Fluctuation properties of compound Poisson-Erlang Lévy processes

Richard B. Paris, Vladimir Vinogradov

Research output: Contribution to journalArticle

Abstract

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
Pages (from-to)283-302
Number of pages20
JournalCommunications on Stochastic Analysis
Volume7
Issue number2
Publication statusPublished - Jun 2013

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Compound Poisson
Fluctuations
Siméon Denis Poisson
Duality
Wright Function
Passage Time
Overshoot
Reciprocity
Lévy Process
Weak Convergence
Mean Square
Multiplier
Space-time
Analogue
Zero

Cite this

Paris, Richard B. ; Vinogradov, Vladimir. / Fluctuation properties of compound Poisson-Erlang Lévy processes. In: Communications on Stochastic Analysis. 2013 ; Vol. 7, No. 2. pp. 283-302.
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Fluctuation properties of compound Poisson-Erlang Lévy processes. / Paris, Richard B.; Vinogradov, Vladimir.

In: Communications on Stochastic Analysis, Vol. 7, No. 2, 06.2013, p. 283-302.

Research output: Contribution to journalArticle

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AU - Paris, Richard B.

AU - Vinogradov, Vladimir

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AB - 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.

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JO - Communications on Stochastic Analysis

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