Monte Carlo computation of multiple weak singular integrals of spherical and Volterra's type

TL;DR

This paper introduces a Monte Carlo method for efficiently computing multiple weakly singular integrals of spherical and Volterra types, addressing variance issues and enabling effective multidimensional distribution generation.

Contribution

The paper presents a novel Monte Carlo approach that incorporates singularities into the density to eliminate infinite variance in computing weakly singular integrals.

Findings

Variance of the Monte Carlo estimator is significantly reduced.

Method effectively handles multidimensional integrals with weak singularities.

Extension to parameter-dependent integrals demonstrated.

Abstract

We offer a simple method Monte Carlo for computation of Volterra's and spherical type multiple integrals with weak (integrable) singularities. An elimination of infinity of variance is achieved by incorporating singularities in the density, and we offer a highly effective way for generation of appeared multidimensional distribution. We extend offered method onto multiple Volterra's and spherical integrals with weak singularities containing parameter.