Numerical evaluation of renewal equations: applications to risk theory and financial models

Abstract

The so-called Renewal Theory is a frequently used methodology in applied mathematics. Renewal Theory is mainly focussed on solving a Volterra integral equation of the second kind known as Renewal Integral EquationAn interesting problem arises when choosing the appropriate numerical tool in order to approximate the solution of the former integral. The decision will be based on the degree of knowledge of function F(x) and some properties of <I>(u). Three methods based in classical methodologies (simulation, product integration and inverting Laplace transform) will be presented and applied to the calculation of ultimate ruin probabilities in the classical case of Risk Theory. The first one is an original simulation scheme, based on the importance sampling technique, that leads to tight interval estimations of the solution of the Renewal equation. In the second one, the use of the so-called Product Integration technique will be considered and compared with other techniques based on the Newton-Cotes methodology. The last method considered is the Gaver-Stehfest algorithm of inverting Laplace transformo This last one, under certain conditions, could be considered as a very fast and accurate method.