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 [1], 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+ε. 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. MSC 2010. 65C30, 60H35