Fluctuation properties of compound Poisson-Erlang Lévy processes

Richard B. Paris, Vladimir Vinogradov

    Research output: Contribution to journalArticlepeer-review


    We derive an expression in terms of the Wright function for the density of the first-passage times (or FPT’s) for the Poisson-Erlang Levy processes. For Poisson-exponential processes, we establish an analogue of Zolotarev space-time duality between the original process and its FPT process “truncated” at zero. We show that an asymptotic duality holds in the sense of weak convergence, thereby providing an interpretation of the Letac-Mora reciprocity. The corresponding limits in the sense of convergence in mean and in mean square have an additional multiplier, which is also present in the asymptotic relationship between the marginals of a Poisson-Erlang process and its “truncated” FPT process. We prove that for Poisson-exponential processes, the FPT and the overshoot are independent.
    Original languageEnglish
    Article number8
    Pages (from-to)283-302
    Number of pages20
    JournalCommunications on Stochastic Analysis
    Issue number2
    Publication statusPublished - Jun 2013


    • Convergence: L1, L2
    • Frst-passage-time (FPT) process
    • Generalized Mittag-Leffler function
    • Incremental process
    • Letac-Mora reciprocity
    • Overshoot
    • Poisson-gamma Levy process
    • Wright function
    • Zolotarev duality


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