Supermartingale Brenier's Theorem with full-marginals constraint

Paper: arXiv 2212.14174

Authors: Erhan Bayraktar, Shuoqing Deng, Dominykas Norgilas

Abstract

We explicitly construct the supermartingale version of the Fr{é}chet-Hoeffding coupling in the setting with infinitely many marginal constraints. This extends the results of Henry-Labordere et al. obtained in the martingale setting. Our construction is based on the Markovian iteration of one-period optimal supermartingale couplings. In the limit, as the number of iterations goes to infinity, we obtain a pure jump process that belongs to a family of local L{é}vy models introduced by Carr et al. We show that the constructed processes solve the continuous-time supermartingale optimal transport problem for a particular family of path-dependent cost functions. The explicit computations are provided in the following three cases: the uniform case, the Bachelier model and the Geometric Brownian Motion case.

Complexity vs Empirical Score

• Math Complexity: 9.2/10
• Empirical Rigor: 2.5/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous mathematical extension of optimal transport theory (Brenier’s Theorem) to the supermartingale setting with continuous marginal constraints. While exceptionally strong in theoretical novelty and mathematical derivation, it lacks empirical backtesting or real-world market data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Extend Fréchet-Hoeffding Coupling to Supermartingales with Full Marginals] --> B[Inputs: Marginals for Uniform, Bachelier & GBM Models]
    B --> C[Methodology: Markovian Iteration of 1-Period Optimal Supermartingale Couplings]
    C --> D[Computational Limit: N Iterations Go to Infinity]
    D --> E[Key Outcome 1: Pure Jump Process in Local Lévy Model Class]
    D --> F[Key Outcome 2: Explicit Solutions for Path-Dependent Cost Functions]