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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
##### Keywords:
Lévy processes; renewal equations
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##### References:
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