Functional quantization-based stratified sampling methods

TitleFunctional quantization-based stratified sampling methods
Publication TypeJournal Article
Year of Publication2010
AuthorsSylvain Corlay, and Gilles Pagès
JournalPreprint
KeywordsBrownian bridge, Brownian motion, functional quantization, Gaussian process, Karhunen-Loève basis, Monte-Carlo simulation, numerical integration, option pricing, principal component analysis, product quantizer, stratification, Variance reduction, vector quantization, Voronoi diagram
Abstract

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.

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