Risk Models with Stochastic Premium and Ruin Probability Estimation

We investigate the risk model called the random premiums model that generalizes the classical risk process. Within this model, the total claim amount process is the same as in the classical model while the premium income, unlike the classical case, is considered to be a stochastic process. A representation of the ruin probability for the random premiums risk process (i.e. the analog of the Beekman convolution formula) is derived. Some aspects of numerical estimation of the ruin probability are investigated. The Cramér–Lundberg theory is generalized for the random premiums model and results obtained by the other authors are surveyed. Prospects for application of the investigated model in practical problems of financial mathematics are discussed.