## The Annals of Probability

### Canonical supermartingale couplings

#### Abstract

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aop/1537862436

Digital Object Identifier
doi:10.1214/17-AOP1249

Mathematical Reviews number (MathSciNet)
MR3857858

Zentralblatt MATH identifier
06975489

#### Citation

Nutz, Marcel; Stebegg, Florian. Canonical supermartingale couplings. Ann. Probab. 46 (2018), no. 6, 3351--3398. doi:10.1214/17-AOP1249. https://projecteuclid.org/euclid.aop/1537862436

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