On adaptive stratification

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 the subsets 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 which are hyperrectangles 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 (i.e. when both the number of samples and the number of strata are large). It turns out that the limiting variance depends on the directions defining the hyperrectangles but not on the precise abscissae of their boundaries along these directions, which gives a mathematical justification to the common choice of equiprobable strata. So, only the directions are adaptively modified by our algorithm. 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. Acknowledgements This work has been written in honour of R. Rubinstein, for his 70th birthday. Most of the ideas used in this paper to reduce the variance of Monte Carlo estimator have been inspired by the pioneering work of R. Rubinstein on coupling simulation and stochastic optimization. These very fruitful ideas were a constant source of inspiration during this work. This work is supported by the french National Research Agency (ANR) under the program ANR-05-BLAN-0299 and by the “Chair Risques Financiers”, Fondation du Risque. P. Etore CMAP, École Polytechnique, Route de Saclay, 91128 Palaiseau Cedex Tel.: +33 (0)1 69 33 45 67 E-mail: [email protected] G. Fort and E. Moulines Institut des Télécoms, Télécom ParisTech, 46 Rue Barrault, 75634 Paris Cedex 13, France Tel.: +33 (0)1 45 81 77 82 E-mail: [email protected] B. Jourdain Université Paris-Est, CERMICS, Projet MathFi ENPC-INRIA-UMLV, 6 et 8 avenue Blaise Pascal, 77455 Marne La Vallée, Cedex 2, France Tel.: +33 (0)1 64 15 35 67 E-mail: [email protected]