Electronic Journal of Probability

Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme

Aurélien Alfonsi, Benjamin Jourdain, and Arturo Kohatsu-Higa

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Abstract

In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order $2$ in the spatial variables and Hölder continuous with exponent $\gamma$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C(1+\mathbf{1}_{\gamma=1} \sqrt{\ln(N)})N^{-\gamma}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio et al. to compute the time derivative of the Wasserstein distance between the time-marginals. We deduce a stability inequality for the Wasserstein distance which finally leads to the desired estimation.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 70, 31 pp.

Dates
Accepted: 23 June 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067176

Digital Object Identifier
doi:10.1214/EJP.v20-4195

Mathematical Reviews number (MathSciNet)
MR3361258

Zentralblatt MATH identifier
06471548

Subjects
Primary: 65C30: Stochastic differential and integral equations
Secondary: 60H35: Computational methods for stochastic equations [See also 65C30] 49K99: None of the above, but in this section

Keywords
Euler scheme Wasserstein distance Optimal transport

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Alfonsi, Aurélien; Jourdain, Benjamin; Kohatsu-Higa, Arturo. Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme. Electron. J. Probab. 20 (2015), paper no. 70, 31 pp. doi:10.1214/EJP.v20-4195. https://projecteuclid.org/euclid.ejp/1465067176


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