## Apr 2022 - Mar 2023

### Math-Fi seminar on 7 Jul.

2022.07.06 Wed up
• Date: 7 Jul. (Thu.)
• Place: W.W. 6th-floor, Colloquium Room and on the Web (Zoom)
• Time: 16:30-18:00
• Speaker: Pierre Bras (Sorbonne Université)
• Title: Asymptotics for the total variation distance between an SDE and its Euler-Maruyama scheme in small time
• Abstract:
We give bounds for the total variation distance between the law of an SDE and the law of its one-step Euler-Maruyama scheme as $t \to 0$. The case of the total variation is more complex to deal with than the classic case of Wasserstein ($L^p$) distances. We show that this distance is of order $t^{1/3}$, and more generally of order $t^{r/(2r+1)}$ for any $r \in \mathbb{N}$. Improving the bounds from $1/3$ to $r/(2r+1)$ relies on a weighted multi-level Richardson-Romberg extrapolation which consists in linear combination annealing the terms of a Taylor expansion, up to some order. This method was introduced for bias reduction in practical problems, but is used here for theoretical purposes.