# Multilevel Richardson-Romberg extrapolation

Abstract : We propose and analyze a Multilevel Richardson-Romberg ($MLRR$) estimator which combines the higher order bias cancellation of the Multistep Richardson-Romberg ($MSRR$) method introduced in [Pages 07] and the variance control resulting from the stratification in the Multilevel Monte Carlo ($MLMC$) method (see [Heinrich, 01] and [Giles, 08]). Thus we show that in standard frameworks like discretization schemes of diffusion processes an assigned quadratic error $\varepsilon$ can be obtained with our ($MLRR$) estimator with a global complexity of $\log(1/\varepsilon)/\varepsilon^2$ instead of $(\log(1/\varepsilon))^2/\varepsilon^2$ with the standard ($MLMC$) method, at least when the weak error $\esp{Y_h}-\esp{Y_0}$ of the biased implemented estimator $Y_h$ can be expanded at any order in $h$. We analyze and compare these estimators on two numerical problems: the classical vanilla and exotic option pricing by Monte Carlo simulation and the less classical Nested Monte Carlo simulation.
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Type de document :
Rapport
[University works] UPMC LPMA. 2014
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https://hal.archives-ouvertes.fr/hal-00920660
Contributeur : Gilles Pagès <>
Soumis le : vendredi 19 décembre 2014 - 15:21:21
Dernière modification le : mardi 11 octobre 2016 - 13:48:13
Document(s) archivé(s) le : lundi 23 mars 2015 - 18:26:01

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MultilevelRR_3.pdf
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• HAL Id : hal-00920660, version 3

### Citation

Vincent Lemaire, Gilles Pagès. Multilevel Richardson-Romberg extrapolation. [University works] UPMC LPMA. 2014. <hal-00920660v3>

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