×

The maximum surplus before ruin and related problems in a jump-diffusion renewal risk process. (English) Zbl 1268.91085

Summary: We investigate a Sparre Andersen risk model perturbed by diffusion with phase-type inter-claim times. We mainly study the distribution of maximum surplus prior to ruin. A matrix form of the integro-differential equation for this quantity is derived, and its solution can be expressed as a linear combination of particular solutions of the corresponding homogeneous integro-differential equations. By using the divided differences technique and nonnegative real part roots of Lundberg’s equation, the explicit Laplace transforms of particular solutions are obtained. Specially, we can deduce closed-form results as long as the individual claim size is rationally distributed. We also give a concise matrix expression for the expected discounted dividend payments under a barrier dividend strategy. Finally, we give some examples to present our main results.

MSC:

91B30 Risk theory, insurance (MSC2010)
60G15 Gaussian processes
60K10 Applications of renewal theory (reliability, demand theory, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andersen, E. S.: On the collective theory of risk in case of contagion between claims. Bull. Inst. Math. Appl., 12, 275–279 (1957)
[2] Dickson, D. C. M., Hipp, C.: Ruin probability for Erlang(2) risk process. Insurance Math. Econom., 22, 251–262 (1998) · Zbl 0907.90097 · doi:10.1016/S0167-6687(98)00003-1
[3] Cheng, Y., Tang, Q.: Moments of surplus before ruin and deficit at ruin in the Erlang(2) risk process. N. Am. Actuar. J., 7(1), 1–12 (2003) · Zbl 1084.60544 · doi:10.1080/10920277.2003.10596073
[4] Li, S., Garrido, J.: On ruin for Erlang(n) risk process. Insurance Math. Econom., 34(3), 391–408 (2004) · Zbl 1188.91089 · doi:10.1016/j.insmatheco.2004.01.002
[5] Gerber, H. U., Shiu, E. S. W.: The time value of ruin in a Sparre Andersen model. N. Am. Actuar. J., 9(2), 49–69 (2005) · Zbl 1085.62508 · doi:10.1080/10920277.2005.10596197
[6] Avram, F., Usábel, M.: Ruin probabilities and deficit for the renewal risk model with phase-type interarrival times. Astin Bull., 34(2), 315–332 (2004) · Zbl 1274.91244 · doi:10.2143/AST.34.2.505146
[7] Schmidli, H.: Discussion of ”The time value of ruin in a Sparre Andersen model”. N. Am. Actuar. J., 9(2), 74–77 (2005) · doi:10.1080/10920277.2005.10596200
[8] Albrecher, H., Boxma, O. J.: On the discounted penalty function in a Markov-dependent risk model. Insurance Math. Econom., 37, 650–672, (2005) · Zbl 1129.91023 · doi:10.1016/j.insmatheco.2005.06.007
[9] Ren, J.: The discounted joint distribution of the surplus prior to ruin and the deficit at ruin in a Sparre Andersen model. N. Am. Actuar. J., 11(3), 128–136 (2007) · doi:10.1080/10920277.2007.10597471
[10] Li, S.: Discussion of ”The discounted joint distribution of the surplus prior to ruin and the deficit at ruin in a Sparre Andersen model”. N. Am. Actuar. J., 12(2), 208–210 (2008a) · doi:10.1080/10920277.2008.10597512
[11] Li, S.: The time of recovery and the maximum severity of ruin in a Sparre Andersen model. N. Am. Actuar. J., 12(4), 1–13, (2008b) · doi:10.1080/10920277.2008.10597497
[12] Li, S., Lu, Y.: The distribution of total dividend payments in a Sparre Andersen model. Statist. Probab. Lett., 79, 1246–1251 (2009) · Zbl 1160.62359 · doi:10.1016/j.spl.2009.01.018
[13] Gerber, H. U.: An extension of the renewal equation and its application in the collective theory of risk. Skand. Aktuarietidskrift, 205–210 (1970) · Zbl 0229.60062
[14] Dufresne, F., Gerber, H. U.: Risk theory for the compound Poisson process that is perturbed by diffusion. Insurance Math. Econom., 10, 51–59 (1991) · Zbl 0723.62065 · doi:10.1016/0167-6687(91)90023-Q
[15] Furrer, H. J., Schmidli, H.: Exponential inequalities for ruin probabilities of risk processes perturbed by diffusion. Insurance Math. Econom., 15, 23–36 (1994) · Zbl 0814.62066 · doi:10.1016/0167-6687(94)00017-4
[16] Schmidli, H.: Cramer-Lundberg approximations for ruin probabilities of risk processes perturbed by diffusion. Insurance Math. Econom., 16, 135–149 (1995) · Zbl 0837.62087 · doi:10.1016/0167-6687(95)00003-B
[17] Gerber, H. U., Landry, B.: On the discounted penalty at ruin in a jump-diffusion and the perpetual put option. Insurance Math. Econom., 22, 263–276 (1998) · Zbl 0924.60075 · doi:10.1016/S0167-6687(98)00014-6
[18] Wang, G. J., Wu, R.: Some distributions for classical risk processes that is perturbed by diffusion. Insurance Math. Econom., 26, 15–24, (2000) · Zbl 0961.62095 · doi:10.1016/S0167-6687(99)00035-9
[19] Zhang, C. S., Wang, G. J.: The joint density function of three characteristics on jump-diffusion risk process. Insurance Math. Econom., 32, 445–455 (2003) · Zbl 1066.91063 · doi:10.1016/S0167-6687(03)00133-1
[20] Li, S., Garrido, J.: The Gerber-Shiu function in a Sparre Andersen risk process perturbed by diffusion. Scand. Actuar. J., 26(3), 161–186 (2005) · Zbl 1092.91049 · doi:10.1080/03461230510006955
[21] Song, M.: Joint distribution and duration of negative surplus for some kinds of risk process. In: Ph.D. Thesis Edition, Nankai University, arXiv: 0803.0906v1 [math.PR], 2008
[22] Gerber, H. U., Shiu, E. S. W.: On the time value of ruin. N. Am. Actuar. J., 2(1), 48–78 (1998) · Zbl 1081.60550 · doi:10.1080/10920277.1998.10595671
[23] Li, S., Dickson, D. C. M.: The maximum surplus before ruin in an Erlang(n) risk process and related problems. Insurance Math. Econom., 38, 529–539 (2006) · Zbl 1168.60363 · doi:10.1016/j.insmatheco.2005.11.005
[24] Li, S., Lu, Y.: The decompositions of the discounted penalty functions and dividends-penalty identity in a Markov-modulated risk model. Astin Bull., 38(1), 53–71 (2008) · Zbl 1169.91390 · doi:10.2143/AST.38.1.2030402
[25] Asmussen, S.: Ruin Probabilities, World Scientific, Singapore, 2000 · Zbl 0960.60003
[26] Neuts, M. F.: Matrix-Geometric Solutions in StochasticModels, Johns Hopkins University Press, Baltimore, 1981 · Zbl 0469.60002
[27] Cheung, E. C. K.: Discussion of ”Moments of the dividend payments and related problems in a Markovmodulated risk model”. N. Am. Actuar. J., 11(4), 145–148 (2007) · doi:10.1080/10920277.2007.10597494
[28] Jacobson, M.: The time to ruin for a class of Markov additive risk process with twosided jumps. Adv. in Appl. Probab., 37, 963–992 (2005) · Zbl 1100.60021 · doi:10.1239/aap/1134587749
[29] Lu, Y., Li, S.: The Markovian regime-switching risk model with a threshold dividend strategy. Insurance Math. Econom., 44(2), 296–303 (2009) · Zbl 1163.91438 · doi:10.1016/j.insmatheco.2008.04.004
[30] Li, S.: The distribution of the dividend payments in the compound Poisson risk model perturbed by Diffusion. Scand. Actuar. J., 2, 73–85 (2006) · Zbl 1143.91032 · doi:10.1080/03461230600589237
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.