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On a general class of renewal risk process: analysis of the Gerber-Shiu function. (English) Zbl 1077.60063
Summary: We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a \(K_n\) distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most \(n\in \mathbb{N})\). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by G. E. Willmot [J. Appl. Probab. 36, 570–584 (1999; Zbl 0942.60086)] and H. U. Gerber and E. S. W. Shiu [N. Amer. Actuarial J. 9, 49–69 (2005)]. Finally, explicit results are given for rationally distributed claim severities.

MSC:
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60K05 Renewal theory
91B30 Risk theory, insurance (MSC2010)
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