Point processes with finite-dimensional conditional probabilities

We study the structure of point processes N with the property that the P(θtN∈· | Ft) vary in a finite-dimensional space where θt is the shift and Ft the σ-field generated by the counting process up to time t. This class of point processes is strictly larger than Neuts’ class of Markovian arrival processes. On the one hand, it allows for more general features like interarrival distributions which are matrix-exponential rather than phase type, on the other the probabilistic interpretation is a priori less clear. Nevertheless, the properties are very similar. In particular, finite-dimensional distributions of interarrival times, moments, Laplace transforms, Palm distributions, etc., are shown to be given by two fundamental matrices C, D just as for the Markovian arrival process. We also give a probabilistic interpretation in terms of a piecewise deterministic Markov process on a compact convex subset of Rp, whose jump times are identical to the epochs of N.