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

TL;DR

This paper establishes a new rate of convergence in Wasserstein distance between a one-dimensional diffusion process and its Euler scheme, providing insights into pathwise approximation errors.

Contribution

It proves a novel bound on the Wasserstein distance between the diffusion and Euler scheme laws, bridging strong and weak error estimates.

Findings

Wasserstein distance between diffusion and Euler scheme is O(N^{-2/3+ε})

Supremum Wasserstein distance over time behaves like O(√log(N) N^{-1})

Rate is intermediate between strong and weak error bounds

Abstract

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 $O(N^{-2/3+\varepsilon})$ where $\varepsilon$ 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 $t\in[0,T]$ 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 $O(\sqrt{\log(N)}N^{-1})$.