Risk Theory in a Periodic Environment: The Cramér-Lundberg Approximation and Lundberg's Inequality

A risk process with the claim arrival intensity β(t), the claim size distribution β^(t) and the premium rate p(t) at time t being periodic functions of t is considered. It is shown that the adjustment coefficient γ* 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 ψ^(s)(u) 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 γ_n → γ* 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.