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On a risk model with dependence between interclaim arrivals and claim sizes. (English) Zbl 1145.91030

The authors consider the risk model with time-dependent claim sizes. Let \(N_{t}, t\geq0,\) be the claim number process. It is assumed that \(N_{t}\) is a Poisson process with i.i.d. exponential interclaim time random variables \(\{W_{j}, j\in N^{+}\}\). Let us denote the claim arrival times \(\{T_{j}, j\in N^{+}\}\) by \(T_{j}=W_1+\dots+W_{j}\). The individual claim amount random variables \(\{X_{j}, j\in N^{+}\}\) are assumed to be a sequence of strictly positive i.i.d. random variables. It is assumed that the bivariate random vectors \((W_{j},X_{j})\) are mutually independent but that the random variables \(W_{j}\) and \(X_{j}\) are not independent. The total claim amount process is defined as \(S_{t}=\sum_{j=1}^{N_{t}}X_{j}\). The surplus process is defined as \(U_{t}=u+ct-S_{t}\), where \(u\) is the initial surplus level and \(c\) is the level premium rate. The authors derive the defective renewal equation satisfied by the expected discounted penalty function. Based on the compound geometric tail representation of the Laplace transform of the time of ruin, an explicit expression for this Laplace transform for a large class of claim size distributions is obtained. The authors measure the impact of the various dependence structures in the risk model on the ruin probability via the comparison of their Lundberg adjustment coefficients.

MSC:

91B30 Risk theory, insurance (MSC2010)
60K15 Markov renewal processes, semi-Markov processes
60G40 Stopping times; optimal stopping problems; gambling theory
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