Paper: arXiv 2310.08492

Authors: Benjamin Jourdain, Kexin Shao

Abstract

In this note, we complete the analysis of the Martingale Wasserstein Inequality started in arXiv:2011.11599 by checking that this inequality fails in dimension $d\ge 2$ when the integrability parameter $ρ$ belongs to $[1,2)$ while a stronger Maximal Martingale Wasserstein Inequality holds whatever the dimension $d$ when $ρ\ge 2$.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 1.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: This paper presents rigorous theoretical work in probability and optimal transport, extending the Martingale Wasserstein Inequality for higher dimensions. It lacks any empirical backtesting or real-world data validation, focusing purely on formal mathematical proofs.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Analyze Martingale Wasserstein Inequality for d ≥ 2 & ρ ≥ 1"] --> B["Input Parameters:<br>Dimension d ≥ 2, Integrability ρ ∈ [1, ∞)"]
    B --> C["Mathematical Analysis & Counterexample Construction"]
    C --> D{"Evaluate Integrability<br>Parameter ρ"}
    D -- "ρ ∈ [1, 2)" --> E["Outcome 1:<br>Martingale Wasserstein Inequality Fails"]
    D -- "ρ ≥ 2" --> F["Outcome 2:<br>Stronger Maximal Martingale Wasserstein Inequality Holds"]