English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In the present paper, we prove that the Wasserstein distance on the space ofcontinuous sample-paths equipped with the supremum norm between the laws of auniformly elliptic one-dimensional diffusion process and its Eulerdiscretization with $N$ steps is smaller than $O(N^{-2/3+\varepsilon})$ where$\varepsilon$ is an arbitrary positive constant. This rate is intermediatebetween the strong error estimation in $O(N^{-1/2})$ obtained when coupling thestochastic differential equation and the Euler scheme with the same Brownianmotion and the weak error estimation $O(N^{-1})$ obtained when comparing theexpectations of the same function of the diffusion and of the Euler scheme atthe terminal time $T$. We also check that the supremum over $t\in[0,T]$ of theWasserstein distance on the space of probability measures on the real linebetween the laws of the diffusion at time $t$ and the Euler scheme at time $t$behaves like $O(\sqrt{\log(N)}N^{-1})$.

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