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 [Monte Carlo Methods Appl. 13 (2007) 37-70] and the variance control resulting from Multilevel Monte Carlo (MLMC) paradigm (see [Ann. AppL Probab. 24 (2014) 1585-1620, In Large-Scale Scientific Computing (2001) 58-67 Berlin]). Thus, in standard frameworks like discretization schemes of diffusion processes, the root mean squared error (RMSE) epsilon > 0 can be achieved with our ML2R estimator with a global complexity of epsilon(-2) log(1/epsilon) instead of epsilon(-2)(log(1/epsilon))(2) with the standard MLMC method, at least when the weak error E[Y-h] - E[Y-0] of the biased implemented estimator Yh can be expanded at any order in h and parallel to Y-h - Y-0 parallel to(2) = 0(h(1/2)). The ML2R estimator is then halfway between a regular MLMC and a virtual unbiased Monte Carlo. When the strong error parallel to Y-h - Y-0 parallel to(2) = 0(h(beta/2)), beta < 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.