  Important models in insurance, for example the Carm{\'e}r--Lundberg theory
and the Sparre Andersen model, essentially rely on the Poisson process. The
process is used to model arrival times of insurance claims.
  This paper extends the classical framework for ruin probabilities by
proposing and involving the fractional Poisson process as a counting process
and addresses fields of applications in insurance.
  The interdependence of the fractional Poisson process is an important feature
of the process, which leads to initial stress of the surplus process. On the
other hand we demonstrate that the average capital required to recover a
company after ruin does not change when switching to the fractional Poisson
regime. We finally address particular risk measures, which allow simple
evaluations in an environment governed by the fractional Poisson process.
