An Antithetic Approach of Multilevel Richardson-Romberg Extrapolation Estimator for Multidimensional SDES

Abstract

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 $O(\varepsilon ^{-2})$ 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.

Appendix: Main Results for Antithetic Estimators

Table 1. Best-of-Call option: parameters and results of ML2R estimator.

Table 2. Best-of-Call option: parameters and results of MLMC estimator.

Optimal Parameters for the Antithetic Estimators. In all the numerical experiments presented in the paper, we set $\mathbf {h}=T$ in the definition of the parameters of the simulation for both MLMC and ML2R estimators. In the table below, keep in mind that $n_0=0$ by convention (Table 3).

Table 3. Optimal parameters for antithetic multilevel estimators.

Note that in the table above, $\mu ^*$ is defined such that $\sum \limits _{j=1}^Rq_j=1$. For the optimal parameters R and h, their formulas remain the same as in [1].