An Insurance Company Model with Random Premiums and Claims

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