Change of numeraire in the two-marginals martingale transport problem

Campi, Luciano; Laachir, Ismail; Martini, Claude

doi:10.1007/s00780-016-0322-2

Cited by 41 publications

(45 citation statements)

References 21 publications

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“…Referred to as martingale optimal transport (MOT), they were introduced by [2,8] to study the mathematical finance question of computing model-independent no-arbitrage price bounds, see [11] for a survey, and have been studied in many papers since, e.g. [13,7,6,14]. They are however of much wider mathematical interest.…”

Section: Introductionmentioning

confidence: 99%

Dual attainment for the martingale transport problem

Beiglböck¹,

Lim²,

Obłój³

2019

Bernoulli

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We investigate existence of dual optimizers in one-dimensional martingale optimal transport problems. While [5] established such existence for weak (quasi-sure) duality, [2] showed existence for the natural stronger (pointwise) duality may fail even in regular cases. We establish that (pointwise) dual maximizers exist when y → c(x, y) is convex, or equivalent to a convex function. It follows that when marginals are compactly supported, the existence holds when the cost c(x, y) is twice continuously differentiable in y. Further, this may not be improved as we give examples with c(x, ·) ∈ C 2−ε , ε > 0, where dual attainment fails. Finally, when measures are compactly supported, we show that dual optimizers are Lipschitz if c is Lipschitz.

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Section: Introductionmentioning

confidence: 99%

Dual attainment for the martingale transport problem

Beiglböck¹,

Lim²,

Obłój³

2019

Bernoulli

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“…[7,14,25]). Early papers to investigate such problems include [18,4,12,11,9], and this topic is commonly referred to as martingale optimal transport. In mathematical finance, transport techniques complement the Skorokhod embedding approach to model-independent/robust finance (we refer to [24,16] for an overview, see [2] for a link with optimal transport theory).…”

Section: Introductionmentioning

confidence: 99%

Monotone martingale transport plans and Skorokhod embedding

Beiglböck

Henry-Labordère²,

Touzi

2017

Stochastic Processes and their Applications

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“…Optimal transport problems where the transport plan must satisfy additional martingale constraints have recently been investigated, e.g. in [36,4,7,22,20,11].…”

Section: • Martingale Optimal Transportmentioning

confidence: 99%

The geometry of multi-marginal Skorokhod Embedding

Beiglböck

Cox

Huesmann

2019

Probab. Theory Relat. Fields

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The Skorokhod Embedding Problem (SEP) is one of the classical problems in the study of stochastic processes, with applications in many different fields (cf. the surveys [48,34]). Many of these applications have natural multi-marginal extensions leading to the (optimal) multi-marginal Skorokhod problem (MSEP). Some of the first papers to consider this problem are [32,10,44]. However, this turns out to be difficult using existing techniques: only recently a complete solution was be obtained in [14] establishing an extension of the Root construction, while other instances are only partially answered or remain wide open.In this paper, we extend the theory developed in [2] to the multi-marginal setup which is comparable to the extension of the optimal transport problem to the multi-marginal optimal transport problem. As for the one-marginal case, this viewpoint turns out to be very powerful. In particular, we are able to show that all classical optimal embeddings have natural multi-marginal counterparts. Notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases.Moreover, our results also have consequences for the study of the martingale transport problem as well as the peacock problem.

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Change of numeraire in the two-marginals martingale transport problem

Cited by 41 publications

References 21 publications

Dual attainment for the martingale transport problem

Dual attainment for the martingale transport problem

Monotone martingale transport plans and Skorokhod embedding

The geometry of multi-marginal Skorokhod Embedding

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