Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation

Ahmed Kebaier () and Jérôme Lelong ()
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Ahmed Kebaier: LAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - UP13 - Université Paris 13 - Institut Galilée - CNRS - Centre National de la Recherche Scientifique
Jérôme Lelong: DAO - Données, Apprentissage et Optimisation - LJK - Laboratoire Jean Kuntzmann - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - Inria - Institut National de Recherche en Informatique et en Automatique - CNRS - Centre National de la Recherche Scientifique - UGA [2016-2019] - Université Grenoble Alpes [2016-2019]

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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: Uniform strong large law of numbers; Central limit theorem; variance reduction; Importance Sampling; Multilevel Monte Carlo (search for similar items in EconPapers)
Date: 2018-06
Note: View the original document on HAL open archive server: https://hal.science/hal-01214840v4
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Citations: View citations in EconPapers (6)

Published in Methodology and Computing in Applied Probability, 2018, 20 (2), pp.611-641. ⟨10.1007/s11009-017-9579-y⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01214840

DOI: 10.1007/s11009-017-9579-y

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