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

Abstract

In this paper, we discuss the Cramér-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 $2a/{\sigma }^2\le 1$.