On a problem of optimal transport under marginal martingale constraints

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 $T_1,T_2:\mathbb{R}\to \mathbb{R}$.

Published at http://dx.doi.org/10.1214/14-AOP966 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• The Skorokhod embedding problem and its offspring
• A stochastic control approach to no-arbitrage bounds given marginals, with an application to lookback options

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