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arXiv · 1611.05275

Limit theorems for weighted and regular Multilevel estimators

Daphné Giorgi·Vincent Lemaire·Gilles Pagès
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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.

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Daphné Giorgi, Vincent Lemaire, Gilles Pagès. 2016-11-16. Limit theorems for weighted and regular Multilevel estimators. https://doi.org/10.1515/mcma-2017-0102

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