ESAIM: P&S, August 2007, Vol. 11, pp. 427-447
DOI: 10.1051/ps:2007028

Minimum variance importance sampling via Population Monte Carlo

R. Douc¹, A. Guillin², J.-M. Marin³ and C.P. Robert<sup>4

1</sup>  CMAP, École Polytechnique, Palaiseau, France; douc@cmapx.polytechnique.fr
²  École Centrale Marseille and LATP, France; guillin@cmi.univ-mrs.fr
³  Projet , INRIA Futurs, Université Paris-Sud, France; jean-michel.marin@inria.fr
⁴  CEREMADE, Université Paris Dauphine and CREST, INSEE, Paris, France; xian@ceremade.dauphine.fr

(Received January 19, 2007. Published online 17 August 2007.)

Abstract
Variance reduction has always been a central issue in Monte Carlo experiments. Population Monte Carlo can be used to this effect, in that a mixture of importance functions, called a D-kernel, can be iteratively optimized to achieve the minimum asymptotic variance for a function of interest among all possible mixtures. The implementation of this iterative scheme is illustrated for the computation of the price of a European option in the Cox-Ingersoll-Ross model. A Central Limit theorem as well as moderate deviations are established for the D-kernel Population Monte Carlo methodology.

Mathematics Subject Classification. 60F05, 62L12, 65-04, 65C05, 65C40, 65C60

Key words: Adaptivity, Cox-Ingersoll-Ross model, Euler scheme, importance sampling, mathematical finance, mixtures, moderate deviations, population Monte Carlo, variance reduction.


© EDP Sciences, SMAI 2007