Cramér–Lundberg model for some classes of extremal Markov sequences

Abstract

The classical Cramér–Lundberg model was the first attempt to describe the financial condition of the insurance company. The incomes were approximated by a steady stream of money, and insurance payments were not limited and could take any value from zero to infinity. The society did not invest any part of its money and does not have any employees, shareholders, or enterprise maintenance costs. There exist many modifications of the Cramér–Lundberg model that cover at least some of the problems described here but usually require insight into the internal financial policy of the insurance company. We propose another modification based on Markov processes defined by generalized convolutions. Thanks to the generalized convolutions, we can stochastically approximate the internal financial policy of the company based on publicly available data. In this paper, we focus on computing the ruin probability in the Cramér–Lundberg model for an infinite time horizon for the Markov processes where the transition probabilities are defined by generalized convolutions, in particular, by the α-convolution, maximal convolution, or Kendall convolution.