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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.
##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60K10 Applications of renewal theory (reliability, demand theory, etc.) 62P05 Applications of statistics to actuarial sciences and financial mathematics
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