# Cramer-Lundberg model for some classes of extremal Markov sequences

## TL;DR

This paper extends the classical Cramer-Lundberg model using Markov processes with generalized convolutions to better approximate an insurance company's financial dynamics and compute ruin probabilities over an infinite horizon.

## Contribution

It introduces a novel modification of the Cramer-Lundberg model based on Markov processes with generalized convolutions, enabling stochastic approximation of internal financial policies.

## Key findings

- Derived ruin probabilities for models with $ ext{alpha}$-convolution, maximal convolution, and Kendall convolution.
- Provided a framework for approximating internal financial policies using publicly available data.
- Extended classical risk models to include generalized convolution-based Markov processes.

## Abstract

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