The Cramér–Lundberg Model
We consider the simplest possible model for the surplus of an insurance portfolio in continuous time (besides the Brownian motion). Even though this model is far from reality, it offers an excellent possibility to study the effect of a decision to the risk involved. We are particularly interested in ruin probabilities. We show how many characteristics of the model can be calculated, including the capital prior and immediately after ruin. We use both basic methods as well as martingales. For large initial capital, we approximate the ruin probabilities both for small and for large claims. Finally, we discuss an alternative measure for the risk, taking into account the event of ruin as well as the time to ruin and the deficit at ruin.