Abstract
In this paper, the generalized Erlang(n) risk model with two-sided jumps and a constant dividend barrier is considered. We assume that the downward jump sizes follow an arbitrary distribution and the upward jump sizes follow the mixed Erlang distribution. An integro-differential equation with boundary conditions for the expected discounted penalty function is derived and the solution is provided. The defective renewal equation for the expected discounted penalty function with no barrier is derived. We also give an example to obtain the expression of the expected discounted penalty function when the claim amounts are exponentially distributed.
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Albrecher, H., Claramunt, M.M., Mármol, M.: On the distribution of dividend payments in a Sparre Andersen model with generalized Erlang(\(n\)) interclaim times. Insur. Math. Econ. 37(2), 324–334 (2005)
Albrecher, H., Hartinger, J., Thonhauser, S.: On exact solutions for dividend strategies of threshold and linear barrier type in a Sparre Andersen model. ASTIN Bull. 37(2), 203–233 (2007)
Asmussen, S., Avram, F., Pistorius, M.R.: Russian and American put options under exponential phase-type Lévy models. Stoch. Process. Appl. 109(1), 79–111 (2004)
Bo, L., Song, R., Tang, D., Wang, Y., Yang, X.: Lévy risk model with two-sided jumps and a barrier dividend strategy. Insur. Math. Econ. 50(2), 280–291 (2012)
Boucheire, R.J., Boxma, O.J., Sigman, K.: A note on negative customers, GI/G/1 workload, and risk processes. Probab. Eng. Inform. Syst. 11, 305–311 (1997)
Boutsikas, M.V., Rakitzis, A.C., Antzoulakos, D.L.: On the number of claims until ruin in a two-barrier renewal risk model with Erlang mixtures. J. Comput. Appl. Math. 294(C), 124–137 (2016)
Breuer, L.: First passage times for Markov additive processes with positive jumps of phase type. J. Appl. Probab. 45(3), 779–799 (2008)
Cai, N.: On first passage times of a hyper-exponential jump diffusion process. Oper. Res. Lett. 37(2), 127–134 (2009)
Chadjiconstantinidis, S., Papaioannou, A.D.: On a perturbed by diffusion compound Poisson risk model with delayed claims and multi-layer dividend strategy. J. Comput. Appl. Math. 253(Complete), 26–50 (2013)
Cheung, E.C.K.: On a class of stochastic models with two-sided jumps. Queueing Syst. 69(1), 1–28 (2011)
Cheung, E.C.K., Liu, H., Willmot, G.E.: Joint moments of the total discounted gains and losses in the renewal risk model with two-sided jumps. Appl. Math. Comput. 331, 358–377 (2018)
Cossette, H., Marceau, E., Marri, F.: On a compound Poisson risk model with dependence and in the presence of a constant dividend barrier. Appl. Stoch. Models Bus. Ind. 30(2), 82–98 (2014)
De Finetti, B.: Su un’impostazione alternativa dell teoria colletiva del rischio. In: Transactions of the XV International Congress of Actuaries, vol. 2, pp. 433–443 (1957)
Dickson, D.C.M., Hipp, C.: On the time to ruin for Erlang(2) risk processes. Insur. Math. Econ. 29, 333–344 (2001)
Gerber, H.U., Shiu, E.S.W.: The time value of ruin in a Sparre Andersen model. N. Am. Actuar. J. 9, 49–69 (2005)
Li, S., Garrido, J.: On ruin for the Erlang(\(n\)) risk process. Insur. Math. Econ. 34(3), 391–408 (2004a)
Li, S., Garrido, J.: On a class of renewal risk models with a constant dividend barrier. Insur. Math. Econ. 35, 691–701 (2004b)
Lin, X.S., Willmot, G.E., Drekic, S.: The classical risk model with a constant dividend barrier: analysis of the Gerber-Shiu discounted penalty function. Insur. Math. Econ. 33, 551–566 (2003)
Lin, X.S., Pavlova, K.P.: The compound Poisson risk model with a threshold dividend strategy. Insur. Math. Econ. 38, 57–80 (2006)
Petrovski, I.G.: Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs (1966)
Perry, D., Stadje, W., Zacks, S.: First-exit time for the compound Poisson processes for some types of positive and negative jumps. Stoch. Models 18(1), 139–157 (2002)
Siegl, T., Tichy, R.F.: A process with stochastic claim frequency and a linear dividend barrier. Insur. Math. Econ. 24(1–2), 51–65 (1999)
Wang, W., Chen, P., Li, S.: Generalized expected discounted penalty function at general drawdown for Lévy risk processes. Insur. Math. Econ. 91, 12–25 (2020)
Zhang, Z., Yang, H.: A generalized penalty function in the Sparre-Andersen risk model with two-sided jumps. Stat. Probab. Lett. 80(7–8), 597–607 (2010)
Zhang, Z., Yang, H., Li, S.: The perturbed compound Poisson risk model with two-sided jumps. J. Comput. Appl. Math. 233(8), 1773–1784 (2010)
Zhou, J., Wu, L., Bai, Y.: Occupation times of Lévy-driven Ornstein–Uhlenbeck processes with two-sided exponential jumps and applications. Stat. Probab. Lett. 125, 80–90 (2017)
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Communicated by Mahmoud Hadizadeh-Yazdi.
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Zhang, L. The Expected Discounted Penalty Function in the Generalized Erlang (n) Risk Model with Two-Sided Jumps and a Constant Dividend Barrier. Bull. Iran. Math. Soc. 47, 569–583 (2021). https://doi.org/10.1007/s41980-020-00399-1
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DOI: https://doi.org/10.1007/s41980-020-00399-1