Existence, duality, and cyclical monotonicity for weak transport costs
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In this article, the authors provide general existence and duality results for weak transport problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity.Abstract:
The optimal weak transport problem has recently been introduced by Gozlan et al. (J Funct Anal 273(11):3327–3405, 2017). We provide general existence and duality results for these problems on arbitrary Polish spaces, as well as a necessary and sufficient optimality criterion in the spirit of cyclical monotonicity. As an application we extend the Brenier–Strassen Theorem of Gozlan and Juillet (On a mixture of brenier and strassen theorems. arXiv:1808.02681, 2018) to general probability measures on $$\mathbb {R}^d$$ under minimal assumptions. A driving idea behind our proofs is to consider the set of transport plans with a new (‘adapted’) topology which seems better suited for the weak transport problem and allows to carry out arguments which are close to the proofs in the classical setup.read more
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Adapted Wasserstein Distances and Stability in Mathematical Finance
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Adapted Wasserstein distances and stability in mathematical finance
TL;DR: In this article, a suitable adapted version of the Wasserstein distance which takes the temporal structure of pricing models into account is proposed, which allows to establish Lipschitz properties of hedging strategies for semimartingale models in discrete and continuous time.
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On a mixture of brenier and strassen theorems
Nathael Gozlan,Nicolas Juillet +1 more
TL;DR: In this paper, the authors give a characterization of optimal transport plans for a variant of the usual quadratic transport cost introduced in [33], where the optimal plans are composition of a deterministic transport given by the gradient of a continuously differentiable convex function followed by a martingale coupling.
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Stability of martingale optimal transport and weak optimal transport.
TL;DR: 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.
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The geometry of optimal transportation
TL;DR: In this paper, the existence and uniqueness of optimal maps are discussed. But the uniqueness of the optimal map is not discussed. And the role of the map in finding the optimal solution is left open.