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

Abstract

We give bounds for the total variation distance between the solution of a stochastic differential equation in Rd and its one-step Euler-Maruyama scheme in small time. We show that for small t, the total variation distance is of order t1/3, and more generally of order tr/(2r+1) if the noise coefficient σ of the SDE is elliptic and C2rb, r∈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.

Publication
In arXiv e-prints

Gilles Pagès’ website Fabien Panloup’s website

Pierre Bras
Pierre Bras
PhD Student in Applied Mathematics

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