A proof of a conjecture in the Cramér-Lundberg model with investments

Details A proof of a conjecture in the Cramér-Lundberg model with investments A proof of a conjecture in the Cramér-Lundberg model with investments

2010-02-27

arxiv.org/abs/1003.0135v1

Mathematics

Probability

Scientific paper

In this paper, we discuss the Cram\'er-Lundberg model with investments, where the price of the invested risk asset follows a geometric Brownian motion with drift $a$ and volatility $\sigma> 0.$ By assuming there is a cap on the claim sizes, we prove that the probability of ruin has at least an algebraic decay rate if $2a/\sigma^2 > 1$. More importantly, without this assumption, we show that the probability of ruin is certain for all initial capital $u$, if $2a/\sigma^2 \le 1$.

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