An antithetic approach of multilevel Richardson-Romberg extrapolation estimator for multidimensional SDES

Mbaye, Cheikh [UCL] Pagès, Gilles [Université Pierre et Marie Curie, Paris, France] Vrins, Frédéric [UCL]

The Multilevel Richardson-Romberg (ML2R) estimator was introduced by Pagès & Lemaire in [1] in order to remove the bias of the standard Multilevel Monte Carlo (MLMC) estimator in the 1D Euler scheme. Milstein scheme is however preferable to Euler scheme as it allows to reach the optimal complexity O(ε^{−2}) for each of these estimators. Unfortunately, Milstein scheme requires the simulation of Lévy areas when the SDE is driven by a multidimensional Brownian motion, and no efficient method is currently available to this purpose so far (except in dimension 2). Giles and Szpruch [2] recently introduced an antithetic multilevel correction estimator avoiding the simulation of these areas without affecting the second order complexity. In this work, we revisit the ML2R and MLMC estimators in the framework of the antithetic approach, thereby allowing us to remove the bias whilst preserving the optimal complexity when using Milstein scheme.