The Annals of Probability

On a problem of optimal transport under marginal martingale constraints

Mathias Beiglböck and Nicolas Juillet

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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 $T_{1},T_{2}:\mathbb{R} \to\mathbb{R}$.

Article information

Ann. Probab., Volume 44, Number 1 (2016), 42-106.

Received: November 2012
Revised: August 2014
First available in Project Euclid: 2 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Optimal transport convex order martingales model-independence


Beiglböck, Mathias; Juillet, Nicolas. On a problem of optimal transport under marginal martingale constraints. Ann. Probab. 44 (2016), no. 1, 42--106. doi:10.1214/14-AOP966.

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