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Stability of martingale optimal transport and weak optimal transport.
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In this paper, the authors give a positive answer and establish stability of the martingale transport problem, and they also apply to the weak transport problem introduced by Gozlan, Roberto, Samson and Tetali.Abstract:
Under mild regularity assumptions, the transport problem is stable in the following sense: if a sequence of optimal transport plans $\pi_1, \pi_2, \ldots$ converges weakly to a transport plan $\pi$, then $\pi$ is also optimal (between its marginals).
Alfonsi, Corbetta and Jourdain asked whether the same property is true for the martingale transport problem. This question seems particularly pressing since martingale transport is motivated by robust finance where data is naturally noisy. On a technical level, stability in the martingale case appears more intricate than for classical transport since optimal transport plans $\pi$ are not characterized by a `monotonicity'-property of their support.
In this paper we give a positive answer and establish stability of the martingale transport problem. As a particular case, this recovers the stability of the left curtain coupling established by Juillet. An important auxiliary tool is an unconventional topology which takes the temporal structure of martingales into account. Our techniques also apply to the the weak transport problem introduced by Gozlan, Roberto, Samson and Tetali.read more
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References
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Topics in Optimal Transportation
TL;DR: In this paper, the metric side of optimal transportation is considered from a differential point of view on optimal transportation, and the Kantorovich duality of the optimal transportation problem is investigated.
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Concentration of measure and isoperimetric inequalities in product spaces
TL;DR: The concentration of measure phenomenon in product spaces roughly states that, if a set A in a product ΩN of probability spaces has measure at least one half, "most" of the points of Ωn are "close" to A as mentioned in this paper.
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New concentration inequalities in product spaces
TL;DR: In this article, the authors introduce three new ways to measure the distance from a point to a subset of a product space and prove corresponding concentration inequalities, each of which allows to control the fluctuation of a new class of random variables.
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Bounding $\bar{d}$-distance by informational divergence: a method to prove measure concentration
TL;DR: This paper considers probability measures on sequences taken from countable alphabets, and derives bounds on the d-distance by informational divergence, which can be used to prove the concentration of measure phenomenon for some nonproduct distributions.