Cossette, Hélène; Marceau, Etienne; Marri, Fouad Constant dividend barrier in a risk model with a generalized Farlie-Gumbel-Morgenstern copula. (English) Zbl 1232.91343 Methodol. Comput. Appl. Probab. 13, No. 3, 487-510 (2011). Summary: In this paper, we consider the classical surplus process with a constant dividend barrier and a dependence structure between the claim amounts and the inter-claim times. We derive an integro-differential equation with boundary conditions. Its solution is expressed as the Gerber-Shiu discounted penalty function in the absence of a dividend barrier plus a linear combination of a finite number of linearly independent particular solutions to the associated homogeneous integro-differential equation. Finally, we obtain an explicit solution when the claim amounts are exponentially distributed and we investigate the effects of dependence on ruin quantities. Cited in 7 Documents MSC: 91B30 Risk theory, insurance (MSC2010) 60K05 Renewal theory 62P05 Applications of statistics to actuarial sciences and financial mathematics Keywords:compound Poisson risk model; copula; generalized Farlie-Gumbel-Morgenstern copulas; constant dividend barrier; ruin theory; dependence models; Gerber-Shiu discounted penalty function PDFBibTeX XMLCite \textit{H. Cossette} et al., Methodol. Comput. Appl. Probab. 13, No. 3, 487--510 (2011; Zbl 1232.91343) Full Text: DOI References: [1] Albrecher H, Boxma O (2004) A ruin model with dependence between claim sizes and claim intervals. Insurance: Mathematics and Economics 35:245–254 · Zbl 1079.91048 [2] Albrecher H, Teugels J (2006) Exponential behavior in the presence of dependence in risk theory. J Appl Probab 43(1):265–285 · Zbl 1097.62110 [3] Albrecher H, Hartinger J, Tichy RF (2005) On the distribution of dividend payments and the discounted penalty function in a risk model with linear dividend barrier. Scand Actuar J 2005(2):103–126 · Zbl 1092.91036 [4] Boudreault M, Cossette H, Landriault D, Marceau E (2006) On a risk model with dependence between interclaim arrivals and claim sizes. Scand Actuar J 2006(5):301–323 · Zbl 1145.91030 [5] Bühlmann H (1970) Mathematical methods in risk theory. Springer-Verlag, Berlin 4 · Zbl 0209.23302 [6] Cossette H, Marceau E, Marri F (2008) On the compound Poisson risk model with dependence based on a generalized Farlie–Gumbel–Morgenstern copula. Insurance: Mathematics and Economics 43:444–455 · Zbl 1151.91565 [7] De Finetti B (1957) Su un’impostazione alternativa dell teoria colletiva del rischio. Transactions of the XV International Congress of Actuaries 2:433–443 [8] Gerber HU (1979) An introduction to mathematical risk theory. S.S. Huebner Foundation Monographs, University of Pennsylvania · Zbl 0431.62066 [9] Gerber HU, Shiu ESW (2006) On optimal dividend strategies in the compound Poisson risk model. N Am Actuar J 10:76–93 [10] Grandell J (1991) Aspects of risk theory. Springer-Verlag, New York · Zbl 0717.62100 [11] Landriault D (2008) Constant dividend barrier in a risk model with interclaim-dependent claim sizes. Insurance: Mathematics and Economics 42:31–38 · Zbl 1141.91523 [12] Li S, Garrido J (2004a) On a class of renewal risk models with a constant dividend barrier. Insurance: Mathematics and Economics 35:691–701 · Zbl 1122.91345 [13] Li S, Garrido J (2004b) On ruin for Erlang(n) risk processes. Insurance: Mathematics and Economics 34:391–408 · Zbl 1188.91089 [14] Lin XS, Pavlova K (2006) The compound Poisson risk model with a threshold dividend strategy. Insurance: Mathematics and Economics 38:57–80 · Zbl 1157.91383 [15] Lin XS, Sendova K (2008) The compound Poisson risk model with multiple thresholds. Insurance: Mathematics and Economics 42:617–627 · Zbl 1152.91592 [16] Lin XS, Willmot GE, Drekic S (2003) The classical risk model with a dividend barrier: analysis of the Gerber-Shiu discounted penalty function. Insurance: Mathematics and Economics 33:551–566 · Zbl 1103.91369 [17] Rodriguez-Lallena JA, Úbeda-Flores M (2004) A new class of bivariate copulas. Stat Probab Lett 66:315–325 · Zbl 1102.62054 [18] Rolski T, Schmidli H, Schmidt V, Teugels J (1999) Stochastic processes for insurance and finance. Wiley, New York · Zbl 0940.60005 [19] Willmot GE, Lin XS (2001) Lundberg approximations for compound distributions with insurance applications. Springer series in statistics. Springer-Verlag, New York 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.