On the one-dimensional distributions of counting processes with stochastic intensities†

Abstract

Consider a counting process {N(t)>0}. In the one-dimensional case it is shown that the set of distributions of N(t) for t>0. This result is a "marginal"version of the characterization of a point process by its stochastic intensity. It is shown that the correspondng results have to be modifed in the multi-dimensional case. These results are particularized to the case of point processes with Poisson marginals.