Multilevel Richardson-Romberg extrapolation

Vincent Lemaire (1) , Gilles Pagès (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 [Pagès 07] and the variance control resulting from the stratification in the Multilevel Monte Carlo (MLMC) method (see [Heinrich 01]). 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/epsilon) epsilon^{-2} instead of (log(1/epsilon))^2 epsilon^{-2} with the standard MLMC method, at least when the weak error E(Y_h) - EY_0 of the biased implemented estimator of Y_h can be expanded at any order in h. We analyze and compare these estimators on two numerical problems: the classical vanilla option pricing by Monte Carlo simulation and the less classical Nested Monte Carlo simulation.

Mots clés: Multilevel Monte Carlo methods; Richardson-Romberg Extrapolation; Multi-Step; Euler scheme; Nested Monte Carlo method; Stratification; Option pricing

Domaines: Probabilités [math.PR]

Vincent Lemaire, Gilles Pagès. Multilevel Richardson-Romberg extrapolation. 2013. ⟨hal-00920660v1⟩