  We show that deliberately introducing a nested simulation stage can lead to
significant variance reductions when comparing two stopping times by Monte
Carlo. We derive the optimal number of nested simulations and prove that the
algorithm is remarkably robust to misspecifications of this number. The method
is applied to several problems related to Bermudan/American options. In these
applications, our method allows to substantially increase the efficiency of
other variance reduction techniques, namely, Quasi-Control Variates and
Multilevel Monte Carlo.
