On the equivalence of various definitions of mixed poisson processes

Abstract

Under mild assumptions the equivalence of the mixed Poisson process with mixing parameter a real-valued random variable to the one with mixing distribution as well as to the mixed Poisson process in the sense of Huang is obtained, and a characterization of each one of the above mixed Poisson processes in terms of disintegrations is provided. Moreover, some examples of "canonical" probability spaces admitting counting processes satisfying the equivalence of all above statements are given.Finally, it is shown that our assumptions are essential for the characterization of mixed Poisson processes in terms of disintegrations.MSC 2010: Primary 60G55 ; secondary 60A10, 28A50, 60G05, 60K05, 60J27, 91B30.

Some characterizations for Markov processes as mixed renewal processes

“…In Section 3, a method for the construction of non-trivial probability spaces admitting extended MRPs is given, providing concrete examples of probability spaces and extended MRPs satisfying the assumptions of the main result and allowing us to check whether a extended MRP has the Markov property or not. Further applications of our results, concerning the equivalence of the existing definitions of MPPs, are given in the forthcoming paper [7].…”

Applications of a change of measures technique for compound mixed renewal processes to the ruin problem

“…The most common definition encountered in the literature is that of a mixed Poisson process with mixing probability distribution U on B(Υ ) (cf., e.g., [12], Definition 4.2) (written MPP(U ) for short). Since every MPP(Θ) is a MPP(U ) with U = P Θ (see [18], Theorem 3.1) all the previous results can be transferred to that case. On the other hand, existing results for the class of MPP(U ) (cf., e.g., [12], Subsection 9.2.1, for ruin probabilities and mixed Poisson processes) cannot, in general, be transferred to the case of MPP(Θ), since it is not always possible, given a MPP(U ), to construct a positive real-valued random variable Θ such that P Θ = U .…”

“…A rcp {P θ } θ∈D of P over P Θ consistent with Θ is essentially unique, if for any other rcp { P θ } θ∈D of P over P Θ consistent with Θ there exists a P Θ -null set N ∈ B(D) such that for any θ / ∈ N the equality P θ = P θ holds true. Regular conditional probabilities seem to have a bad reputation when it comes to applications, and that is probably due to the facts that their own existence is not always guaranteed (see [18], Examples 4 and 5) and their construction usually involves manipulations with Radon-Nikodým derivatives. Nevertheless, as the spaces used in applied Probability Theory are mainly Polish ones, such rcps always exist (see [11], Theorem 6), and in fact they can be explicitly constructed for the class of (compound) mixed renewal processes (see [31], Proposition 4.1).…”

Applications of a change of measures technique for compound mixed renewal processes to the ruin problem

“…Regular conditional probabilities seem to have a bad reputation when it comes to applications, and that is probably due to the fact that their own existence is not always guaranteed (see [13], Examples 4 and 5). Nevertheless, as the spaces used in applied Probability Theory are mainly Polish ones, such rcps always exist (see [8], Theorem 6), and in fact they can be explicitly constructed for the class of (compound) mixed renewal processes (see [24], Proposition 4.1).…”