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Minimum variance importance sampling via Population Monte Carlo

Published online by Cambridge University Press:  17 August 2007

R. Douc ,
A. Guillin ,
J.-M. Marin  and
C. P. Robert
R. Douc
Affiliation:
CMAP, École Polytechnique, Palaiseau, France; douc@cmapx.polytechnique.fr
A. Guillin
Affiliation:
École Centrale Marseille and LATP, France; guillin@cmi.univ-mrs.fr
J.-M. Marin
Affiliation:
Projet , INRIA Futurs, Université Paris-Sud, France; jean-michel.marin@inria.fr
C. P. Robert
Affiliation:
CEREMADE, Université Paris Dauphine and CREST, INSEE, Paris, France; xian@ceremade.dauphine.fr
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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.


Keywords

AdaptivityCox-Ingersoll-Ross modelEuler schemeimportance samplingmathematical financemixturesmoderate deviationspopulation Monte Carlovariance reduction.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 11: Special Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthdaySpecial Issue: "Stochastic analysis and mathematical finance" in honor of Nicole El Karoui's 60th birthday , June 2007 , pp. 427 - 447
Copyright
© EDP Sciences, SMAI, 2007

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References

Arouna, B., Robbins-Monro algorithms and variance reduction in Finance. J. Computational Finance 7 (2003) 1245–1255.
Arouna, B., Adaptative Monte Carlo method, A variance reduction technique. Monte Carlo Methods Appl. 10 (2004) 1–24. CrossRef
Bally, V. and Talay, D., The law of the Euler scheme for stochastic differential equations (i): convergence rate of the distribution function. Probability Theory and Related Fields 104 (1996a) 43–60. CrossRef
Bally, V. and Talay, D., The law of the Euler scheme for stochastic differential equations (ii): convergence rate of the density. Probability Theory and Related Fields 104 (1996b) 98–128. CrossRef
Bossy, M., Gobet, E. and Talay, D., Symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 41 (2004) 877–889.
J. Bucklew, Large Deviation Techniques in Decision, Simulation and Estimation. John Wiley, New York (1990).
Cappé, O., Guillin, A., Marin, J.-M. and Robert, C., Population Monte Carlo. J. Comput. Graph. Statist. 13 (2004) 907–929. CrossRef
O. Cappé, E. Moulines and T. Rydèn, Inference in Hidden Markov Models. Springer-Verlag, New York (2005).
Cox, J., Ingersoll, J., and Ross, A., A theory of the term structure of interest rates. Econometrica 53 (1985) 385–408. CrossRef
Del Moral, P., Doucet, A., and Jasra, A., Sequential Monte Carlo samplers. J. Royal Statist. Soc. Series B 68 (2006) 411–436. CrossRef
A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Jones and Barlett Publishers, Inc., Boston (1993).
R. Douc, A. Guillin, J.-M. Marin and C. Robert Convergence of adaptive mixtures of importance sampling schemes. Ann. Statist. 35 (2007).
P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer-Verlag (2003)
Iba, Y., Population-based Monte Carlo algorithms. Trans. Japanese Soc. Artificial Intell. 16 (2000) 279–286. CrossRef
P. Jackel Monte Carlo Methods in Finance. John Wiley and Sons (2002).
B. Lapeyre, E. Pardoux and R. Sentis Méthodes de Monte Carlo pour les équations de transport et de diffusion. Mathématiques et Applications, Vol. 29. Springer Verlag (1998).
C .Robert and G. Casella, Monte Carlo Statistical Methods. Springer-Verlag, New York, second edition (2004).
D. Rubin, A noniterative sampling importance resampling alternative to the data augmentation algorithm for creating a few imputations when fractions of missing information are modest: the SIR algorithm. (In the discussion of Tanner and Wong paper) J. American Statist. Assoc. 82 (1987) 543–546.
D. Rubin, Using the SIR algorithm to simulate posterior distributions. In Bernardo, J., Degroot, M., Lindley, D., and Smith, A. Eds., Bayesian Statistics 3: Proceedings of the Third Valencia International Meeting, June 1–5, 1987. Clarendon Press (1988).
R. Rubinstein, Simulation and the Monte Carlo Method. J. Wiley, New York (1981).
R. Rubinstein and D. Kroese, The Cross-Entropy Method. Springer-Verlag, New York (2004).
Su, Y. and Optimal, M. Fu importance sampling in securities pricing. J. Computational Finance 5 (2002) 27–50. CrossRef