Weak monotone rearrangement on the line

Backhoff-Veraguas, Julio; Beiglböck, Mathias; Pammer, Gudmund

doi:10.1214/20-ecp292

Cited by 11 publications

(14 citation statements)

References 26 publications

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“…Proof The first statement can be found in Theorem 3.1 of Backhoff‐Veraguas et al. (2020) and the second in Theorem 3.1.2 of Rachev and Rüschendorf (1998).

□

…”

Section: Weak Optimal Transport Problemmentioning

confidence: 85%

“…(2020) and Backhoff‐Veraguas et al. (2020) for details): Briefly, by using

*

to indicate a push‐forward measure, and

\leq_{c}

to denote convex order between measures 4 , the family

{T : R \to R, T is monotone, 1-Lip, T_{*} false(μ false) \leq_{c} ν}

has a unique element

\overset{T}{true}

for which the 1‐Wasserstein distance

{scriptW}_{1} false(μ, {\hat{T}}_{*} (μ) false)

is minimized. (19) is then minimized if we put

Y_{1} = \hat{T} false(X_{1} false)

and couple

Y_{1}

and

Y_{2}

by an arbitrary martingale coupling of

T_{*} false(μ false)

and

ν

.…”

Section: Robust Pricing Problemmentioning

confidence: 99%

See 1 more Smart Citation

Weak transport for non‐convex costs and model‐independence in a fixed‐income market

Acciaio

Beiglböck

Pammer

2021

Mathematical Finance

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We consider a model‐independent pricing problem in a fixed‐income market and show that it leads to a weak optimal transport problem as introduced by Gozlan et al. We use this to characterize the extremal models for the pricing of caplets on the spot rate and to establish a first robust super‐replication result that is applicable to fixed‐income markets. Notably, the weak transport problem exhibits a cost function which is non‐convex and thus not covered by the standard assumptions of the theory. In an independent section, we establish that weak transport problems for general costs can be reduced to equivalent problems that do satisfy the convexity assumption, extending the scope of weak transport theory. This part could be of its own interest independent of our financial application, and is accessible to readers who are not familiar with mathematical finance notation.

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“…Proof The first statement can be found in Theorem 3.1 of Backhoff‐Veraguas et al. (2020) and the second in Theorem 3.1.2 of Rachev and Rüschendorf (1998).

□

…”

Section: Weak Optimal Transport Problemmentioning

confidence: 85%

“…(2020) and Backhoff‐Veraguas et al. (2020) for details): Briefly, by using

*

to indicate a push‐forward measure, and

\leq_{c}

to denote convex order between measures 4 , the family

{T : R \to R, T is monotone, 1-Lip, T_{*} false(μ false) \leq_{c} ν}

has a unique element

\overset{T}{true}

for which the 1‐Wasserstein distance

{scriptW}_{1} false(μ, {\hat{T}}_{*} (μ) false)

is minimized. (19) is then minimized if we put

Y_{1} = \hat{T} false(X_{1} false)

and couple

Y_{1}

and

Y_{2}

by an arbitrary martingale coupling of

T_{*} false(μ false)

and

ν

.…”

Section: Robust Pricing Problemmentioning

confidence: 99%

Weak transport for non‐convex costs and model‐independence in a fixed‐income market

Acciaio

Beiglböck

Pammer

2021

Mathematical Finance

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“…Indeed, particular costs of the form (OWT) were already considered by Marton [40,39] and Talagrand [48,49]. The theory for problem (OWT) has been further developed in [1,34,33,45,44,46,29,32,6,9,7]: Basic results of existence and duality are established in the articles [34,2,6]. The notion of C-monotonicity was developed in [4,32,6] as an analogue of classical c-cyclical montonicity in order to provide a characterization of optimizers to the weak transport problem.…”

Section: Literature Connected To the Weak Transport Problemmentioning

confidence: 99%

“…A weak transport analogue to the case of quadratic costs in classical optimal transport, is the case of barycentric costs. This case has received particular attention, and we refer to [33,45,44,46,29,32,9,7,1].…”

Section: Literature Connected To the Weak Transport Problemmentioning

confidence: 99%

Applications of weak transport theory

Backhoff-Veraguas¹,

Pammer²

2020

Preprint

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Motivated by applications to geometric inequalities, Gozlan, Roberto, Samson, and Tetali [34] introduced a transport problem for 'weak' cost functionals. Basic results of optimal transport theory can be extended to this setup in remarkable generality.In this article we collect several problems from different areas that can be recast in the framework of weak transport theory, namely: the Schrödinger problem, the Brenier-Strassen theorem, optimal mechanism design, linear transfers, semimartingale transport. Our viewpoint yields a unified approach and often allows to strengthen the original results.

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“…The properties of backward and forward mapping and their relation in item (4) are remarkable, however it seems to be unique to the convex order case. In one dimension, these properties are obtained by [1,5] via methods specific to one dimension.…”

mentioning

confidence: 99%

Backward and Forward Wasserstein Projections in Stochastic Order

Kim,

Ruan

2021

Preprint

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We study metric projections onto cones in the Wasserstein space of probability measures, defined by stochastic orders. Dualities for backward and forward projections are established under general conditions. Dual optimal solutions and their characterizations require study on a case-by-case basis. Particular attention is given to convex order and subharmonic order. While backward and forward cones possess distinct geometric properties, strong connections between backward and forward projections can be obtained in the convex order case. Compared with convex order, the study of subharmonic order is subtler. In all cases, Brenier-Strassen type polar factorization theorems are proved, thus providing a full picture of the decomposition of optimal couplings between probability measures given by deterministic contractions (resp. expansions) and stochastic couplings. Our results extend to the forward convex order case the decomposition obtained by Gozlan and Juillet, which builds a connection with Caffarelli's contraction theorem. A further noteworthy addition to the early results is the decomposition in the subharmonic order case where the optimal mappings are characterized by volume distortion properties. To our knowledge, this is the first time in this occasion such results are available in the literature.

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Weak monotone rearrangement on the line

Cited by 11 publications

References 26 publications

Weak transport for non‐convex costs and model‐independence in a fixed‐income market

Weak transport for non‐convex costs and model‐independence in a fixed‐income market

Applications of weak transport theory

Backward and Forward Wasserstein Projections in Stochastic Order

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