Asymptotics of ruin probabilities for controlled risk processes in the small claims case

We consider a risk process with the possibility of investment into a risky asset. The aim of the paper is to obtain the asymptotic behaviour of the ruin probability under the optimal investment strategy in the small claims case. In addition we prove convergence of the optimal investment level as the initial capital tends to infinity. Keywords: Ruin Probability, Change Of Measure, Optimal Control, Cramer-Lundberg Approximation, Adjustment Coefficient, Lundberg Bounds, Martingale, Geometric Browninan Motion