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) ε > 0 can be achieved with our ML2R estimator with a global complexity of ε−2log(1/ε) instead of ε−2(log(1/ε))2 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(h½). The ML2R estimator is then halfway between a regular MLMC and a virtual unbiased Monte Carlo. When the strong error $\Vert {\mathrm{Y}}_{\mathrm{h}}-{\mathrm{Y}}_{0}{\Vert }_{2}=\mathrm{O}\left({\mathrm{h}}^{\frac{\mathrm{\beta }}{2}}\right),\mathrm{\beta }