On the continuity of martingale optimal transport in higher dimensions

Paper: arXiv 2611.16429

Authors: Francesco Mattesini, Johannes Wiesel

Abstract

We investigate qualitative and quantitative continuity of martingale optimal transport under perturbations of both marginals in dimensions $d\geq2$. We assume that each pair of marginals admits a reference martingale coupling whose density relative to the product of its marginals is bounded above and away from zero, uniformly along the sequence. If the limiting target measure is not carried by a hyperplane, we prove continuity under $W_1$-convergence for continuous costs of at most linear growth, assuming only finite first moments. For Lipschitz costs, a uniform $p$-moment bound on the targets, $p>1$, yields an explicit stability estimate with exponent $(1-1/p)/2$; uniformly bounded target supports give exponent $1/2$. The proof constructs a correction of couplings with small martingale bias while preserving both marginals. Truncated moment corrections allow the target measure to have unbounded support. The construction also shows that every martingale coupling of the limiting marginals can be approximated in $W_1$ by martingale couplings of the approximating marginals. We also discuss counterexamples showing that neither the uniform density condition nor non-degeneracy of the limiting target measure can be omitted in general.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper is highly theoretical, focusing on advanced mathematical proofs related to martingale optimal transport. It lacks any empirical validation or data analysis, placing it firmly in the ‘Lab Rats’ quadrant. The novelty lies in extending continuity results to higher dimensions under specific conditions.

Research Flowchart

  flowchart TD
    A[Research Goal: Investigate continuity of Martingale Optimal Transport under perturbations] --> B{Key Methodology: Construct correction of couplings with small martingale bias};
    B --> C{Inputs/Assumptions: W1-convergence of marginals, bounded density of reference coupling, finite 1st moments};
    C --> D{Core Analysis: Preserve marginals, truncated moment corrections for unbounded support};
    D --> E{Outcomes (General Costs): Continuity under W1 for continuous costs with linear growth};
    D --> F{Outcomes (Lipschitz Costs): Stability estimate with exponent (1-1/p)/2 for uniform p-moment bounds (p>1)};
    D --> G{Outcomes (Bounded Support): Stability estimate with exponent 1/2 for uniformly bounded target supports};
    D --> H{Additional Finding: Every limiting martingale coupling can be W1-approximated by approximating couplings};
    E & F & G & H --> I[Conclusion: Conditions for continuity and stability of Martingale Optimal Transport identified, counterexamples discussed];

Cited References (29)

1. Entropic Martingale Optimal Transport with Wasserstein Relaxed Marginal Constraints (2026) — link
2. Exponential convergence of Sinkhorn algorithm for entropy martingale optimal transport (2026) — link
3. Multi-Dimensional Martingales from Mutual Information (2025) — link
4. Weak optimal transport with moment constraints: constraint qualification, dual attainment and entropic regularization (2025) — link
5. Convergence of Sinkhorn’s Algorithm for Entropic Martingale Optimal Transport Problem (2024) — link
6. On the Martingale Schr"odinger Bridge between Two Distributions — → 2401.05209
7. Martingale Schrödinger bridges and optimal semistatic portfolios (2022) — link
8. Limits of semistatic trading strategies (2022) — link

1 of 29 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2570 linked papers.