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Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes

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  • Veillette, Mark
  • Taqqu, Murad S.

Abstract

Let {D(s),s>=0} be a Lévy subordinator, that is, a non-decreasing process with stationary and independent increments and suppose that D(0)=0. We study the first-hitting time of the process D, namely, the process E(t)=inf{s:D(s)>t}, t>=0. The process E is, in general, non-Markovian with non-stationary and non-independent increments. We derive a partial differential equation for the Laplace transform of the n-time tail distribution function P[E(t1)>s1,...,E(tn)>sn]. This PDE can be used to derive all n-time moments of the process E. As an application, we give a recursive formula for multiple-time moments of the local time of a Markov process in terms of its transition density.

Suggested Citation

  • Veillette, Mark & Taqqu, Murad S., 2010. "Using differential equations to obtain joint moments of first-passage times of increasing Lévy processes," Statistics & Probability Letters, Elsevier, vol. 80(7-8), pages 697-705, April.
    • Handle: RePEc:eee:stapro:v:80:y:2010:i:7-8:p:697-705
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    References listed on IDEAS

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    1. Meerschaert, Mark M. & Scheffler, Hans-Peter, 2006. "Stochastic model for ultraslow diffusion," Stochastic Processes and their Applications, Elsevier, vol. 116(9), pages 1215-1235, September.
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    Cited by:

    1. Chi, Zhiyi, 2016. "On exact sampling of the first passage event of a Lévy process with infinite Lévy measure and bounded variation," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1124-1144.
    2. Ascione, Giacomo & Leonenko, Nikolai & Pirozzi, Enrica, 2020. "Fractional Erlang queues," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3249-3276.
    3. Shantanu Awasthi & Indranil SenGupta, 2020. "First exit-time analysis for an approximate Barndorff-Nielsen and Shephard model with stationary self-decomposable variance process," Papers 2006.07167, arXiv.org, revised Jan 2021.
    4. Choe, Geon Ho & Lee, Dong Min, 2016. "Numerical computation of hitting time distributions of increasing Lévy processes," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 289-294.
    5. Foucart, Clément & Möhle, Martin, 2022. "Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 510-531.
    6. K. K. Kataria & M. Khandakar, 2021. "On the Long-Range Dependence of Mixed Fractional Poisson Process," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1607-1622, September.
    7. Meerschaert, Mark M. & Nane, Erkan & Xiao, Yimin, 2013. "Fractal dimension results for continuous time random walks," Statistics & Probability Letters, Elsevier, vol. 83(4), pages 1083-1093.

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