An extension of martingale transport and stability in robust finance

While many questions in robust finance can be posed in the martingale optimal transport framework or its weak extension, others like the subreplication price of VIX futures, the robust pricing of American options or the construction of shadow couplings necessitate additional information to be incorporated into the optimization problem beyond that of the underlying asset. In the present paper, we take into account this extra information by introducing an additional parameter to the weak martingale optimal transport problem. We prove the stability of the resulting problem with respect to the risk neutral marginal distributions of the underlying asset, thus extending the results in \cite{BeJoMaPa21b}. A key step is the generalization of the main result in \cite{BJMP22} to include the extra parameter into the setting. This result establishes that any martingale coupling can be approximated by a sequence of martingale couplings with specified marginals, provided that the marginals of this sequence converge to those of the original coupling. Finally, we deduce stability of the three previously mentioned motivating examples.