Minimum variance importance sampling via population Monte Carlo
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 427-447.

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.

DOI : 10.1051/ps:2007028
Classification : 60F05, 62L12, 65-04, 65C05, 65C40, 65C60
Mots-clés : adaptivity, Cox-Ingersoll-Ross model, Euler scheme, importance sampling, mathematical finance, mixtures, moderate deviations, population Monte Carlo, variance reduction
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     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},
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     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/ps:2007028/}
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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://archive.numdam.org/articles/10.1051/ps:2007028/

[1] B. Arouna, Robbins-Monro algorithms and variance reduction in Finance. J. Computational Finance 7 (2003) 1245-1255.

[2] B. Arouna, Adaptative Monte Carlo method, A variance reduction technique. Monte Carlo Methods Appl. 10 (2004) 1-24. | Zbl

[3] V. Bally and D. Talay, 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] V. Bally and D. Talay, 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] M. Bossy, E. Gobet and D. Talay, Symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 41 (2004) 877-889. | Zbl

[6] J. Bucklew, Large Deviation Techniques in Decision, Simulation and Estimation. John Wiley, New York (1990). | MR

[7] O. Cappé, A. Guillin, J.-M. Marin and C. Robert, Population Monte Carlo. J. Comput. Graph. Statist. 13 (2004) 907-929.

[8] O. Cappé, E. Moulines and T. Rydèn, Inference in Hidden Markov Models. Springer-Verlag, New York (2005). | MR | Zbl

[9] J. Cox, J. Ingersoll, and A. Ross, A theory of the term structure of interest rates. Econometrica 53 (1985) 385-408.

[10] P. Del Moral, A. Doucet, and A. Jasra, Sequential Monte Carlo samplers. J. Royal Statist. Soc. Series B 68 (2006) 411-436. | MR | Zbl

[11] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Jones and Barlett Publishers, Inc., Boston (1993). | MR | Zbl

[12] R. Douc, A. Guillin, J.-M. Marin and C. Robert Convergence of adaptive mixtures of importance sampling schemes. Ann. Statist. 35 (2007). | MR | Zbl

[13] P. Glasserman, Monte Carlo Methods in Financial Engineering. Springer-Verlag (2003) | MR | Zbl

[14] Y. Iba, Population-based Monte Carlo algorithms. Trans. Japanese Soc. Artificial Intell. 16 (2000) 279-286.

[15] P. Jackel Monte Carlo Methods in Finance. John Wiley and Sons (2002).

[16] 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). | MR | Zbl

[17] C .Robert and G. Casella, Monte Carlo Statistical Methods. Springer-Verlag, New York, second edition (2004). | MR | Zbl

[18] 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.

[19] 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). | MR | Zbl

[20] R. Rubinstein, Simulation and the Monte Carlo Method. J. Wiley, New York (1981). | MR | Zbl

[21] R. Rubinstein and D. Kroese, The Cross-Entropy Method. Springer-Verlag, New York (2004). | MR | Zbl

[22] Y. Su and M. Fu, Optimal importance sampling in securities pricing. J. Computational Finance 5 (2002) 27-50.

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