A point process \({\mathcal N}\) on \({\mathbb{R}}_ +\) can be represented by the associated counting process \((\xi _ t;t\in {\mathbb{R}}_ +)\) or by the associated sequence of jump times \((\tau _ n;n\in {\mathbb{Z}}_ +)\) and in accordance may possess two types of Markov property. The present paper first clarifies their mutual dependence, leading in particular to the notion of ”weak multiplicativity” for the joint distribution of two consecutive jump times. Then, by means of results from a previous paper, a uniquely determined ”Markov variant” \(\tilde {\mathcal N}\) is assigned to \({\mathcal N}\) without changing the one-dimensional marginals. This provides in particular a new characterization of the Poisson process by these marginals and the adequate Markov property. Further applications concern the explicit construction of the compensator and certain transition probabilities of \(\tilde {\mathcal N}\).