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

Aurélien Alfonsi (), Benjamin Jourdain () and Arturo Kohatsu-Higa
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Aurélien Alfonsi: MATHRISK - Mathematical Risk handling - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en Automatique - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École nationale des ponts et chaussées, CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École nationale des ponts et chaussées
Benjamin Jourdain: CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École nationale des ponts et chaussées, MATHRISK - Mathematical Risk handling - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en Automatique - UPEM - Université Paris-Est Marne-la-Vallée - ENPC - École nationale des ponts et chaussées
Arturo Kohatsu-Higa: Department of Mathematical Sciences, Ritsumeikan Universtiy - Ritsumeikan University

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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 $O(\sqrt{\log(N)}N^{-1})$.

Date: 2014
Note: View the original document on HAL open archive server: https://enpc.hal.science/hal-00727430v1
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Citations: View citations in EconPapers (5)

Published in The Annals of Applied Probability, 2014, http://dx.doi.org/10.1214/13-AAP941

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