Ruin probabilities in the presence of regularly varying tails and optimal investment

We study the infinite time ruin probability in the classical Cramer–Lundberg model, where the company is allowed to invest their money in a stock, which is described by geometric Brownian motion. Starting from an integro-differential equation for the maximal survival probability, we analyze the case of claim sizes, which have distribution functions F with regularly varying tails. Our result is: if 1−F is regularly varying with index ρ<−1, then the ruin probability ψ is also regularly varying with index ρ<−1. This holds under the assumption of zero interest rates.