2015/12/01 by Enzo Orsingher, Bruno Toaldo · 5 citations
Mathematics · Computer Science · #Mathematical functions and polynomials #Bayesian Methods and Mixture Models #Fractional Differential Equations Solutions
paper · doi:10.1239/jap/1450802751
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 v. 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 of jumps with height l j ( ) under the condition N ( t ) = k for all these special processes is investigated in detail.