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**Ruin probabilities in the presence of regularly varying tails and optimal investment.**
*(English)*
Zbl 1055.91049

Summary: We study the infinite time ruin probability in the classical Cramer-Lundberg model, where the company is allowed to invest their money in a stock, which is described by geometric Brownian motion. Starting from an integro-differential equation for the maximal survival probability, we analyze the case of claim sizes, which have distribution functions \(F\) with regularly varying tails. Our result is: if \(1-F\) is regularly varying with index \(\rho<-1\), then the ruin probability \(\psi\) is also regularly varying with index \(\rho<-1\). This holds under the assumption of zero interest rates.

### MSC:

91B30 | Risk theory, insurance (MSC2010) |

91B28 | Finance etc. (MSC2000) |

60J70 | Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) |

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\textit{J. Gaier} and \textit{P. Grandits}, Insur. Math. Econ. 30, No. 2, 211--217 (2002; Zbl 1055.91049)

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### References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.