Canonical supermartingale couplings

Nutz, Marcel; Stebegg, Florian

doi:10.1214/17-aop1249

Cited by 30 publications

(45 citation statements)

References 67 publications

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“…There, the martingale transport problem is studied for particular families of costs satisfying the cross derivative condition

\partial_{x} \partial_{y}^{2} ρ < 0

giving rise to the left‐curtain coupling (on this coupling, see also ) and for the cost functions

ρ : (x, y) \mapsto \pm | y - x |

, generalizing results by Hobson and his coauthors, Neuberger and Klimmek , respectively. The supermartingale problem in dimension 1 with cross‐derivative condition is studied in . In higher dimension,

ρ : (x, y) \mapsto \pm ∥ y - x ∥

is studied in , more general costs are considered in .…”

Section: Introductionmentioning

confidence: 99%

On a mixture of Brenier and Strassen Theorems

Gozlan

Juillet

2020

Proc. London Math. Soc.

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We give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in Gozlan, Roberto, Samson and Tetali (J. Funct. Anal. 273 (2017) 3327–3405). Optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling. We also establish some connections with Caffarelli's contraction theorem (Caffarelli, Comm. Math. Phys. 214 (2000) 547–563).

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“…There, the martingale transport problem is studied for particular families of costs satisfying the cross derivative condition

\partial_{x} \partial_{y}^{2} ρ < 0

giving rise to the left‐curtain coupling (on this coupling, see also ) and for the cost functions

ρ : (x, y) \mapsto \pm | y - x |

ρ : (x, y) \mapsto \pm ∥ y - x ∥

is studied in , more general costs are considered in .…”

Section: Introductionmentioning

confidence: 99%

On a mixture of Brenier and Strassen Theorems

Gozlan

Juillet

2020

Proc. London Math. Soc.

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“…For general initial and target laws Hobson and Norgilas [15] constructed the upper and lower functions that characterise the generalised (or lifted ) left-curtain martingale coupling using weak approximation of measures. Several other authors further investigate the properties and extensions of the left-curtain coupling, see Henry-Labordère et al [11], Beiglböck et al [3,1], Juillet [16,17], Nutz et al [20,21] and Brückerhoff at al. [6].…”

Section: Introductionmentioning

confidence: 99%

The potential of the shadow measure

Beiglböck

Hobson

Norgilas

2022

Electron. Commun. Probab.

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It is well known that given two probability measures µ and ν on R in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglböck and Juillet (Ann. Probab. 44 (2016) 42-106) introduced the shadow measure which induces a family of martingale couplings, and solves the optimal martingale transport problem for a class of bivariate objective functions. In this article we extend their (existence and uniqueness) results by providing an explicit construction of the shadow measure and, as an application, give a simple proof of its associativity.

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“…The classical result by Strassen [29] states that, for two probability measures µ and ν on R, the set of supermartingale couplings of X ∼ µ and Y ∼ ν is non-empty if and only if µ ≤ cd ν, i.e., µ is smaller than ν with respect to the convex-decreasing order. The natural question is then whether there is any canonical choice to couple µ and ν. Nutz and Stebegg [25] recently introduced the increasing supermartingale coupling, denoted by π I , and proved that it is canonical in several ways.…”

Section: Introductionmentioning

confidence: 99%

A potential-based construction of the increasing supermartingale coupling

Bayraktar¹,

Deng²,

Norgilas³

2021

Preprint

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The increasing supermartingale coupling, introduced by Nutz and Stebegg (Canonical supermartingale couplings, Annals of Probability, 46(6):3351-3398, 2018) is an extreme point of the set of 'supermartingale' couplings between two real probability measures in convex-decreasing order. In the present paper we provide an explicit construction of a triple of functions, on the graph of which the increasing supermartingale coupling concentrates. In particular, we show that the increasing supermartingale coupling can be identified with the left-curtain martingale coupling and the antitone coupling to the left and to the right of a uniquely determined regime-switching point, respectively.Our construction is based on the concept of the shadow measure. We show how to determine the potential of the shadow measure associated to a supermartingale, extending the recent results of Beiglböck et al. (The potential of the shadow measure, arXiv preprint, 2020) obtained in the martingale setting. Contents 1. Introduction 2. Preliminaries 2.1. Measures and Convex order 2.2. Convex hull 3. The shadow measure and π I 3.1. The maximal element 3.2. The shadow measure 3.3. The increasing supermartingale coupling π I 3.4. Lifted supermartingale transport plans 4. The geometric construction of π I 4.1. The regime-switching point u * 4.2. Proof of Theorem 1.1 5. Appendix References

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Canonical supermartingale couplings

Cited by 30 publications

References 67 publications

On a mixture of Brenier and Strassen Theorems

On a mixture of Brenier and Strassen Theorems

The potential of the shadow measure

A potential-based construction of the increasing supermartingale coupling

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