Stability of supermartingale optimal transport problems

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.