On a problem of optimal transport under marginal martingale constraints

Abstract

The basic problem of optimal transportation consists in minimizing the expected costs E[c(X1, X2)] by varying the joint distribution (X1, X2) where the marginal distributions of the random variables X1 and X2 are fixed.Inspired by recent applications in mathematical finance and connections with the peacock problem, we study this problem under the additional condition that (Xi)i=1,2 is a martingale, that is,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 T1, T2 : R → R.

Optimal Skorokhod Embedding Under Finitely Many Marginal Constraints

“…Further development enriches this literature, such as Beiglböck & Juillet [4], , Henry-Labordère & Touzi [32], Henry-Labordère, Tan & Touzi [31], etc. A remarkable contribution for the continuous-time martingale optimal transport is due to Dolinsky & Soner [21,22].…”

Conditional convex orders and measurable martingale couplings

“…However, similar results related to martingale couplings have appeared recently in the context of optimal transport. Beiglböck and Juillet [6] consider the problem of finding an optimal transport plan under the constraint that the transport plan is a martingale. The work of Fontbona, Guérin and Méléard [10] has the most similarities with our developments.…”

“…We next turn into so-called martingale optimal transport problem [6] which is linked to applications in mathematical finance, e.g., [5,9,11,15]. Optimal transport problems, in general, mean finding a coupling of two probability measures μ such that the 'cost' μ(c) := c(x, y)μ(dx × dy) is minimised.…”

Change of numeraire in the two-marginals martingale transport problem

“…Both notions were introduced in Beiglböck and Juillet [1], who show their existence and uniqueness for convex ordered marginals, and prove that they solve the maximization and the minimization problem in (2.4) for a specific set of payoffs of the form C(x, y) = h(y − x) with h differentiable with strictly convex first derivative. Henry-Labordère and Touzi [7] extend these results to a wider set of payoffs.…”

“…This complements, using a different method, the results in Hobson and Klimmek [10] on forward start straddles of type II. On the other hand, regarding the Beiglböck and Juillet [1] and Henry-Labordère and Touzi [7] left-and right-monotone optimal transport plans, the change of numeraire can be viewed as a mirror coupling for positive martingales. More precisely, we show that the rightmonotone transport plan can be obtained with no effort from its left-monotone counterpart by suitably changing numeraire.…”