×

zbMATH — the first resource for mathematics

On the Gerber-Shiu discounted penalty function in the Sparre Andersen model with an arbitrary interclaim time distribution. (English) Zbl 1152.91591
Summary: We consider the Sparre Andersen risk model with an arbitrary interclaim time distribution and a fairly general class of distributions for the claim sizes. Via a two-step procedure which involves a combination of a probabilitic and an analytic argument, an explicit expression is derived for the Gerber-Shiu discounted penalty function, subject to some restrictions on its form. A special case of Sparre Andersen risk models is then further analyzed, whereby the claim sizes’ distribution is assumed to be a mixture of exponentials. Finally, a numerical example follows to determine the impact on various ruin related quantities of assuming a heavy-tail distribution for the interclaim times.

MSC:
91B30 Risk theory, insurance (MSC2010)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cohen, J.W., The single server queue, (1982), North-Holland Amsterdam · Zbl 0481.60003
[2] De Smit, J.H.A., Explicit wiener – hoff factorization for the analysis of multi-dimensional queues, (), 293-310
[3] Dickson, D.C.M.; Hipp, C., On the time to ruin for Erlang(2) risk process, Insurance: mathematics and economics, 29, 333-344, (2001) · Zbl 1074.91549
[4] Gaver, D.P., Observing stochastic processes, and approximate transform inversion, Operations research, 14, 444-459, (1966)
[5] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-78, (1998) · Zbl 1081.60550
[6] Gerber, H.U.; Shiu, E.S.W., The time value of ruin in a sparre Andersen model, North American actuarial journal, 9, 2, 49-69, (2005)
[7] Li, S.; Garrido, J., On ruin for the Erlang(n) risk process, Insurance: mathematics and economics, 34, 391-408, (2004) · Zbl 1188.91089
[8] Li, S.; Garrido, J., On a general class of renewal risk process: analysis of the gerber – shiu function, Advances in applied probability, 37, 836-856, (2005) · Zbl 1077.60063
[9] Willmot, G., On the discounted penalty function in the renewal risk model with general interclaim times, Insurance: mathematics and economics, 41, 17-31, (2007) · Zbl 1119.91058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.