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

TL;DR

This paper derives bounds on the total variation distance between solutions of two stochastic differential equations with close coefficients, especially comparing an SDE solution to its Euler-Maruyama approximation in small time, even with unbounded drift.

Contribution

It provides new bounds for the total variation distance in small time between SDE solutions with unbounded drift and their Euler-Maruyama schemes, extending previous results.

Findings

Total variation distance is of order t^{r/(2r+1)} for small t under certain regularity conditions.

Boundedness of the drift is not required for the bounds.

A counterexample shows the bound cannot generally be better than t^{1/2}.

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 $\sigma$ of the SDE is elliptic and $\mathcal{C}^{2r}_b$, $r\in \mathbb{N}$ and if the drift is $C^1$ 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.