Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme

In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order 2 in the spatial variables and Holder continuous with exponent gamma with respect to the time variable and its Euler scheme with N uniform time-steps is smaller than C (1 vertical bar 1(gamma) = 1 root ln(N)) N-gamma. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio et al. [2] to compute the time derivative of the Wasserstein distance between the time-marginals. We deduce a stability inequality for the Wasserstein distance which finally leads to the desired estimation.