Existence, duality, and cyclical monotonicity for weak transport costs

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

doi:10.1007/s00526-019-1624-y

Cited by 55 publications

(73 citation statements)

References 45 publications

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“…4 and 6, but which also might be of independent interest. Specifically, in [9,10] the two-step version of Lemma 1.7 is used as a crucial tool in the investigation of the weak transport problem. A more detailed investigation of compactness in P( ) with the weak adapted topology is the topic of the companion paper to this one, [24].…”

Section: Preservation Of Compactnessmentioning

confidence: 99%

All adapted topologies are equal

Backhoff-Veraguas

Bartl

Beiglböck

et al. 2020

Probab. Theory Relat. Fields

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A number of researchers have introduced topological structures on the set of laws of stochastic processes. A unifying goal of these authors is to strengthen the usual weak topology in order to adequately capture the temporal structure of stochastic processes. Aldous defines an extended weak topology based on the weak convergence of prediction processes. In the economic literature, Hellwig introduced the information topology to study the stability of equilibrium problems. Bion–Nadal and Talay introduce a version of the Wasserstein distance between the laws of diffusion processes. Pflug and Pichler consider the nested distance (and the weak nested topology) to obtain continuity of stochastic multistage programming problems. These distances can be seen as a symmetrization of Lassalle’s causal transport problem, but there are also further natural ways to derive a topology from causal transport. Our main result is that all of these seemingly independent approaches define the same topology in finite discrete time. Moreover we show that this ‘weak adapted topology’ is characterized as the coarsest topology that guarantees continuity of optimal stopping problems for continuous bounded reward functions.

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Section: Preservation Of Compactnessmentioning

confidence: 99%

All adapted topologies are equal

Backhoff-Veraguas

Bartl

Beiglböck

et al. 2020

Probab. Theory Relat. Fields

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“…Hence, the optimal transport map

x \mapsto \nabla φ (x)

is pointwise well defined and there is no ambiguity on the range of small sets. We stress that the optimal transport map from

μ

\overset{μ}{true}

is continuous and even 1‐Lipschitz continuous, which is not automatic for quadratic optimal transport between arbitrary measures of

{scriptP}_{2} false(R^{d} false)

. In a previous version of this paper, Theorem appeared with the additional assumption that

μ

and

ν

are compactly supported. Soon after this first version has been released, a paper by Backhoff‐Veraguas, Beiglböck and Pammer proposed an improved version of Items (b) and (c) of our main result removing this compactness assumption . Their approach is based on a clever combination of generalized cyclical monotonicity arguments and on the first compact version of our Theorem .…”

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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“…≥ θ m x − yπ x (dy) µ(dx), by [8,Proposition 2.8]. Thus the claim follows by taking the supremum in m.…”

Section: Sufficiency Of the Geometric Characterizationmentioning

confidence: 83%

“…Proof of Theorem 1.5. Lower-semicontinuity of the map (µ, ν) → V θ (µ, ν) follows from [8,Theorem 1.3]. By [8, Lemma 6.1] we have…”

Section: Stability Of Barycentric Weak Transport Problems In Multiplementioning

confidence: 97%

Weak monotone rearrangement on the line

Backhoff-Veraguas¹,

Beiglböck²,

Pammer³

2020

Electron. Commun. Probab.

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Weak optimal transport has been recently introduced by Gozlan et al. The original motivation stems from the theory of geometric inequalities; further applications concern numerics of martingale optimal transport and stability in mathematical finance.In this note we provide a complete geometric characterization of the weak version of the classical monotone rearrangement between measures on the real line, complementing earlier results of Alfonsi, Corbetta, and Jourdain. arXiv:1902.05763v1 [math.PR]

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Existence, duality, and cyclical monotonicity for weak transport costs

Cited by 55 publications

References 45 publications

All adapted topologies are equal

All adapted topologies are equal

On a mixture of Brenier and Strassen Theorems

Weak monotone rearrangement on the line

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