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On the expected discounted dividends in the Cramér-Lundberg risk model with more frequent ruin monitoring than dividend decisions. (English) Zbl 1306.91072
Summary: In this paper, we further extend the insurance risk model in [H. Albrecher et al., Astin Bull. 41, No. 2, 645–672 (2011; Zbl 1239.91072)], who proposed to only intervene in the compound Poisson risk process at the discrete time points \(\{L_k \}_{k = 0}^\infty\) where the event of ruin is checked and dividend decisions are made. In practice, an insurance company typically balances its books (and monitors its solvency) more frequently than deciding on dividend payments. This motivates us to propose a generalization in which ruin is monitored at \(\{L_k \}_{k = 0}^\infty\) whereas dividend decisions are only made at \(\{L_{j k} \}_{k = 0}^\infty\) for some positive integer \(j\). Assuming that the intervals between the time points \(\{L_k \}_{k = 0}^\infty\) are Erlang(\(n\)) distributed, the Erlangization technique (e.g. [S. Asmussen et al., ibid. 32, No. 2, 267–281 (2002; Zbl 1081.60028)]) allows us to model the more realistic situation with the books balanced e.g. monthly and dividend decisions made e.g. quarterly or semi-annually. Under a dividend barrier strategy with the above randomized interventions, we derive the expected discounted dividends paid until ruin. Numerical examples about dividend maximization with respect to the barrier \(b\) and/or the value of \(j\) are given.

MSC:
91B30 Risk theory, insurance (MSC2010)
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[1] Albrecher, H.; Bäuerle, N.; Thonhauser, S., Optimal dividend-payout in random discrete time, Statist. Risk Model., 28, 3, 251-276, (2011) · Zbl 1233.91139
[2] Albrecher, H.; Cheung, E. C.K.; Thonhauser, S., Randomized observation periods for the compound Poisson risk model: dividends, ASTIN Bull., 41, 2, 645-672, (2011) · Zbl 1239.91072
[3] Albrecher, H.; Cheung, E. C.K.; Thonhauser, S., Randomized observation periods for the compound Poisson risk model: the discounted penalty function, Scand. Actuar. J., 2013, 6, 424-452, (2013) · Zbl 1401.91089
[4] Albrecher, H.; Gerber, H. U.; Shiu, E. S.W., The optimal dividend barrier in the gamma-omega model, Eur. Actuar. J., 1, 1, 43-55, (2011) · Zbl 1219.91062
[5] Albrecher, H.; Ivanovs, J., A risk model with an observer in a Markov environment, Risks, 1, 3, 148-161, (2013)
[6] Albrecher, H., Ivanovs, J., Zhou, X., 2014. Exit identities for Lévy processes observed at Poisson arrival times. Preprint. · Zbl 1338.60125
[7] Albrecher, H.; Lautscham, V., From ruin to bankruptcy for compound Poisson surplus processes, ASTIN Bull., 43, 2, 213-243, (2013) · Zbl 1283.91084
[8] Albrecher, H.; Thonhauser, S., Optimality results for dividend problems in insurance, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 103, 2, 295-320, (2009) · Zbl 1187.93138
[9] Asmussen, S.; Albrecher, H., Ruin probabilities, (2010), World Scientific New Jersey · Zbl 1247.91080
[10] Asmussen, S.; Avram, F.; Usabel, M., Erlangian approximations for finite-horizon ruin probabilities, ASTIN Bull., 32, 2, 267-281, (2002) · Zbl 1081.60028
[11] Avanzi, B., Strategies for dividend distribution: A review, N. Am. Actuar. J., 13, 2, 217-251, (2009)
[12] Avanzi, B.; Cheung, E. C.K.; Wong, B.; Woo, J.-K., On a periodic dividend barrier strategy in the dual model with continuous monitoring of solvency, Insurance Math. Econom., 52, 1, 98-113, (2013) · Zbl 1291.91088
[13] Avanzi, B.; Gerber, H. U.; Shiu, E. S.W., Optimal dividends in the dual model, Insurance Math. Econom., 41, 1, 111-123, (2007) · Zbl 1131.91026
[14] Avanzi, B.; Tu, V.; Wong, B., On optimal periodic dividend strategies in the dual model with diffusion, Insurance Math. Econom., 55, 210-224, (2014) · Zbl 1296.91143
[15] Carr, P., Randomization and the American put, Rev. Financ. Stud., 11, 3, 597-626, (1998) · Zbl 1386.91134
[16] Chen, X.; Xiao, T.; Yang, X., A Markov-modulated jump-diffusion risk model with randomized observation periods and threshold dividend strategy, Insurance Math. Econom., 54, 76-83, (2014) · Zbl 1289.91074
[17] de Finetti, B., 1957. Su un’impostazione alternativa della teoria collettiva del rischio. In: Transactions of the XVth International Congress of Actuaries 2, pp. 433-443.
[18] Dickson, D. C.M.; Waters, H. R., Some optimal dividends problems, ASTIN Bull., 34, 1, 49-74, (2004) · Zbl 1097.91040
[19] Gerber, H. U., (An Introduction to Mathematical Risk Theory, Huebner Foundation Monograph, vol. 8, (1979), Richard D. Irwin Homewood, IL) · Zbl 0431.62066
[20] Gerber, H. U.; Lin, X. S.; Yang, H., A note on the dividends-penalty identity and the optimal dividend barrier, ASTIN Bull., 36, 2, 489-503, (2006) · Zbl 1162.91374
[21] Gerber, H. U.; Shiu, E. S.W., On the time value of ruin, N. Am. Actuar. J., 2, 1, 48-72, (1998) · Zbl 1081.60550
[22] Gerber, H. U.; Shiu, E. S.W.; Yang, H., The omega model: from bankruptcy to occupation times in the red, Eur. Actuar. J., 2, 2, 259-272, (2012) · Zbl 1256.91057
[23] Kyprianou, A. E.; Pistorius, M. R., Perpetual options and canadization through fluctuation theory, Ann. Appl. Probab., 13, 3, 1077-1098, (2003) · Zbl 1039.60044
[24] Lin, X. S.; Willmot, G. E.; Drekic, S., The compound Poisson risk model with a constant dividend barrier: analysis of the gerber-shiu discounted penalty function, Insurance Math. Econom., 33, 3, 551-566, (2003) · Zbl 1103.91369
[25] Loeffen, R. L., On optimality of the barrier strategy in de finetti’s dividend problem for spectrally negative Lévy processes, Ann. Appl. Probab., 18, 5, 1669-1680, (2008) · Zbl 1152.60344
[26] Ramaswami, V.; Woolford, D. G.; Stanford, D. A., The erlangization method for Markovian fluid flows, Ann. Oper. Res., 160, 1, 215-225, (2008) · Zbl 1140.60357
[27] Stanford, D. A.; Avram, F.; Badescu, A. L.; Breuer, L.; Da Silva Soares, A.; Latouche, G., Phase-type approximations to finite-time ruin probabilities in the sparre-Anderson and stationary renewal risk models, ASTIN Bull., 35, 1, 131-144, (2005) · Zbl 1123.62078
[28] Stanford, D. A.; Yu, K.; Ren, J., Erlangian approximation to finite time ruin probabilities in perturbed risk models, Scand. Actuar. J., 2011, 1, 38-58, (2011) · Zbl 1277.60128
[29] Zhang, Z., On a risk model with randomized dividend-decision times, J. Ind. Manag. Optim., 10, 4, 1041-1058, (2014) · Zbl 1282.91164
[30] Zhang, Z.; Cheung, E. C.K., The Markov additive risk process under an erlangized dividend barrier strategy, Methodol. Comput. Appl. Probab., (2014), (in press)
[31] Zhang, Z., Cheung, E.C.K., 2014b. A note on a Lévy insurance risk model under periodic dividend decisions. Preprint.
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