We consider an insurance company whose risk reserve is given by a Brownian motion with drift and which is able to invest the money into a Black-Scholes financial market. As optimization criteria, we treat mean-variance problems, problems with other risk measures, exponential utility and the probability of ruin. Following recent research, we assume that investment strategies have to be deterministic. This leads to deterministic control problems, which are quite easy to solve. Moreover, it turns out that there are some interesting links between the optimal investment strategies of these problems. Finally, we also show that this approach works in the L'evy process framework.
deterministic control problem mean-variance risk measure Lévy process