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The expected time to ruin in a risk process with constant barrier via martingales. (English) Zbl 1117.91381
Summary: Two risk models with a constant dividend barrier are considered. In the two models claims arrive according to a Poisson process. In the first model the claim size has a phase type distribution. In the second model the claim size is exponentially distributed, but the arrival rate, the mean claim size, and the premium rate are governed by a random environment, which changes according to a Markov process. O. Kella and W. Whitt [J. Appl. Probab. 29, 396–403 (1992; Zbl 0761.60065)] martingale is applied in the first model. S. Asmussen and S. Kella [Adv. Appl. Probab. 32, 376–393 (2000; Zbl 0961.60081)] multi-dimensional martingale is applied in the second model. The expected time to ruin and the amount of dividends paid until ruin occurs are obtained for both models.

MSC:
91B30 Risk theory, insurance (MSC2010)
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[1] Abate, J.; Whitt, W., Numerical inversion of Laplace transforms of probability distributions, ORSA J. computing, 7, 36-43, (1995) · Zbl 0821.65085
[2] Asmussen, S., Ruin probabilities, (2000), World Scientific
[3] Asmussen, S., Applied probability and queues, (2003), Springer · Zbl 1029.60001
[4] Asmussen, S.; Højgaard, B.; Taksar, M., Optimal risk process and dividend distribution policies. example of excess-of loss reinsurance for an insurance corporation, Finance stochastic, 4, 299-324, (2000) · Zbl 0958.91026
[5] Asmussen, S.; Jobmann, M.; Schwefel, H.P., Exact buffer overflow calculations for queues via martingales, Queueing sys., 42, 63-90, (2002) · Zbl 1035.90018
[6] Asmussen, S.; Kella, O., A multi-dimensional martingale for Markov additive processes and its applications, Adv. appl. probability, 32, 376-393, (2000) · Zbl 0961.60081
[7] Asmussen, S.; Kella, O., On optional stopping of time exponential martingales for Lévy processes with and without reflection, Stochastic processes appl., 91, 47-55, (2001) · Zbl 1047.60038
[8] Bühlmann, H., Mathematical methods in risk theory, (1970), Springer-Verlag Berlin, Heidelberg, New York · Zbl 0209.23302
[9] De Finetti, B., Su un impostazione alternativa Della teoria collectiva del rischio, Trans. XV int. congr. actuaries, 2, 433-443, (1957)
[10] Dickson, D.C.M.; Gray, J.R., Approximations to ruin probability in the presence of an upper absorbing barrier, Scand. actuarial J., 2, 105-115, (1984) · Zbl 0584.62174
[11] Dickson, D.C.M.; Gray, J.R., Exact solutions for ruin probability in the presence of an upper absorbing barrier, Scand. actuarial J., 3, 174-186, (1984) · Zbl 0557.62086
[12] Dickson, D.C.M.; Waters, H.R., Some optimal dividends problems, ASTIN bull., 34, 49-74, (2004) · Zbl 1097.91040
[13] Gerber, H.U., 1979. An Introduction to Mathematical Risk Theory. S.S. Hubner Foundation Monographs, University of Pennsylvania. · Zbl 0431.62066
[14] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, N. am. actuarial J., 2, 1, 48-78, (1998) · Zbl 1081.60550
[15] Højgaard, B., Optimal dynamic premium control in non-life insurance. maximizing dividends pay-out, Scand. actuarial J., 4, 225-245, (2002) · Zbl 1039.91042
[16] Irbäck, J., Asymptotic theory for a risk process with high dividend barrier, Scand. actuarial J., 2, 97-118, (2003) · Zbl 1092.91043
[17] Kella, O.; Perry, D.; Stadje, W., A stochastic clearing model with a Brownian and a compound Poisson components, Probability eng. informational sci., 17, 1-22, (2003) · Zbl 1064.60186
[18] Kella, O.; Whitt, W., Useful martingales for stochastic storage processes with Lévy input, J. appl. probability, 29, 396-403, (1992) · Zbl 0761.60065
[19] Lin, S.X.; Willmot, G.; Drekic, S., The classical risk model with constant dividend barrier: analysis of the gerber-shiu discounted penalty function, Ins.: math. econ., 33, 551-566, (2003) · Zbl 1103.91369
[20] Perry, D.; Stadje, W., Risk analysis for a cash management model with two types of customers, Ins.: math. econ., 26, 25-36, (2000) · Zbl 0990.91028
[21] Segerdahl, C., On some distributions in time connected with the collective theory of risk, Scand. actuarial J., 167-192, (1970) · Zbl 0229.60063
[22] Wang, N.; Politis, K., Some characteristic of a surplus process in the presence of an upper barrier, Ins.: math. econo., 30, 231-241, (2002) · Zbl 1055.91058
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