Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average ApproximationReport as inadecuate

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1 LAGA - Laboratoire Analyse, Géométrie et Applications 2 DAO - Données, Apprentissage et Optimisation LJK - Laboratoire Jean Kuntzmann 3 MATHRISK - Mathematical Risk Handling UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech, Inria de Paris

Abstract : In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo MLMC. We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to compute the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error the bias and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure among the class of changes we consider, which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.

Keywords : Importance Sampling Multilevel Monte Carlo variance reduction Central limit theorem Uniform strong large law of numbers

Author: Ahmed Kebaier - Jérôme Lelong -



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