Total variation convergence of the Euler-Maruyama scheme in small time with unbounded drift
Résumé
We give bounds for the total variation distance between the solution of a stochastic differential equation in $\mathbb{R}^d$ and its one-step Euler-Maruyama scheme in small time. We show that for small $t$, the total variation distance is of order $t^{1/3}$, and more generally of order $t^{r/(2r+1)}$ if the noise coefficient $\sigma$ of the SDE is elliptic and $\mathcal{C}^{2r}_b$, $r\in \mathbb{N}$, using multi-step Richardson-Romberg extrapolation. We also extend our results to the case where the drift is not bounded. Then we prove with a counterexample that we cannot achieve better bounds in general.