An Analogue of the Cramér–Lundberg Approximation in the Optimal Investment Case

We consider ruin probabilities for an insurance company, which canalso invest in the stock market. The risk process is modeled by a compound Poissonprocess and the stock price by geometric Brownian motion. We show that if the tailsof the claims are light tailed, then the optimal strategy is asymptotically given byholding a constant $-value in the stock position. Furthermore, we show that a kind ofCramér–Lundberg approximation holds for the minimal ruin probability. Everythingis shown under assumptions, which are analogous to the assumptions in the case of theclassical Cramér–Lundberg approximation without investment.