Paper: arXiv 2603.27940

Authors: Shuoqing Deng, Gaoyue Guo, Dominykas Norgilas

Abstract

We investigate stability properties of weak supermartingale optimal transport (WSOT) problems on $\mathbb{R}$. For probability measures $μ,ν\in\mathcal{P}r$ satisfying $μ\leq{cd} ν$ (equivalently, $Π_S(μ,ν)\neq\emptyset$), we consider supermartingale couplings $π=μ(d x)π_x(d y)$ and the weak transport functional [ V_S^C(μ,ν) := \inf_{π\inΠ_S(μ,ν)} \int_\mathbb{R} C(x,π_x),μ(d x), ] for some appropriate cost function $C:\mathbb{R}\times\mathcal{P}r\to\mathbb{R}$. Our first main contribution is an approximation result in adapted Wasserstein distance: under $W_r$-convergence of marginals $(μ^k,ν^k)\to(μ,ν)$ with $μ^k\leq{cd} ν^k$, any $π\inΠ_S(μ,ν)$ can be approximated by $π^k\inΠ_S(μ^k,ν^k)$ such that $A\mathcal{W}_r(π^k,π)\to0$. As a consequence, we obtain the continuity of the functional $(μ,ν) \mapsto V_S^C(μ,ν)$, and the monotonicity principle for WSOT.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents a highly rigorous mathematical analysis of weak supermartingale optimal transport (WSOT) and adapted Wasserstein distances. It is purely theoretical with advanced measure-theoretic proofs, lacking empirical implementation or backtesting.

Research Flowchart

  flowchart TD
    A[Research Goal: Analyze Stability of WSOT Problems] --> B[Input: Marginals μ^k, ν^k in convex order & Cost Function C]
    B --> C[Method: Approximate couplings π in Π_S via Adapted Wasserstein Distance AW_r]
    C --> D[Process: Prove convergence AW_r π^k, π to 0 under W_r convergence of marginals]
    D --> E[Outcome 1: Continuity of Optimal Transport Functional V_S^C μ, ν]
    D --> F[Outcome 2: Monotonicity Principle for WSOT]

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