Multilevel Richardson–Romberg extrapolation

Abstract

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) $\varepsilon>0$ can be achieved with our ML2R 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 $\Vert Y_{h}-Y_{0}\Vert_{2}=O(h^{\frac{1}{2}})$. The ML2R estimator is then halfway between a regular MLMC and a virtual unbiased Monte Carlo. When the strong error $\Vert Y_{h}-Y_{0}\Vert_{2}=O(h^{\frac{\beta}{2}})$, $\beta<1$, 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.