Before revision:  Much like the Hoeffding-Fr\'echet coupling of classical transport and its symmetric counterpart, the Antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions, and through no-crossing conditions on their supports . However , our two couplings have asymmetric geometries due to the directed nature of the supermartingale constraint .
After revision:  Much like the Hoeffding-Fr\'echet coupling of classical transport and its symmetric counterpart, the Antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions, and through no-crossing conditions on their supports ; however , our two couplings have asymmetric geometries due to the directed nature of the supermartingale constraint .