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On the joint distribution of surplus before and after ruin under a Markovian regime switching model. (English) Zbl 1093.60051
The paper considers the Markov-modulated risk model introduced by S. Asmussen [Scand. Actuar. J. 1989, No. 2, 69–100 (1989; Zbl 0684.62073)], the joint distribution of the surplus before and after ruin assuming that the claim sizes are phase-type distributed. If the problem can be solved in the case of phase-type distribution, the problem in the general case can be approximated by using a sequence of phase-type distributions which converge to the desired probability distribution. The authors show when the initial surplus is zero or the claim size distributions are phase-type, it is possible to obtain a closed form solution to the joint distribution being considered. For the study of ruin probability, joint distribution of surplus, moments of surplus at moments of the ruin and the time of ruin, a set of integro-differential equations satisfied by the H. U. Gerber and E. S. W. Shiu expected discounted penalty function [N. Am. Actuar. J. 2, No. 1, 48–78 (1998; Zbl 1081.60550)] is derived.

MSC:
60J27 Continuous-time Markov processes on discrete state spaces
91B30 Risk theory, insurance (MSC2010)
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[1] Asmussen, S., Risk theory in a Markovian environment, Scand. actuar. J., 1989, 69-100, (1989) · Zbl 0684.62073
[2] Asmussen, S., Ruin probabilities, (2000), World Scientific Singapore
[3] Asmussen, S.; Perry, D., On cycle maxima, first passage problems and extreme value theory for queues, Stochastic models, 8, 421-458, (1992) · Zbl 0762.60086
[4] Dickson, D.C.M., On the distribution of the surplus prior to ruin, Insurance: math. econom., 11, 191-207, (1992) · Zbl 0770.62090
[5] H.U. Gerber, An Introduction to Mathematical Risk Theory, S.S. Huebner Foundation Monograph Series No. 8, R. Irwin, Homewood, IL, 1979.
[6] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuar. J., 2, 1, 48-78, (1998) · Zbl 1081.60550
[7] Graham, A., Kronecker products and matrix calculus with applications, (1981), Ellis Horwood Series in Mathematics and its Applications Chichester, Horwood · Zbl 0497.26005
[8] Grandell, J., Aspects of risk theory, (1990), Springer New York
[9] Hardy, M.R., A regime-switching model of long-term stock returns, North American actuar. J., 5, 2, 41-53, (2001) · Zbl 1083.62530
[10] Li, S.; Garrido, J., On ruin for the Erlang(n) risk process, Insurance: math. econom., 34, 391-408, (2004) · Zbl 1188.91089
[11] S. Li, J. Garrido, On a general class of renewal risk process: analysis of the Gerber-Shiu function, Adv. Appl. Probab. 37 (2005) 836-856. · Zbl 1077.60063
[12] Neuts, M.F., Matrix-geometric solutions in stochastic models, (1981), Johns Hopkins University Press Baltimore, London · Zbl 0469.60002
[13] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic processes for insurance and finance, (1999), Wiley New York · Zbl 0940.60005
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