Multilevel Richardson-Romberg extrapolation

We propose and analyze a Multilevel Richardson-Romberg (ML2R) estimator which combines the higher order bias cancellation of the Multistep Richardson-Romberg method introduced in [Pag07] and the variance control resulting from the stratification introduced in the Multilevel Monte Carlo (MLMC) method (see [Gil08, Hei01]). Thus, in standard frameworks like discretization schemes of diffusion processes, the root mean squared error (RMSE) ε > 0 can be achieved with our ML2R estimator with a global complexity of ε log(1/ε) instead of ε(log(1/ε)) with the standard MLMC method, at least when the weak error E [Yh]−E [Y0] of the biased implemented estimator Yh can be expanded at any order in h and ∥Yh − Y0 ∥∥ 2 = O(h 1 2 ). The ML2R estimator is then halfway between a regular MLMC and a virtual unbiased Monte Carlo. When the strong error ∥Yh − Y0 ∥∥ 2 = O(h β 2 ), β < 1, the gain of ML2R 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.