Integral equations, quasi-Monte Carlo methods and risk modelling

TL;DR

This paper surveys quasi-Monte Carlo methods for integral equations, introduces a new isotropic discrepancy concept, and applies these techniques to risk modeling, providing error bounds and numerical validation.

Contribution

It introduces a novel isotropic discrepancy measure and applies QMC methods to risk models, offering rigorous error bounds and practical numerical examples.

Findings

Derived a new error bound for risk expectations

Introduced isotropic discrepancy for numerical integration

Validated methods with numerical experiments

Abstract

We survey a QMC approach to integral equations and develop some new applications to risk modeling. In particular, a rigorous error bound derived from Koksma-Hlawka type inequalities is achieved for certain expectations related to the probability of ruin in Markovian models. The method is based on a new concept of isotropic discrepancy and its applications to numerical integration. The theoretical results are complemented by numerical examples and computations.