  We propose and analyze a Multilevel Richardson-Romberg (MLRR) estimator which
combines the higher order bias cancellation of the Multistep Richardson-Romberg
method introduced in [Pa07] and the variance control resulting from the
stratification introduced in the Multilevel Monte Carlo (MLMC) method (see
[Hei01, Gi08]). Thus, in standard frameworks like discretization schemes of
diffusion processes, the root mean squared error (RMSE) $\varepsilon > 0$ can
be achieved with our MLRR estimator with a global complexity of
$\varepsilon^{-2} \log(1/\varepsilon)$ instead of $\varepsilon^{-2}
(\log(1/\varepsilon))^2$ with the standard MLMC method, at least when the weak
error $\mathbf{E}[Y_h]-\mathbf{E}[Y_0]$ of the biased implemented estimator
$Y_h$ can be expanded at any order in $h$ and $\|Y_h - Y_0\|_2 =
O(h^{\frac{1}{2}})$. The MLRR estimator is then halfway between a regular MLMC
and a virtual unbiased Monte Carlo. When the strong error $\|Y_h - Y_0\|_2 =
O(h^{\frac{\beta}{2}})$, $\beta < 1$, the gain of MLRR over MLMC becomes even
more striking. We carry out numerical simulations to compare these estimators
in two settings: vanilla and path-dependent option pricing by Monte Carlo
simulation and the less classical Nested Monte Carlo simulation.
