This is a joint work with V. Lemaire
(LPMA-UPMC). We propose and analyze a Multilevel Richardson-Romberg
(MLRR) estimator which combines the higher order bias cancellation of
the Multistep Richardson-Romberg ($MSRR$) method introduced
in~[Pag\`es 07] and the variance control resulting from the
stratification in the Multilevel Monte Carlo (MLMC) method (see~$e.g.$
[Heinrich 01, M. Giles 08]). Thus we show that in standard frameworks
like discretization schemes of diffusion processes, an assigned
quadratic error $\varepsilon$ can be obtained with our (MLRR)
estimator with a global complexity of
$\log(1/\varepsilon)/\varepsilon^2$ instead of
$(\log(1/\varepsilon))^2/\varepsilon^2$ with the standard (MLMC)
method, at least when the weak error $\E Y_h-\EY_0}$ induced by the
biased implemented estimator $Y_h$ can be expanded at any order in
$h$. We analyze and compare these estimators on several numerical
problems: option pricing (vanilla or exotic) using $MC$ simulation and
the less classical Nested Monte Carlo simulation (see~[Gordy \& Juneja
2010]).