# zbMATH — the first resource for mathematics

On the analysis of a general class of dependent risk processes. (English) Zbl 1284.91277
Summary: A generalized Sparre Andersen risk process is examined, whereby the joint distribution of the interclaim time and the ensuing claim amount is assumed to have a particular mathematical structure. This structure is present in various dependency models which have previously been proposed and analyzed. It is then shown that this structure in turn often implies particular functional forms for joint discounted densities of ruin related variables including some or all of the deficit at ruin, the surplus immediately prior to ruin, and the surplus after the second last claim. Then, employing a fairly general interclaim time structure which involves a combination of Erlang type densities, a complete identification of a generalized Gerber-Shiu function is provided. An application is given applying these results to a situation involving a mixed Erlang type of claim amount assumption. Various examples and special cases of the model are then considered, including one involving a bivariate Erlang mixture model.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60K10 Applications of renewal theory (reliability, demand theory, etc.) 62H20 Measures of association (correlation, canonical correlation, etc.)
Full Text:
##### References:
 [1] Albrecher, H.; Teugels, J.L., Exponential behavior in the presence of dependence in risk theory, Journal of applied probability, 43, 1, 257-273, (2006) · Zbl 1097.62110 [2] Asmussen, S.; Albrecher, H., Ruin probabilities, (2010), World Scientific · Zbl 1247.91080 [3] Bargès, M.; Cossette, H.; Marceau, E., Tvar-based capital allocation with copulas, Insurance: mathematics and economics, 45, 3, 348-361, (2009) · Zbl 1231.91141 [4] Block, H.W.; Basu, A.P., A continuous bivariate exponential distribution, Journal of the American statistical association, 69, 1031-1037, (1974) · Zbl 0299.62027 [5] Boudreault, M.; Cossette, H.; Laudriault, D.; Marceau, E., On a risk model with dependence between interclaim arrivals and claim sizes, Scandinavian actuarial journal, 5, 265-285, (2006) · Zbl 1145.91030 [6] Cheung, E.C.K., A generalized penalty function in sparre Andersen risk models with surplus-dependent premium, Insurance: mathematics and economics, 48, 3, 384-397, (2011) · Zbl 1229.91157 [7] Cheung, E.C.K.; Laudriault, D.; Willmot, G.E.; Woo, J.-K., Structural properties of gerber – shiu functions in dependent sparre Andersen models, Insurance: mathematics and economics, 46, 1, 117-126, (2010) · Zbl 1231.91157 [8] Cheung, E.C.K.; Laudriault, D.; Willmot, G.E.; Woo, J.-K., Gerber – shiu analysis with a generalized penalty function, Scandinavian actuarial journal, 3, 185-199, (2010) · Zbl 1226.60123 [9] Cheung, E.C.K.; Laudriault, D.; Willmot, G.E.; Woo, J.-K., On orderings and bounds in a generalized sparre Andersen risk model, Applied stochastic models in business and industry, 27, 1, 51-60, (2011) · Zbl 1274.60050 [10] Cossette, H.; Marceau, E.; Marri, F., On the compound Poisson risk model with dependence based on a generalized farlie – gumble – morgenstern copula, Insurance: mathematics and economics, 43, 3, 444-455, (2008) · Zbl 1151.91565 [11] Cossette, H.; Marceau, E.; Marri, F., Analysis of ruin measures for the classical compound Poisson risk model with dependence, Scandinavian actuarial journal, 3, 221-245, (2010) · Zbl 1226.91024 [12] D’Este, G.M., A morgenstern-type bivariate gamma distribution, Biometrika, 68, 339-340, (1981) [13] Farlie, D.J.G., The performance of some correlation coefficients for a general bivariate distribution, Biometrika, 47, 307-323, (1960) · Zbl 0102.14903 [14] Freund, J., A bivariate extension of the exponential distribution, Journal of the American statistical association, 56, 971-977, (1961) · Zbl 0106.13304 [15] Gerber, H.U.; Shiu, E.S.W., On the time value at ruin, North American actuarial journal, 2, 1, 48-72, (1998), Discussions: pp. 72-78 [16] Gerber, H.U.; Shiu, E.S.W., The time value of ruin in a sparre Andersen model, North American actuarial journal, 9, 2, 49-84, (2005) · Zbl 1085.62508 [17] Gupta, A.K.; Wong, C.F., On a morgenstern-type bivariate gamma distribution, Metrika, 31, 327-332, (1989) · Zbl 0576.62021 [18] Kotz, S.; Balakrishnan, N.; Johnson, N.L., Continuous multivariate distributions vol. 1: models and applications, (2000), Wiley-Interscience · Zbl 0946.62001 [19] Lee, S.C.K., Lin, X.S., 2012. Modeling dependent risks with multivariate Erlang mixtures. ASTIN Bulletin (in press). · Zbl 1277.62255 [20] Li, S.; Garrido, J., On a general class of renewal risk process: analysis of the gerber shiu function, Advances in applied probability, 37, 3, 836-856, (2005) · Zbl 1077.60063 [21] Li, S.; Garrido, J., On ruin for the Erlang($$n$$) risk process, Insurance: mathematics and economics, 34, 3, 391-408, (2004) · Zbl 1188.91089 [22] Rodriguez-Lallena, J.A.; Ubena-Flores, M., A new class of bivariate copulas, Statistics and probability letters, 66, 315-325, (2004) · Zbl 1102.62054 [23] Willmot, G.E., On the discounted penalty function in the renewal risk model with general interclaim times, Insurance: mathematics and economics, 41, 1, 17-31, (2007) · Zbl 1119.91058 [24] Willmot, G.E.; Lin, X.S., Lundberg approximations for compound distributions with insurance applications, (2001), Springer-Verlag New York [25] Willmot, G.E.; Lin, X.S., Risk modelling with the mixed Erlang distribution, Applied stochastic models in business and industry, 27, 1, 2-26, (2011) [26] Willmot, G.E.; Woo, J.-K., Surplus analysis for a class of Coxian interclaim time distributions with applications to mixed Erlang claim amounts, Insurance: mathematics and economics, 46, 1, 32-41, (2010) · Zbl 1231.91250 [27] Willmot, G.E.; Woo, J.-K., On the class of Erlang mixtures with risk theoretic applications, North American actuarial journal, 11, 2, 99-115, (2007) [28] Zhang, Z.; Yang, H.; Yang, H., On a sparre Andersen risk model with time-dependent claim sizes and jump-diffusion perturbation, Methodology and computing in applied probability, 1-23, (2011) [29] Zhao, Kui., 2008. On the expected discounted penalty function—a bivariate exponential distribution. Master Thesis. University of Waterloo.
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.