Insurance company model with random premiums and claims

Abstract

The article discusses the stochastic Cramer-Lundberg model, in which premiums and insurance claims (claims) are random and independent. The bonuses are equally distributed and obey the exponential law. The claims are also equally distributed according to the exponential law, which has a positive shift from the origin. A homogeneous Poisson process is introduced, the jumps of which are interpreted as the moments of receipt of premiums, and the intensity corresponds to the average number of premiums per year. The Poisson process does not depend on random variables representing premiums and claims. Insurance cases occur at the same moments at which premiums are received, but with less intensity. The probabilities of the company’s ruin in the first three moments of the occurrence of claims are found, and a scheme is given for sequentially calculating the probabilities of ruin at the moments of insured claims. Examples are given.

Keywords:

ruin probability, risk model, stochastic model of an insurance company, exponential distribution with loss shift