Total variation convergence of the Euler-Maruyama scheme in small time with unbounded drift - Archive ouverte HAL Access content directly
Journal Articles Electronic Journal of Probability Year : 2022

## Total variation convergence of the Euler-Maruyama scheme in small time with unbounded drift

, (1) ,
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Pierre Bras
• Function : Author
Gilles Pagès
• Function : Author
• PersonId : 856726
• IdHAL : gilpag
Fabien Panloup
• Function : Author

#### Abstract

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.

#### Domains

Mathematics [math] Probability [math.PR]

### Dates and versions

hal-03890514 , version 1 (08-12-2022)

### Identifiers

• HAL Id : hal-03890514 , version 1
• ARXIV :

### Cite

Pierre Bras, Gilles Pagès, Fabien Panloup. Total variation convergence of the Euler-Maruyama scheme in small time with unbounded drift. Electronic Journal of Probability, 2022, 27, pp.1-19. ⟨hal-03890514⟩

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