Ruin probabilities and optimal investment when the stock price follows an exponential Lévy process

This paper investigates the infinite and finite time ruin probability under the condition that the company is allowed to invest a certain amount of money in some stock market, and the remaining reserve in the bond with constant interest force. The total insurance claim amount is modeled by a compound Poisson process and the price of the risky asset follows a general exponential Levy process. Exponential type upper bounds for the ultimate ruin probability are derived when the investment is a fixed constant, which can be calculated explicitly. This constant investment strategy yields the optimal asymptotic decay of the ruin probability under some mild assumptions. Finally, we provide an approximation of the optimal investment strategy, which maximizes the expected wealth of the insurance company under a risk constraint on the Value-at-Risk.