Variance of the simplest Monte Carlo estimators in the sign-changing case

The variance minimization problem is very important in the theory of Monte Carlo methods. In the case of integral equations with sign-changing kernel this problem is comparatively little studied. In this article the author points out some methods which allow to avoid difficulties emerging in the sign-changing case. The simplest (in the computational sense) estimators for functionals of the solutions to second type integral equations play an important role in theory and applications of the Monte Carlo method. We consider in detail the most simple, the so-called "estimator on absorbtion", and establish some new facts about reduction of its variance in the sign-changing case. In conclusion some observations considering other types of estimators are made which allow to improve some known results in the sigh-changing case.