Asymptotic ruin probabilities and optimal investment

We study the infinite time ruin probability for an insurance company in the classical Cramér--Lundberg model with finite exponential moments. The additional nonclassical feature is that the company is also allowed to invest in some stock market, modeled by geometric Brownian motion. We obtain an exact analogue of the classical estimate for the ruin probability without investment, that is, an exponential inequality. The exponent is larger than the one obtained without investment, the classical Lundberg adjustment coefficient, and thus one gets a sharper bound on the ruin probability. A surprising result is that the trading strategy yielding the optimal asymptotic decay of the ruin probability simply consists in holding a fixed quantity (which can be explicitly calculated) in the risky asset, independent of the current reserve. This result is in apparent contradiction to the common believe that "rich" companies should invest more in risky assets than "poor" ones. The reason for this seemingly paradoxical result is that the minimization of the ruin probability is an extremely conservative optimization criterion, especially for "rich" companies.