Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme

In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum norm between the laws of a uniformly elliptic one-dimensional diffusion process and its Euler discretization with N steps is smaller than O(N.2/3+.) where . is an arbitrary positive constant. This rate is intermediate between the strong error estimation in O(N.1/2) obtained when coupling the stochastic differential equation and the Euler scheme with the same Brownian motion and the weak error estimation O(N.1) obtained when comparing the expectations of the same function of the diffusion and of the Euler scheme at the terminal time T. We also check that the supremum over t.[0,T] of the Wasserstein distance on the space of probability measures on the real line between the laws of the diffusion at time t and the Euler scheme at time t behaves like O(.log(N)N.1).