On convergence and optimization of functional Monte Carlo estimators in Sobolev spaces

The paper deals with the study of convergence and optimization of unbiased functional Monte Carlo estimators. We have obtained the estimators, that are optimal in Sobolev's Hilbert spaces, for calculating the integrals dependent on a parameter and for calculating the families of functionals of the solution to the integral equation of the second kind. The results have been obtained in the context of the new concept proposed by the author in order to compare the efficiency of functional estimators in Monte Carlo methods. New conditions of estimators convergence in the space of continuous functions have been proved.