The Annals of Applied Probability

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

A. Alfonsi, B. Jourdain, and A. Kohatsu-Higa

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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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Ann. Appl. Probab., Volume 24, Number 3 (2014), 1049-1080.

First available in Project Euclid: 23 April 2014

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Zentralblatt MATH identifier

Primary: 65C30: Stochastic differential and integral equations 60H35: Computational methods for stochastic equations [See also 65C30]

Euler scheme Wasserstein distance weak trajectorial error diffusion bridges


Alfonsi, A.; Jourdain, B.; Kohatsu-Higa, A. Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme. Ann. Appl. Probab. 24 (2014), no. 3, 1049--1080. doi:10.1214/13-AAP941.

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