# Total variation distance between two diffusions in small time with unbounded drift: application to the Euler-Maruyama scheme

Pierre Bras, Gilles Pagès, Fabien Panloup

November 2021

### Abstract

We give bounds for the total variation distance between the solutions to two stochastic differential equations starting at the same point and with close coefficients, which applies in particular to the distance between an exact solution and its Euler-Maruyama scheme in small time. We show that for small t, the total variation distance is of order t r/(2r+1) if the noise coefficient σ of the SDE is elliptic and C 2r b*, r ∈ N and if the drift is C1 with bounded derivatives, using multi-step Richardson-Romberg extrapolation. We do not require the drift to be bounded. Then we prove with a counterexample that we cannot achieve a bound better than t 1/2 in general.

Publication

In *Electron. J. Probab.* 27, 1-19 (2022)

###### PhD Student in Applied Mathematics

I am a PhD student under the direction of Gilles Pagès, interested in Machine Learning, Stochastic Optimization and Numerical Probability.