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Monte Carlo Methods and Applications

Managing Editor: Sabelfeld, Karl K.

Editorial Board: Binder, Kurt / Bouleau, Nicolas / Chorin, Alexandre J. / Dimov, Ivan / Dubus, Alain / Egorov, Alexander D. / Ermakov, Sergei M. / Halton, John H. / Heinrich, Stefan / Kalos, Malvin H. / Lepingle, D. / Makarov, Roman / Mascagni, Michael / Mathe, Peter / Niederreiter, Harald / Platen, Eckhard / Sawford, Brian R. / Schmid, Wolfgang Ch. / Schoenmakers, John / Simonov, Nikolai A. / Sobol, Ilya M. / Spanier, Jerry / Talay, Denis

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Volume 21, Issue 1


Functional quantization-based stratified sampling methods

Sylvain Corlay / Gilles Pagès
  • Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Université Paris 6, case 188, 4, pl. Jussieu, F-75252 Paris Cedex 5, France
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Published Online: 2015-02-06 | DOI: https://doi.org/10.1515/mcma-2014-0010


In this article, we propose several quantization-based stratified sampling methods to reduce the variance of a Monte Carlo simulation. Theoretical aspects of stratification lead to a strong link between optimal quadratic quantization and the variance reduction that can be achieved with stratified sampling. We first put the emphasis on the consistency of quantization for partitioning the state space in stratified sampling methods in both finite and infinite-dimensional cases. We show that the proposed quantization-based strata design has uniform efficiency among the class of Lipschitz continuous functionals. Then a stratified sampling algorithm based on product functional quantization is proposed for path-dependent functionals of multi-factor diffusions. The method is also available for other Gaussian processes such as Brownian bridge or Ornstein–Uhlenbeck processes. We derive in detail the case of Ornstein–Uhlenbeck processes. We also study the balance between the algorithmic complexity of the simulation and the variance reduction factor.

Keywords: Functional quantization; vector quantization; stratification; variance reduction; Monte Carlo simulation; Karhunen–Loève; Gaussian process; Brownian motion; Brownian bridge; Ornstein–Uhlenbeck process; Ornstein–Uhlenbeck bridge; principal component analysis; numerical integration; option pricing; Voronoi diagram; product quantizer; path-dependent option

MSC: 65C05; 65C30; 60H10; 91G60

About the article

Received: 2014-10-04

Accepted: 2015-01-10

Published Online: 2015-02-06

Published in Print: 2015-03-01

Citation Information: Monte Carlo Methods and Applications, Volume 21, Issue 1, Pages 1–32, ISSN (Online) 1569-3961, ISSN (Print) 0929-9629, DOI: https://doi.org/10.1515/mcma-2014-0010.

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