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

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+.. 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.