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A note on deficit analysis in dependency models involving Coxian claim amounts. (English) Zbl 1401.91157
Summary: In this paper, we consider a fairly large class of dependent Sparre Andersen risk models where the claim sizes belong to the class of Coxian distributions. We analyze the Gerber-Shiu discounted penalty function when the penalty function depends on the deficit at ruin. We show that the system of equations needed to solve for this quantity is surprisingly simple. Various applications of this result are also considered.

MSC:
91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[24] Sancetta, A., and Satchell, S. 2004. The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory 20: 535-562. · Zbl 1061.62080
[25] ]bib17 Pitts, S. M., Politis, K. 2007. The joint density of the surplus before and after ruin in the Sparre Andersen model. Journal of Applied Probability 44(3): 695-712. · Zbl 1132.60061
[26] Lin, XS. 2000. The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance: Mathematics and Economics, 27(1): 19-44. · Zbl 0971.91031
[27] Lin, XS and Willmot, GE. 2000. The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance: Mathematics and Economics, 27(1): 19-44. · Zbl 0971.91031
[28] Lin, XS and Willmot, GE. 2000. The moments of the time of ruin, the surplus before ruin, and the deficit at ruin. Insurance: Mathematics and Economics, 27(1): 19-44. · Zbl 0971.91031
[29] Nelsen, R.B. 1994. A Characterization of Farlie-Gumbel-Morgenstern Distributions via Spearman’s Rho and Chi-Square Divergence. Sankhyā: The Indian Journal of Statistics, Series A, 56(3): 476-479. · Zbl 0846.62040
[30] Nelsen, R.B. 2006. An Introduction to Copulas. Springer Series in Statistics, 2nd edition, Springer-Verlag: New York. · Zbl 1152.62030
[31] Pitts, SM and Politis, K. 2007. The joint density of the surplus before and after ruin in the Sparre Andersen model. Journal of Applied Probability, 44(3): 695-712. · Zbl 1132.60061
[32] Sancetta, A and Satchell, S. 2004. The Bernstein copula and its applications to modeling and approximations of multivariate distributions. Econometric Theory, 20: 535-562. · Zbl 1061.62080
[33] Tang, Q., and Wei, L. 2010. Asymptotic aspects of the Gerber-Shiu function in the renewal risk model using Wiener-Hopf factorization and convolution equivalence. Insurance: Mathematics and Economics 46(1): 19-31. · Zbl 1231.91243
[34] Willmot, G.E., and Lin, X.S. 2000. Lundberg Approximations for Compound Distributions with Insurance Applications. Lecture Notes in Statistics 156. Springer-Verlag: New York. · Zbl 0962.62099
[35] Willmot, G.E. 2007. On the discounted penalty function in the renewal risk model with general interclaim times. Insurance: Mathematics and Economics 41: 17-31. · Zbl 1119.91058
[36] Willmot, G.E., and Woo, J.-K. 2012. On the analysis of a general class of dependent risk processes. Insurance: Mathematics and Economics 51(1): 134-141. · Zbl 1284.91277
[37] Zhang, Z., Yang, H., and Yang, H. 2011. On a Sparre Andersen risk model with time- dependent claim sizes and jump-diffusion perturbation. Methodology and Computing in Applied Probability 1-23. · Zbl 1253.91090
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