Approximations for solutions of renewal-type equations

Building on and extending the results of Grübel (1989a), (J. Appl. Probab. 26, 296–303), approximation formulae for solutions of renewal-type equations are derived. These are obtained by finding the first and higher Fréchet derivatives of the functional that has the underlying lifetime density as input and a normalised version of the solution of the renewal-type equation as output. By approximating a density whose output is not known analytically by another density with easy ouput, we obtain explicit formulae for our approximations, which in many cases can be easily implemented on computer algebra software.