Integrated insurance risk models with exponential Lévy investment

We consider an insurance risk model for the cashflow of an insurance company, which invests its reserve into a portfolio consisting of risky and riskless assets. The price of the risky asset is modeled by an exponential Lévy process. We derive the integrated risk process and the corresponding discounted net loss process. We calculate certain quantities as characteristic functions and moments. We also show under weak conditions stationarity of the discounted net loss process and derive the left and right tail behavior of the model. Our results show that the model carries a high risk, which may originate either from large insurance claims or from the risky investment.