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Numerical Computation of First-Passage Times of Increasing Lévy Processes

Author

Listed:
  • Mark Veillette

    (Boston University)

  • Murad S. Taqqu

    (Boston University)

Abstract

Let {D(s), s ≥ 0} be a non-decreasing Lévy process. The first-hitting time process {E(t), t ≥ 0} (which is sometimes referred to as an inverse subordinator) defined by $E(t) = \inf \{s: D(s) > t \}$ is a process which has arisen in many applications. Of particular interest is the mean first-hitting time $U(t)=\mathbb{E}E(t)$ . This function characterizes all finite-dimensional distributions of the process E. The function U can be calculated by inverting the Laplace transform of the function $\widetilde{U}(\lambda) = (\lambda \phi(\lambda))^{-1}$ , where ϕ is the Lévy exponent of the subordinator D. In this paper, we give two methods for computing numerically the inverse of this Laplace transform. The first is based on the Bromwich integral and the second is based on the Post-Widder inversion formula. The software written to support this work is available from the authors and we illustrate its use at the end of the paper.

Suggested Citation

  • Mark Veillette & Murad S. Taqqu, 2010. "Numerical Computation of First-Passage Times of Increasing Lévy Processes," Methodology and Computing in Applied Probability, Springer, vol. 12(4), pages 695-729, December.
    • Handle: RePEc:spr:metcap:v:12:y:2010:i:4:d:10.1007_s11009-009-9158-y
    DOI: 10.1007/s11009-009-9158-y
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    References listed on IDEAS

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    1. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
    2. 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. A. Maheshwari & P. Vellaisamy, 2019. "Fractional Poisson Process Time-Changed by Lévy Subordinator and Its Inverse," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1278-1305, September.
    2. 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.
    3. Dhar Soma & Mahanta Lipi B. & Das Kishore Kumar, 2019. "Formulation Of The Simple Markovian Model Using Fractional Calculus Approach And Its Application To Analysis Of Queue Behaviour Of Severe Patients," Statistics in Transition New Series, Polish Statistical Association, vol. 20(1), pages 117-129, March.
    4. Dexter O. Cahoy & Federico Polito & Vir Phoha, 2015. "Transient Behavior of Fractional Queues and Related Processes," Methodology and Computing in Applied Probability, Springer, vol. 17(3), pages 739-759, September.

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