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January 2016 On a problem of optimal transport under marginal martingale constraints
Mathias Beiglböck, Nicolas Juillet
Ann. Probab. 44(1): 42-106 (January 2016). DOI: 10.1214/14-AOP966

Abstract

The basic problem of optimal transportation consists in minimizing the expected costs $\mathbb{E} [c(X_{1},X_{2})]$ by varying the joint distribution $(X_{1},X_{2})$ where the marginal distributions of the random variables $X_{1}$ and $X_{2}$ are fixed.

Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that $(X_{i})_{i=1,2}$ is a martingale, that is, $\mathbb{E} [X_{2}|X_{1}]=X_{1}$.

We establish a variational principle for this problem which enables us to determine optimal martingale transport plans for specific cost functions. In particular, we identify a martingale coupling that resembles the classic monotone quantile coupling in several respects. In analogy with the celebrated theorem of Brenier, the following behavior can be observed: If the initial distribution is continuous, then this “monotone martingale” is supported by the graphs of two functions $T_{1},T_{2}:\mathbb{R} \to\mathbb{R}$.

Citation

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Mathias Beiglböck. Nicolas Juillet. "On a problem of optimal transport under marginal martingale constraints." Ann. Probab. 44 (1) 42 - 106, January 2016. https://doi.org/10.1214/14-AOP966

Information

Received: 1 November 2012; Revised: 1 August 2014; Published: January 2016
First available in Project Euclid: 2 February 2016

zbMATH: 1348.49045
MathSciNet: MR3456332
Digital Object Identifier: 10.1214/14-AOP966

Subjects:
Primary: 49N05 , 60G42

Keywords: Convex order , Martingales , model-independence , Optimal transport

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 1 • January 2016
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