Risk theory in a periodic environment: The Cramér-Lundberg approximation and Lundberg's inequality

A risk process with the claim arrival intensity , the claim size distribution and the premium rate p(t) at time t being periodic functions of t is considered. It is shown that the adjustment coefficient is the same as for the standard time-homogeneous compound Poisson risk process obtained by averaging the parameters over a period, and a suitable version of the Cramér-Lundberg approximation for the ruin probability with initial reserve u and initial season s is derived. An approximation in terms of a Markovian environment model with n states is studied, and limit theorems describing the rate of convergence are given. Finally, various upper and lower bounds of Lundberg type for the ruin probabilities are derived for both the periodic and the Markov-modulated model. By time-reversion, the results apply also to periodic M/G/1 queues.