Limit theorems for weighted and regular Multilevel estimators

Giorgi Daphné (), Lemaire Vincent () and Pagès Gilles ()
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Giorgi Daphné: Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, UPMC Paris 6 (Sorbonne Université), France
Lemaire Vincent: Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, UPMC Paris 6 (Sorbonne Université), France
Pagès Gilles: Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, UPMC Paris 6 (Sorbonne Université), France

Monte Carlo Methods and Applications, 2017, vol. 23, issue 1, 43-70

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 [7] and of its weighted version, the Multilevel Richardson–Romberg estimator (ML2R), introduced in [12]. These two estimators permit to compute a very accurate approximation of I0=𝔼⁢[Y0]${I_{0}=\mathbb{E}[Y_{0}]}$ by a Monte Carlo-type estimator when the (non-degenerate) random variable Y0∈L2⁢(ℙ)${Y_{0}\in L^{2}(\mathbb{P})}$ cannot be simulated (exactly) at a reasonable computational cost whereas a family of simulatable approximations (Yh)h∈ℋ${(Y_{h})_{h\in\operatorname{\mathcal{H}}}}$ 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.

Keywords: Multilevel Monte Carlo methods; law of large numbers; Central Limit Theorem; Richardson–Romberg extrapolation; Euler scheme; nested Monte Carlo (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1515/mcma-2017-0102

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