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Gerber-Shiu functionals for classical risk processes perturbed by an \(\alpha\)-stable motion. (English) Zbl 1348.91159
Summary: We study the Gerber-Shiu functional of the classical risk process perturbed by a spectrally negative \(\alpha\)-stable motion. We provide representations of the scale functions of the process as an infinite series of convolutions of given functions. This, together with a result from E. Biffis and A. E. Kyprianou [Insur. Math. Econ. 46, No. 1, 85–91 (2010; Zbl 1231.91145)], allows us to obtain a representation of the Gerber-Shiu functional as an infinite series of convolutions. Moreover, we calculate the Laplace transform and derive a defective renewal equation for the Gerber-Shiu functional, thus extending previous work of H. Furrer [Scand. Actuarial J. 1998, No. 1, 59–74 (1998; Zbl 1026.60516)] and of C. C. L. Tsai and G. E. Willmot [Insur. Math. Econ. 30, No. 1, 51–66 (2002; Zbl 1074.91563)]. We also obtain asymptotic expressions for the joint tail distribution of the severity of ruin and the surplus before ruin.
91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
Full Text: DOI
[1] Bertoin, J., Lévy processes, (1996), Cambridge University Press · Zbl 0861.60003
[2] Biffis, E.; Kyprianou, A. E., A note on scale functions and the time value of ruin for Lévy insurance risk processes, Insurance Math. Econom., 46, 1, 85-91, (2010) · Zbl 1231.91145
[3] Biffis, E.; Morales, M., On a generalization of the gerber-shiu function to path-dependent penalties, Insurance Math. Econom., 46, 1, 92-97, (2010) · Zbl 1231.91146
[4] Dickson, D. C.M.; Hipp, C., On the time to ruin for Erlang(2) risk processes. 4th IME conference (Barcelona, 2000), Insurance Math. Econom., 29, 3, 333-344, (2001) · Zbl 1074.91549
[5] Dufresne, F.; Gerber, H. U., Risk theory for the compound Poisson process that is perturbed by diffusion, Insurance Math. Econom., 10, 1, 51-59, (1991) · Zbl 0723.62065
[6] Embrechts, P.; Goldie, C. M.; Veraverbeke, N., Subexponentiality and infinite divisibility, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 49, 335-347, (1979) · Zbl 0397.60024
[7] Feller, W., An Introduction to Probability Theory and its Applications, vol. II, (1971), John Wiley & Sons · Zbl 0219.60003
[8] Furrer, H., Risk processes perturbed by \(\alpha\)-stable Lévy motion, Scand. Actuar. J., 1, 59-74, (1998) · Zbl 1026.60516
[9] Furrer, H.; Michna, Z.; Weron, A., Stable Lévy motion approximation in collective risk theory, Insurance Math. Econom., 20, 2, 97-114, (1997) · Zbl 0901.90068
[10] Gerber, H. U., An extension of the renewal equation and its application in the collective theory of risk, Skand. Aktuar., 205-210, (1970) · Zbl 0229.60062
[11] 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
[12] Kuznetsov, A.; Kyprianou, A. E.; Rivero, V., Lévy matters II: lecture notes in mathematics 2013, 97-186, (2013)
[13] Kyprianou, A., Introductory lectures on fluctuations of Lévy processes with applications, (2006), Springer-Verlag Berlin, Heidelberg · Zbl 1104.60001
[14] Kyprianou, A. E., (Gerber-Shiu Risk Theory, European Actuarial Academy (EAA) Series, (2013), Springer Cham)
[15] Rolski, T.; Schmidli, H.; Schmidt, V.; Teugels, J., Stochastic processes for insurance and finance, (1999), Wiley & Sons · Zbl 0940.60005
[16] Sato, K., Lévy processes and infinitely divisible distributions, (1999), Cambridge University Press · Zbl 0973.60001
[17] Tsai, C. C.; Willmot, G. E., A generalized defective renewal equation for the surplus process perturbed by diffusion, Insurance Math. Econom., 30, 1, 51-66, (2002) · Zbl 1074.91563
[18] Zolotarev, V. M., (One-Dimensional Stable Distributions, Translations of Mathematical Monographs, vol. 65, (1986), American Mathematical Society Providence, RI)
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