On adaptive stratification

Pierre Etoré (1, 2) , Gersende Fort (3) , Benjamin Jourdain (4) , Éric Moulines (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 MATHFI - Mathématiques financières
3 LTCI - Laboratoire Traitement et Communication de l'Information
4 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique

This paper investigates the use of stratified sampling as a variance reduction technique for approximating integrals over large dimensional spaces. The accuracy of this method critically depends on the choice of the space partition, the strata, which should be ideally fitted to thesubsets where the functions to integrate is nearly constant, and on the allocation of the number of samples within each strata. When the dimension is large and the function to integrate is complex, finding such partitions and allocating the sample is a highly non-trivial problem. In this work, we investigate a novel method to improve the efficiency of the estimator "on the fly", by jointly sampling and adapting the strata and the allocation within the strata. The accuracy of estimators when this method is used is examined in detail, in the so-called asymptotic regime (\ie\ when both the number of samples and the number of strata are large). We illustrate the use of the method for the computation of the price of path-dependent options in models with both constant and stochastic volatility. The use of this adaptive technique yields variance reduction by factors sometimes larger than 1000 compared to classical Monte Carlo estimators.

Domaines: Probabilités [math.PR]

Pierre Etoré, Gersende Fort, Benjamin Jourdain, Éric Moulines. On adaptive stratification. Annals of Operations Research, 2011, 189 (1), pp.127-154. ⟨10.1007/s10479-009-0638-9⟩. ⟨hal-00319157v3⟩