The Annals of Applied Probability

An application of the KMT construction to the pathwise weak error in the Euler approximation of one-dimensional diffusion process with linear diffusion coefficient

Emmanuelle Clément and Arnaud Gloter

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


It is well known that the strong error approximation in the space of continuous paths equipped with the supremum norm between a diffusion process, with smooth coefficients, and its Euler approximation with step $1/n$ is $O(n^{-1/2})$ and that the weak error estimation between the marginal laws at the terminal time $T$ is $O(n^{-1})$. An analysis of the weak trajectorial error has been developed by Alfonsi, Jourdain and Kohatsu-Higa [Ann. Appl. Probab. 24 (2014) 1049–1080], through the study of the $p$-Wasserstein distance between the two processes. For a one-dimensional diffusion, they obtained an intermediate rate for the pathwise Wasserstein distance of order $n^{-2/3+\varepsilon}$. Using the Komlós, Major and Tusnády construction, we improve this bound assuming that the diffusion coefficient is linear and we obtain a rate of order $\log n/n$.

Article information

Ann. Appl. Probab., Volume 27, Number 4 (2017), 2419-2454.

Received: June 2015
Revised: November 2016
First available in Project Euclid: 30 August 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Diffusion process Euler scheme Wasserstein metric quantile coupling technique


Clément, Emmanuelle; Gloter, Arnaud. An application of the KMT construction to the pathwise weak error in the Euler approximation of one-dimensional diffusion process with linear diffusion coefficient. Ann. Appl. Probab. 27 (2017), no. 4, 2419--2454. doi:10.1214/16-AAP1263.

Export citation


  • [1] Alfonsi, A., Jourdain, B. and Kohatsu-Higa, A. (2014). Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme. Ann. Appl. Probab. 24 1049–1080.
  • [2] Alfonsi, A., Jourdain, B. and Kohatsu-Higa, A. (2015). Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme. Electron. J. Probab. 20 Art. ID 70.
  • [3] Bhattacharya, R. N. and Rao, R. R. (2010). Normal Approximation and Asymptotic Expansions. Classics in Applied Mathematics 64. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Updated reprint of the 1986 edition [MR0855460], corrected edition of the 1976 original [MR0436272].
  • [4] Davie, A. (2014). KMT theory applied to approximations of SDE. In Stochastic Analysis and Applications 2014. Springer Proc. Math. Stat. 100 185–201. Springer, Cham.
  • [5] Davie, A. (2014). Pathwise approximation of stochastic differential equations using coupling. Preprint.
  • [6] Doss, H. (1977). Liens entre équations différentielles stochastiques et ordinaires. Ann. Inst. Henri Poincaré B, Calc. Probab. Stat. 13 99–125.
  • [7] Einmahl, U. (1989). Extensions of results of Komlós, Major, and Tusnády to the multivariate case. J. Multivariate Anal. 28 20–68.
  • [8] Flint, G. and Lyons, T. (2015). Pathwise approximation of sdes by coupling piecewise abelian rough paths. arXiv:1505.01298v1.
  • [9] Kanagawa, S. (1988). On the rate of convergence for Maruyama’s approximate solutions of stochastic differential equations. Yokohama Math. J. 36 79–86.
  • [10] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
  • [11] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent RV’s and the sample DF. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • [12] Komlós, J., Major, P. and Tusnády, G. (1976). An approximation of partial sums of independent RV’s, and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 33–58.
  • [13] Major, P. (1976). The approximation of partial sums of independent RV’s. Z. Wahrsch. Verw. Gebiete 35 213–220.
  • [14] Mason, D. and Zhou, H. (2012). Quantile coupling inequalities and their applications. Probab. Surv. 9 439–479.
  • [15] Rachev, S. T. and Rüschendorf, L. (1998). Mass Transportation Problems, Vol. II: Applications. Springer, New York.
  • [16] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [17] Talay, D. and Tubaro, L. (1991). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 483–509.