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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.

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