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First and Last Passage Times of Spectrally Positive Lévy Processes with Application to Reliability

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  • Christian Paroissin

    (Université de Pau et des Pays de l’Adour)

  • Landy Rabehasaina

    (Université de Franche-Comté)

Abstract

We consider a wide class of increasing Lévy processes perturbed by an independent Brownian motion as a degradation model. Such family contains almost all classical degradation models considered in the literature. Classically failure time associated to such model is defined as the hitting time or the first-passage time of a fixed level. Since sample paths are not in general increasing, we consider also the last-passage time as the failure time following a recent work by Barker and Newby (Reliab Eng Syst Saf 94:33–43, 2009). We address here the problem of determining the distribution of the first-passage time and of the last-passage time. In the last section we consider a maintenance policy for such models.

Suggested Citation

  • Christian Paroissin & Landy Rabehasaina, 2015. "First and Last Passage Times of Spectrally Positive Lévy Processes with Application to Reliability," Methodology and Computing in Applied Probability, Springer, vol. 17(2), pages 351-372, June.
    • Handle: RePEc:spr:metcap:v:17:y:2015:i:2:d:10.1007_s11009-013-9360-9
    DOI: 10.1007/s11009-013-9360-9
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    References listed on IDEAS

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    1. Barker, C.T. & Newby, M.J., 2009. "Optimal non-periodic inspection for a multivariate degradation model," Reliability Engineering and System Safety, Elsevier, vol. 94(1), pages 33-43.
    2. Tsai, Cary Chi-Liang & Willmot, Gordon E., 2002. "A generalized defective renewal equation for the surplus process perturbed by diffusion," Insurance: Mathematics and Economics, Elsevier, vol. 30(1), pages 51-66, February.
    3. Biffis, Enrico & Kyprianou, Andreas E., 2010. "A note on scale functions and the time value of ruin for Lévy insurance risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 85-91, February.
    4. Dickson,David C. M., 2010. "Insurance Risk and Ruin," Cambridge Books, Cambridge University Press, number 9780521176750.
    5. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
    6. José Garrido & Manuel Morales, 2006. "On The Expected Discounted Penalty function for Lévy Risk Processes," North American Actuarial Journal, Taylor & Francis Journals, vol. 10(4), pages 196-216.
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    Cited by:

    1. 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.
    2. Landriault, David & Li, Bin & Lkabous, Mohamed Amine & Wang, Zijia, 2023. "Bridging the first and last passage times for Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 308-334.
    3. Yin Shu & Qianmei Feng & David W. Coit, 2015. "Life distribution analysis based on Lévy subordinators for degradation with random jumps," Naval Research Logistics (NRL), John Wiley & Sons, vol. 62(6), pages 483-492, September.

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