Stability of supermartingale optimal transport problems

Abstract

We investigate stability properties of weak supermartingale optimal transport (WSOT) problems on $\mathbb{R}$. For probability measures $\hat{I}¼,\hat{I}½\in {\mathcal{P}}_r$ satisfying $\hat{I}¼{\le }_{cd}\hat{I}½$ (equivalently, ${\hat{I}}_S(\hat{I}¼,\hat{I}½)\ne \varnothing$), we consider supermartingale couplings $\ddot{I}€=\hat{I}¼(dx)\ddot{I}{€}_x(dy)$ and the weak transport functional \[ V_S^C(μ,ν) := \inf_{π\inÎ _S(μ,ν)} \int_\mathbb{R} C(x,π_x)\,μ(d x), \] 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 $(\hat{I}{¼}^k,\hat{I}{½}^k)\to (\hat{I}¼,\hat{I}½)$ with $\hat{I}{¼}^k{\le }_{cd}\hat{I}{½}^k$, any $\ddot{I}€\in {\hat{I}}_S(\hat{I}¼,\hat{I}½)$ can be approximated by $\ddot{I}{€}^k\in {\hat{I}}_S(\hat{I}{¼}^k,\hat{I}{½}^k)$ such that $A{\mathcal{W}}_r(\ddot{I}{€}^k,\ddot{I}€)\to 0$. As a consequence, we obtain the continuity of the functional $(\hat{I}¼,\hat{I}½)\mapsto V_S^C(\hat{I}¼,\hat{I}½)$, and the monotonicity principle for WSOT.

Supermartingale optimal transport, Stability, Monotonicity Principle