In this paper, we remark that any optimal coupling for the quadratic Wasserstein distance $$ between two probability measures μ and ν with finite second order moments on ℝ$$ is the composition of a martingale coupling with an optimal transport map $$. We check the existence of an optimal coupling in which this map gives the unique optimal coupling between μ and $$. Next, we give a direct proof that $$ is differentiable at μ in the Lions (Cours au Collège de France. 2008) sense iff there is a unique optimal coupling between μ and ν and this coupling is given by a map. It was known combining results by Ambrosio, Gigli and Savaré (Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2005) and Ambrosio and Gangbo (Comm. Pure Appl. Math., 61:18–53, 2008) that, under the latter condition, geometric differentiability holds. Moreover, the two notions of differentiability are equivalent according to the recent paper of Gangbo and Tudorascu (J. Math. Pures Appl. 125:119–174, 2019). Besides, we give a self-contained probabilistic proof that mere Fréchet differentiability of a law invariant function F on L2(Ω, ℙ; ℝ$$) is enough for the Fréchet differential at X to be a measurable function of X.
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DOI : 10.1051/ps/2020013
Mots-clés : Optimal transport, Wasserstein distance, differentiability, couplings of probability measures, convex order
@article{PS_2020__24_1_703_0,
author = {Alfonsi, Aur\'elien and Jourdain, Benjamin},
title = {Squared quadratic {Wasserstein} distance: optimal couplings and {Lions} differentiability},
journal = {ESAIM: Probability and Statistics},
pages = {703--717},
publisher = {EDP-Sciences},
volume = {24},
year = {2020},
doi = {10.1051/ps/2020013},
mrnumber = {4174419},
zbl = {1454.90028},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2020013/}
}
TY - JOUR AU - Alfonsi, Aurélien AU - Jourdain, Benjamin TI - Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability JO - ESAIM: Probability and Statistics PY - 2020 SP - 703 EP - 717 VL - 24 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2020013/ DO - 10.1051/ps/2020013 LA - en ID - PS_2020__24_1_703_0 ER -
%0 Journal Article %A Alfonsi, Aurélien %A Jourdain, Benjamin %T Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability %J ESAIM: Probability and Statistics %D 2020 %P 703-717 %V 24 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2020013/ %R 10.1051/ps/2020013 %G en %F PS_2020__24_1_703_0
Alfonsi, Aurélien; Jourdain, Benjamin. Squared quadratic Wasserstein distance: optimal couplings and Lions differentiability. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 703-717. doi : 10.1051/ps/2020013. http://www.numdam.org/articles/10.1051/ps/2020013/
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This research benefited from the support of the “Chaire Risques Financiers”, Fondation du Risque.
