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 -kernel Population Monte Carlo methodology.
Mots-clés : adaptivity, Cox-Ingersoll-Ross model, Euler scheme, importance sampling, mathematical finance, mixtures, moderate deviations, population Monte Carlo, variance reduction
@article{PS_2007__11__427_0,
author = {Douc, R. and Guillin, A. and Marin, J.-M. and Robert, C. P.},
title = {Minimum variance importance sampling via population {Monte} {Carlo}},
journal = {ESAIM: Probability and Statistics},
pages = {427--447},
publisher = {EDP-Sciences},
volume = {11},
year = {2007},
doi = {10.1051/ps:2007028},
mrnumber = {2339302},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2007028/}
}
TY - JOUR AU - Douc, R. AU - Guillin, A. AU - Marin, J.-M. AU - Robert, C. P. TI - Minimum variance importance sampling via population Monte Carlo JO - ESAIM: Probability and Statistics PY - 2007 SP - 427 EP - 447 VL - 11 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2007028/ DO - 10.1051/ps:2007028 LA - en ID - PS_2007__11__427_0 ER -
%0 Journal Article %A Douc, R. %A Guillin, A. %A Marin, J.-M. %A Robert, C. P. %T Minimum variance importance sampling via population Monte Carlo %J ESAIM: Probability and Statistics %D 2007 %P 427-447 %V 11 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2007028/ %R 10.1051/ps:2007028 %G en %F PS_2007__11__427_0
Douc, R.; Guillin, A.; Marin, J.-M.; Robert, C. P. Minimum variance importance sampling via population Monte Carlo. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 427-447. doi : 10.1051/ps:2007028. http://www.numdam.org/articles/10.1051/ps:2007028/
[1] , Robbins-Monro algorithms and variance reduction in Finance. J. Computational Finance 7 (2003) 1245-1255.
[2] , Adaptative Monte Carlo method, A variance reduction technique. Monte Carlo Methods Appl. 10 (2004) 1-24. | Zbl
[3] and, 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. | Zbl
[4] and, 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.
[5] , and, Symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 41 (2004) 877-889. | Zbl
[6] , Large Deviation Techniques in Decision, Simulation and Estimation. John Wiley, New York (1990). | MR
[7] ,, and, Population Monte Carlo. J. Comput. Graph. Statist. 13 (2004) 907-929.
[8] , and, Inference in Hidden Markov Models. Springer-Verlag, New York (2005). | MR | Zbl
[9] ,, and, A theory of the term structure of interest rates. Econometrica 53 (1985) 385-408.
[10] ,, and, Sequential Monte Carlo samplers. J. Royal Statist. Soc. Series B 68 (2006) 411-436. | MR | Zbl
[11] and, Large Deviations Techniques and Applications. Jones and Barlett Publishers, Inc., Boston (1993). | MR | Zbl
[12] ,, and of adaptive mixtures of importance sampling schemes. Ann. Statist. 35 (2007). | MR | Zbl
[13] , Monte Carlo Methods in Financial Engineering. Springer-Verlag (2003) | MR | Zbl
[14] , Population-based Monte Carlo algorithms. Trans. Japanese Soc. Artificial Intell. 16 (2000) 279-286.
[15] in Finance. John Wiley and Sons (2002).
[16] , and pour les équations de transport et de diffusion. Mathématiques et Applications, Vol. 29. Springer Verlag (1998). | MR | Zbl
[17] C .Robert and G. Casella, Monte Carlo Statistical Methods. Springer-Verlag, New York, second edition (2004). | MR | Zbl
[18] , 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.
[19] , 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). | MR | Zbl
[20] , Simulation and the Monte Carlo Method. J. Wiley, New York (1981). | MR | Zbl
[21] and, The Cross-Entropy Method. Springer-Verlag, New York (2004). | MR | Zbl
[22] and, Optimal importance sampling in securities pricing. J. Computational Finance 5 (2002) 27-50.
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