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Asymptotic ordering of risks and ruin probabilities. (English) Zbl 0787.62112

In the framework of risk theory the accumulated claims in \([0,t]\) are \(S(t)=\sum^{N(t)}_{k=1}X_ k\), where \((N(t))_{t\geq 0}\) represents the claim number process, is a Poisson one with parameter \(\lambda\) and the claim sizes \((X_ k)_{k\in\mathbb{N}}\) are i.i.d. nonnegative random variables with common distribution function \(F\) and finite mean \(\mu(F)\). Denoting by \(x(>0)\) the initial risk reserve and by \(c(>0)\) the intensity of the gross risk premium, the safety loading is given by \(\rho=\mu\lambda/c\), with \(0<\rho<1\).
Moreover, the ruin probability and the survival probability are respectively defined as: \[ \psi(x)=P(S(t)-ct>x\text{ for some }t>0),\;\varphi(x)=1-\psi(x). \] In this framework the author obtains bounds for ruin probabilities when it is not possible to have exact estimates. This is obtained by introducing asymptotic orderings, and thus extending a result of A. E. van Heerwaarden, on the claim size distributions in the Cramér case, and then in some intermediate cases until to large claims distributions.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
91B30 Risk theory, insurance (MSC2010)
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