport plans","SOT unifies classical optimal transport and Schrodinger bridges"]},"model":"grok-4.3","cost_usd":0.004269,"raw_usage":{"total_tokens":2162,"prompt_tokens":694,"num_sources_used":0,"completion_tokens":84,"cost_in_usd_ticks":42687000,"prompt_tokens_details":{"text_tokens":694,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1384,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":694,"tokens_out":84,"duration_ms":18388,"temperature":1.0,"reasoning_tokens":1384,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-26T03:33:26.097682+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}