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On the expected discounted penalty function for a perturbed risk process driven by a subordinator. (English) Zbl 1130.91032
The author considers the following perturbed risk model: \(U(t)=u+ct-S(t)+W(t)\), where \(S\) is a subordinator with zero drift and Lévy measure \(\nu\) and \(W\) is a zero-drift Brownian motion with infinitesimal variance \(\sigma^2\). The parameter \(u\) is an initial surplus and \(c\) is a constant premium rate. The purpose of the paper is to show that the expected discounted penalty function for this risk model satisfies a defective renewal equation. The stepping stone to produce the main result is the construction of a family of compound Poisson processes converging weakly to any given subordinator. The results allow for a wide range of models for the aggregate claims process, in particular those closed-form expressions are available like the gamma and inverse Gaussian processes. These results are possible due to the Lévy structure of the risk process.

91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60K05 Renewal theory
Full Text: DOI
[1] Asmussen, S., Ruin probabilities, () · Zbl 0960.60003
[2] Bertoin, J., ()
[3] Bertoin, J.; Doney, R.A., Cramér’s estimate for Lévy processes, Statistics and probability letters, 21, 363-365, (1994) · Zbl 0809.60085
[4] Doney, R.A.; Kyprianou, A.E., Overshoots and undershoots of Lévy processes, Annals of applied probability, 16, 91-106, (2006) · Zbl 1101.60029
[5] Dufresne, F.; Gerber, H., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance: mathematics and economics, 10, 51-59, (1991) · Zbl 0723.62065
[6] Dufresne, F.; Gerber, H.U.; Shiu, E.S.W., Risk theory with the gamma process, ASTIN bulletin, 21, 177-192, (1991)
[7] Garrido, J., Morales, M., 2006. On the Expected Discounted Penalty Function for Lévy Risk Processes. North American Actuarial Journal (in press)
[8] Gerber, H.U.; Landry, B., On a discounted penalty at ruin in a jump – diffusion and the perpetual put option, Insurance: mathematics and economics, 22, 263-276, (1998) · Zbl 0924.60075
[9] Gerber, H.U.; Shiu, E.S.W., The joint distribution of the time of ruin, the surplus immediately before ruin, and the deficit at ruin, Insurance: mathematics and economics, 21, 129-137, (1997) · Zbl 0894.90047
[10] Gerber, H.U.; Shiu, E.S.W., On the time value of ruin, North American actuarial journal, 2, 1, 48-78, (1998) · Zbl 1081.60550
[11] Gerber, H.U.; Shiu, E.S.W., Pricing perpetual options for jump processes, North American actuarial journal, 2, 3, 101-112, (1998) · Zbl 1081.91528
[12] Huzak, M.; Perman, M.; Sikic, H.; Vondracek, Z., Ruin probabilities and decompositions for general perturbed risk processes, The annals of applied probability, 14, 1378-1397, (2004) · Zbl 1061.60075
[13] Kaas, R.; Goovaerts, M.; Dhaene, J.; Denuit, M., Modern actuarial risk theory, (2001), Kluwer Academic Publishers
[14] Li, S.; Garrido, J., On the gerber – shiu functions in a sparre – andersen risk model perturbed by diffusion, Scandinavian actuarial journal, 161-186, (2005) · Zbl 1092.91049
[15] Li, S.; Lu, Y., On the expected discounted penalty function for two classes of risk processes, Insurance: mathematics and economics, 36, 179-193, (2005) · Zbl 1122.91040
[16] Morales, M., Risk theory with the generalized inverse Gaussian Lévy process, ASTIN bulletin, 34, 361-377, (2004) · Zbl 1102.60043
[17] Morales, M.; Schoutens, W., A risk model driven by Lévy processes, Applied stochastic models in business and industry, 19, 147-167, (2003) · Zbl 1051.60051
[18] Sarkar, J.; Sen, A, A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: mathematics and economics, 36, 421-432, (2005)
[19] Sato, K.I., Lévy processes and infinitely divisible distributions, (1999), Cambridge University Press · Zbl 0973.60001
[20] Tsai, C.C.L.; Willmot, G.E., A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance: mathematics and economics, 30, 51-66, (2002) · Zbl 1074.91563
[21] Yang, H.; Zhang, L., Spectrally negative Lévy processes with applications in risk theory, Advances in applied probability, 33, 281-291, (2001) · Zbl 0978.60104
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