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On the probability of ruin in the compound Poisson risk model with potentially delayed claims. (English) Zbl 1307.60065

Authors’ abstract: In this paper, we consider the compound Poisson risk model involving two types of dependent claims, namely main claims and by-claims. The by-claim is induced by the main claim with a certain probability and the occurrence of a by-claim may be delayed depending on the associated main claim amount. Using Rouché’s theorem, both the survival probability with zero initial surplus and the Laplace transform of the survival probability are obtained from a system of integro-differential equations. Then, using the Laplace transform, we derive a defective renewal equation satisfied by the survival probability. An exact representation for the solution of this equation is derived through an associated compound geometric distribution. For exponential claim sizes, we present an explicit formula for the survival probability. We also illustrate the influence of model parameters in the dependent risk model on the survival probability by numerical examples.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60J65 Brownian motion
91B30 Risk theory, insurance (MSC2010)
45J05 Integro-ordinary differential equations
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