Stability of supermartingale optimal transport problems

2026-03-29

AI Summary

This study investigates the stability of optimal weak martingale transport problems on the real line under perturbations of marginal distributions. For marginals satisfying the convex order, the authors introduce an adapted Wasserstein distance and, by combining convex order theory with measure approximation techniques, establish an approximation theorem for martingale couplings under $W_r$-convergence of the marginals. The main contribution is the proof that any target martingale coupling can be approximated by a sequence of couplings corresponding to perturbed marginals, and that the associated weak transport functional is continuous with respect to the marginal measures and satisfies a monotonicity principle. These results provide a rigorous theoretical foundation for the stability analysis of martingale optimal transport problems.

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.

Problem

• supermartingale optimal transport
• stability
• weak transport
• adapted Wasserstein distance
• continuity

Innovation

• supermartingale optimal transport
• adapted Wasserstein distance
• stability
• convex order
• weak transport functional