A Risk Model with Claims by Pure Birth Process

In this paper, we construct a risk model with claims by pure birth process which is the generalization of the classical Poisson risk model. In this model, the occurrence of claims is described by a pure birth process. The ruin probability sequences of a company facing such model that the initial capital is u and the pure birth process started at i are mainly studied. An integral equation satisfied by the ruin probability sequences is obtained and the relationship between the ruin probability sequences and the ruin probability of the Poisson risk model is got. The special cases of tail-Poisson risk model and n-alternate risk model are studied and the ruin probabilities of the exponentially distributed claims are got.