Multiperiod Martingale Transport

Nutz, Marcel; Stebegg, Florian; Tan, Xiaowei

doi:10.48550/arxiv.1703.10588

Cited by 5 publications

(6 citation statements)

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“…It thus seems natural that an improved understanding of an n-marginal martingale transport problem can be obtained based on the multi-marginal Skorokhod embedding problem. Indeed this is exemplified in Theorem 2.16 below, where we use a multi-marginal embedding to establish an n-period version of the martingale monotone transport plan, and recover similar results to recent work of Nutz, Stebegg, and Tan [46].…”

Section: • Martingale Optimal Transportsupporting

confidence: 68%

“…Remark 2.17. In the final stage of writing this article we learned of the work of Nutz, Stebegg, and Tan [46] on multi-period martingale optimal transport which (among various further results) provides an n-marginal version of the monotone martingale transport plan. Their methods are rather different from the ones employed in this article and in particular not related to the Skorokhod problem.…”

Section: 26mentioning

confidence: 99%

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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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Section: • Martingale Optimal Transportsupporting

confidence: 68%

Section: 26mentioning

confidence: 99%

The geometry of multi-marginal Skorokhod Embedding

Beiglböck

Cox

Huesmann

2019

Probab. Theory Relat. Fields

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“…For background on Monge-Kantorovich transport, we refer to [2,55,56,60,61]. Recently, a rich literature has emerged around martingale transport and model uncertainty; see [41,53,59] for surveys and, e.g., [1,10,15,16,17,18,23,27,30,31,32,50,52,62] for models in discrete time, [4,14,20,21,24,25,26,36,37,40,49,51,57,58] for continuous time, and [3,7,8,19,22,33,34,35,39,42,46,54] for related Skorokhod embedding and mimicking problems.…”

Section: Decreasing Supermartingale Transportmentioning

confidence: 99%

Canonical supermartingale couplings

Nutz¹,

Stebegg²

2018

Ann. Probab.

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Two probability distributions µ and ν in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are allowed as transports. Much like the Hoeffding-Fréchet coupling of classical transport and its symmetric counterpart, the antitone coupling, these can be characterized by order-theoretic minimality properties, as simultaneous optimal transports for certain classes of reward (or cost) functions, and through no-crossing conditions on their supports; however, our two couplings have asymmetric geometries. Remarkably, supermartingale optimal transport decomposes into classical and martingale transport in several ways.

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“…The one-dimensional discrete-time martingale optimal transport problem is by now well understood due to the seminal work of [BJ16] for the geometric characterization of optimizers and [BNT17] for a complete duality theory, see also [BLO17]. This was extended to cover the discrete-time multi-marginal problem in [NST17]. In higher dimensions, a complete picture for the discrete-time problem is still missing.…”

Section: Introductionmentioning

confidence: 99%

A Benamou–Brenier formulation of martingale optimal transport

Huesmann¹,

Trevisan²

2019

Bernoulli

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We introduce a Benamou-Brenier formulation for the continuous-time martingale optimal transport problem as a weak length relaxation of its discrete-time counterpart. By the correspondence between classical martingale problems and Fokker-Planck equations, we obtain an equivalent PDE formulation for which basic properties such as existence, duality and geodesic equations can be analytically studied, yielding corresponding results for the stochastic formulation. In the one dimensional case, sufficient conditions for finiteness of the cost are also given and a link between geodesics and porous medium equations is partially investigated.

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Multiperiod Martingale Transport

Cited by 5 publications

References 0 publications

The geometry of multi-marginal Skorokhod Embedding

The geometry of multi-marginal Skorokhod Embedding

Canonical supermartingale couplings

A Benamou–Brenier formulation of martingale optimal transport

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