## 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.)

### Keywords:

Optimal investment; Ruin probabilities; Regular variation
Full Text:

### References:

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