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Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme

Abstract : In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Euler discretization with $N$ steps is smaller than $O(N^{-2/3+\varepsilon})$ where $\varepsilon$ is an arbitrary positive constant. This rate is intermediate between the strong error estimation in $O(N^{-1/2})$ obtained when coupling the stochastic differential equation and the Euler scheme with the same Brownian motion and the weak error estimation $O(N^{-1})$ obtained when comparing the expectations of the same function of the diffusion and of the Euler scheme at the terminal time $T$. We also check that the supremum over $t\in[0,T]$ of the Wasserstein distance on the space of probability measures on the real line between the laws of the diffusion at time $t$ and the Euler scheme at time $t$ behaves like $O(\sqrt{\log(N)}N^{-1})$.
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https://hal-enpc.archives-ouvertes.fr/hal-00727430
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Soumis le : lundi 3 septembre 2012 - 16:16:31
Dernière modification le : lundi 24 août 2020 - 14:14:10
Archivage à long terme le : : mardi 4 décembre 2012 - 03:41:55

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  • HAL Id : hal-00727430, version 1
  • ARXIV : 1209.0576

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Aurélien Alfonsi, Benjamin Jourdain, Arturo Kohatsu-Higa. Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme. Annals of Applied Probability, Institute of Mathematical Statistics (IMS), 2014, http://dx.doi.org/10.1214/13-AAP941. ⟨hal-00727430⟩

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