Stability of supermartingale optimal transport problems

TL;DR

This paper studies the stability and continuity of weak supermartingale optimal transport problems, providing approximation results and establishing key properties like the monotonicity principle.

Contribution

It introduces an approximation result in adapted Wasserstein distance for WSOT problems and proves the continuity of the associated transport functional.

Findings

Approximation of supermartingale couplings under convergence of marginals.

Continuity of the weak supermartingale transport functional.

Establishment of the monotonicity principle for WSOT.

Abstract

We investigate stability properties of weak supermartingale optimal transport (WSOT) problems on $\mathbb{R}$. For probability measures $\mu,\nu\in\mathcal{P}_r$ satisfying $\mu \leq_{cd} \nu$ (equivalently, $\Pi_S(\mu,\nu)\neq\emptyset$), we consider supermartingale couplings $\pi=\mu(d x)\pi_x(d y)$ and the weak transport functional \[ V_S^C(\mu,\nu) := \inf_{\pi\in\Pi_S(\mu,\nu)} \int_\mathbb{R} C(x,\pi_x)\,\mu(d x), \] 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 $(\mu^k,\nu^k)\to(\mu,\nu)$ with $\mu^k\leq_{cd} \nu^k$, any $\pi\in\Pi_S(\mu,\nu)$ can be approximated by $\pi^k\in\Pi_S(\mu^k,\nu^k)$ such that $A\mathcal{W}_r(\pi^k,\pi)\to0$. As a consequence, we obtain the continuity of the functional $(\mu,\nu) \mapsto V_S^C(\mu,\nu)$, and the monotonicity principle for WSOT.