High order expansions for renewal functions and applications to ruin theory

2016/11/28 by Clément Dombry, Dombry, Clément, Landy Rabehasaina +1
Decision Sciences · Economics, Econometrics and Finance · Social Sciences · #FOS: Mathematics #Insurance and Financial Risk Management #Insurance, Mortality, Demography, Risk Management #Probability (math.PR) #Probability and Risk Models

paper · doi:10.48550/arxiv.1611.09197

openalex publication_date 2016/11/28 · openalex created_date 2022/08/15 · openalex updated_date 2026/07/28

Abstract

A high order expansion of the renewal function is provided under the assumption that the inter-renewal time distribution is light tailed with finite moment generating function g on a neighborhood of 0. This expansion relies on complex analysis and is expressed in terms of the residues of the function 1/(1 -- g). Under the assumption that g can be extended into a meromorphic function on the complex plane and some technical conditions, we obtain even an exact expansion of the renewal function. An application to risk theory is given where we consider high order expansion of the ruin probability for the standard compound Poisson risk model. This precises the well known Crámer-Lundberg approximation of the ruin probability when the initial reserve is large.

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