Topological properties of convex order in Wasserstein metric spaces

Martingale optimal transportation has gained significant attention in mathematical finance due to its applications in pricing and hedging. A key distinguishing factor between martingale optimal transportation and traditional optimal transportation is the concept of a peacock, which refers to a sequence of measures satisfying the convex order property. In the realm of traditional optimal transportation, the Wasserstein geometry, induced by a transportation problem with the p-th power of distance as the cost, provides valuable geometric insights. This motivates us to investigate the differences between Wasserstein geometries with and without the martingale constraint. As a first step, this paper focuses on studying the topological properties of convex order, with the aim of establishing a foundational understanding for further exploration of the geometric properties of martingale Wasserstein geometry.