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Multilevel Richardson-Romberg extrapolation

By Vincent Lemaire and Gilles Pagès

Abstract

International audienceWe 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 $E[Y_h]-E[Y_0]$ 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

Topics: Multilevel Monte Carlo methods, Richardson-Romberg Extrapolation, Multi-Step, Euler scheme, Nested Monte Carlo method, Stratification, Option pricing, Primary 65C05, Secondary 65C30, 62P05., [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR]
Publisher: Bernoulli Society for Mathematical Statistics and Probability
Year: 2017
DOI identifier: 10.3150/16-BEJ822
OAI identifier: oai:HAL:hal-00920660v3
Provided by: Hal-Diderot

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