An approximate solution of the integral equation of renewal theory

Abstract

In this study, an approximation to the solution of the renewal integral equation is constructed. Performance of the new method is evaluated and it is shown that the approximation provides an upper bound for the renewal function when the hazard function is a non-increasing function of time.

On the moments of the time to ruin in dependent Sparre Andersen models with emphasis on Coxian interclaim times

“…Since they are often in infinite series form, alternative results in terms of approximate solutions to defective renewal equations (e.g. Deligonul, 1985 and references therein) as well as bounds may be used. For example (e.g.…”

Alternative Analysis of Finite-Time Probability Distributions of Renewal Theory

“…Suppose the support is (0, U), where U ∈ (0, ∞) or U = ∞. If t < U, the difficulties of solving the renewal integral equation are similar to those discussed in Bartholemew [9] and Deligonul [10]. However, if t > U (implying U is finite) solving the renewal equation becomes increasingly difficult as t increases because: (a) the excess life distribution must first be known at nU , n = 1, .…”

“…Bartholemew [9], Deligonul [10], among others, address the need for using approximate solutions of the integral equations. Deligonul [10] mentions authors who have used: Laplace transforms, power series expansion, direct numerical solution of an approximate integral equation, and spline functions.…”

Two-dimensional Renewal Function Approximation

“…Xie (1989Xie ( , 2003 and his team of workers provided approximations by evaluating the integrals using Riemann Stieltjes integration methods. Bartholomew (1963), Deligonul (1985) and others provided approximations by substituting terms in the integral with known expressions. Garg and Kalagnanam (1998) used Pade approximations to provide useful approximation to the renewal function.…”