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 $\mu {\le }_{cd}\nu$ (equivalently, ${\mathrm{\Pi }}_S(\mu ,\nu )\mathrm{\ne }\mathrm{\varnothing }$), we consider supermartingale couplings $\pi =\mu (dx){\pi }_x(dy)$ and the weak transport functional $$V_S^C(\mu ,\nu ):=\underset{\pi \in {\mathrm{\Pi }}_S(\mu ,\nu )}{\inf }{\int }_{\mathbb{R}}C(x,{\pi }_x)\mu (dx),$$ 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 ${\mu }^k{\le }_{cd}{\nu }^k$, any $\pi \in {\mathrm{\Pi }}_S(\mu ,\nu )$ can be approximated by ${\pi }^k\in {\mathrm{\Pi }}_S({\mu }^k,{\nu }^k)$ such that $A{\mathcal{W}}_r({\pi }^k,\pi )\to 0$. As a consequence, we obtain the continuity of the functional $(\mu ,\nu )\mapsto V_S^C(\mu ,\nu )$, and the monotonicity principle for WSOT.