Counting processes with Bernštein intertimes and random jumps

Abstract

In this paper we consider point processes N^f(t), t > 0, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bernštein functions f with Lévy measure ν. We obtain the general expression of the probability generating functions G^f of N^f, the equations governing the state probabilities p_k^f of N^f, and their corresponding explicit forms. We also give the distribution of the first-passage times T_k^f of N^f, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process, and the gamma-Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times τ_j^l_j of jumps with height l_j (∑_j=1^rl_j = k) under the condition N(t) = k for all these special processes is investigated in detail.