Two probability distributions \mu and \nu in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports. 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 .
Two probability distributions \mu and \nu in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports. 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 . Remarkably, supermartingale optimal transport decomposes into classical and martingale transport in several ways .
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Antitone
antitone
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. However
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due to the directed nature of the supermartingale constraint
. Remarkably, supermartingale optimal transport decomposes into classical and martingale transport in several ways