A note on a class of delayed renewal risk processes
type: Item
instance of: scholarly article
title: A note on a class of delayed renewal risk processes
zbMATH Open document ID: 1114.60068
author: Gordon E. Willmot
published in: Insurance Mathematics \& Economics
publication date: 2007-03-02T00:00:00Z
review text: A delayed or modified renewal process is considered where the time until the first renewal (claim) is a positive random variable with density \[ k_1(t) = q \frac{e^{-\alpha t} \int_t^\infty e^{\alpha y} k(y) dy} {\int_0^\infty e^{\alpha y} \bar{K}(y) dy} +(1-q) \alpha e^{-\alpha t} . \] \(K(t)\) is the distribution function of the random time between the \((i-1)\)th and \(i\)th claim, \(i=2,3,\dots\), \(k\) is its probability density, and \(\bar{K}(t) = 1- K(t)\). This choice of \(k_1\) includes two important cases, an equilibrium distribution and the exponential distribution. The Gerber-Shiu discounted penalty function, which is a generalization of the ruin probability, is considered. A mathematically tractable formula is derived for the Gerber-Shiu function (Theorem 2.1). Further, ruin probabilities and those related to the surplus immediately prior to ruin or to the deficit at ruin can be evaluated for specified parameters. An example for the ruin probability is given and it is shown, that it can be expressed in terms of the ruin probability in the ordinary renewal risk model.
reviewed by: Waltraud Kahle
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60K10
Mathematics Subject Classification ID: 91B30
zbMATH DE Number: 5130417
zbMATH Keywords: renewal risk processes
zbMATH Keywords: Gerber-Shiu function
zbMATH Keywords: ruin probability
MaRDI profile type: Publication
full work available at URL: j.insmatheco.2003.12.005
OpenAlex ID: W2014563608
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DOI: 10.1016/J.INSMATHECO.2003.12.005
description: scientific article; zbMATH DE number 5130417