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Ruin probabilities for time-correlated claims in the compound binomial model. (English) Zbl 1074.91032

The paper deals with the ruin probabilities for time-correlated claims in the compound binomial model. Such model is considered in the presence of two classes of correlated risks: every main claim will produce a by-claim, but the occurence of the by-claim may be delayed. This model can be applied to some catastrophic events, as earthquakes, or, within another context, it is useful when the induced claim may be treated as a random portion of the total claims.
If the premium rate is unitary per period, the surplus process at time \(k\) is given by \[ S_k=u+k-U_K^X-U_k^Y,\;k=0,1,\ldots \] where \(u(\geq 0)\) is the initial surplus, \(U_K^X\) and \(U_k^Y\) are, respectively, the total amount of main claims and the total amount of by-claims in the first \(k\) time period.
Then the finite-time survival probability is defined as \[ \Phi (u,k)=Pr(S_j\geq 0; \;j=0,1,\ldots,k), \] while the finite-time ruin probability is \(\psi (u,k)=1-\phi (u,k)\).
The authors provide recursive formulas for the finite-time ruin probabilities and explicit expressions for the ultimate ruin probabilities in some special cases.

MSC:

91B30 Risk theory, insurance (MSC2010)
91B70 Stochastic models in economics
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References:

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