Limit theorems for weighted and regular Multilevel estimators

Abstract

We aim at analyzing in terms of a.s. convergence and weak rate the performances of the Multilevel Monte Carlo estimator (MLMC) introduced in [Gil08] and of its weighted version, the Multilevel Richardson Romberg estimator (ML2R), introduced in [LP14]. These two estimators permit to compute a very accurate approximation of $I_0=\mathbb{E}[Y_0]$ by a Monte Carlo type estimator when the (non-degenerate) random variable $Y_0\in L^2(\mathbb{P})$ cannot be simulated (exactly) at a reasonable computational cost whereas a family of simulatable approximations $(Y_h{)}_{h\in \mathcal{H}}$ is available. We will carry out these investigations in an abstract framework before applying our results, mainly a Strong Law of Large Numbers and a Central Limit Theorem, to some typical fields of applications: discretization schemes of diffusions and nested Monte Carlo.