An analogue of the Cramer-Lundberg approximation in the optimal investment case

We consider ruin probabilities for an insurance company, which can also invest in the stock market. The risk process is modeled by a compound Poisson process and the stock price by geometric Brownian motion. We show that if the tails of the claims are light tailed, then the optimal strategy is asymptotically given by holding a constant $-value in the stock position. Furthermore, we show that a kind of Cramer-Lundberg approximation holds for the minimal ruin probability. Everything is shown under assumptions, which are analogous to the assumptions in the case of the classical Cramer-Lundberg approximation without investment.