The Annals of Probability

Canonical supermartingale couplings

Marcel Nutz and Florian Stebegg

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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.

Article information

Ann. Probab., Volume 46, Number 6 (2018), 3351-3398.

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

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Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter 49N05: Linear optimal control problems [See also 93C05]

Coupling optimal transport Spence–Mirrlees condition


Nutz, Marcel; Stebegg, Florian. Canonical supermartingale couplings. Ann. Probab. 46 (2018), no. 6, 3351--3398. doi:10.1214/17-AOP1249.

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