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

Abstract

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 Hölder continuous with exponent $\gamma$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C(1+\mathbf{1}_{\gamma=1} \sqrt{\ln(N)})N^{-\gamma}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio et al. 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.