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November 2018 Canonical supermartingale couplings
Marcel Nutz, Florian Stebegg
Ann. Probab. 46(6): 3351-3398 (November 2018). DOI: 10.1214/17-AOP1249


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–Fréchet 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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Marcel Nutz. Florian Stebegg. "Canonical supermartingale couplings." Ann. Probab. 46 (6) 3351 - 3398, November 2018.


Received: 1 July 2017; Revised: 1 November 2017; Published: November 2018
First available in Project Euclid: 25 September 2018

zbMATH: 06975489
MathSciNet: MR3857858
Digital Object Identifier: 10.1214/17-AOP1249

Primary: 49N05 , 60G42

Keywords: coupling , Optimal transport , Spence–Mirrlees condition

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 6 • November 2018
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