## The Annals of Applied Probability

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

### 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

#### Abstract

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

**Source**

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

**Dates**

Received: June 2015

Revised: November 2016

First available in Project Euclid: 30 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1504080037

**Digital Object Identifier**

doi:10.1214/16-AAP1263

**Mathematical Reviews number (MathSciNet)**

MR3693530

**Zentralblatt MATH identifier**

06803467

**Subjects**

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

**Keywords**

Diffusion process Euler scheme Wasserstein metric quantile coupling technique

#### Citation

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. https://projecteuclid.org/euclid.aoap/1504080037