A Karamata-type theorem and ruin probabilities for an insurer investing proportionally in the stock market

We study the infinite time ruin probability in the classical Cramér–Lundberg model, where the company invests a constant fraction of its money in a stock, which is described by geometric Brownian motion. We prove that a certain integro-differential equation describes the ruin probability under weak regularity and integrability assumptions on the claim size distribution. Furthermore we show that, within the class of subexponential distributions, the claim size distributions with regularly varying tail distribution are the only ones, for which the ruin probability is proportional to the tail distribution.