Pólya-Aeppli risk models

In this paper we consider three risk models. As counting processes for these risk models we use the Poisson process, the Polya-Aeppli process and the Noncentral Polya-Aeppli process. For the related risk models we define an exponential martingales and the corresponding martingale approximation of the ruin probabilities. In the case of exponentially distributed claims we compare these three models and for specific values of the parameters we make some conclusions for their applications in risk theory.