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 iterativelyoptimized 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 Europeanoption 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.


In deterministic comparisons of breeding schemes it is necessary to take account of variance reduction due to selection. This can take place as multi-stage selection within generations and it takes place across generations when offspring of selected parents are selected. A standard way to deal with this is to set up selection index equations where the parameters are altered as a consequence of selection. It is shown that if the breeding schemes use a univariate or multivariate best linear unbiased prediction (BLUP) animal model for prediction of breeding values this procedure can be simplified. This is done by modelling the distribution of estimated breeding value (EBV) utilizing that changes in EBV of an individual are independent of selection. In the univariate case the variance reduction and the resulting genetic gain can be calculated from the selection intensities and the accuracies in the unselected population. An expression is given for the response in each generation when selection is started in a base population with complete pedigree. This shows that a limiting value is obtained within three to four generations. The asymptotic response for several traits is described in the case where selection is on multitrait BLUP.