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Bridges with Random Length: Gamma Case

Author

Listed:
  • Mohamed Erraoui

    (Université Cadi Ayyad)

  • Astrid Hilbert

    (Linnaeus University)

  • Mohammed Louriki

    (Université Cadi Ayyad)

Abstract

In this paper, we generalize the concept of gamma bridge in the sense that the length will be random, that is, the time to reach the given level is random. The main objective of this paper is to show that certain basic properties of gamma bridges with deterministic length stay true also for gamma bridges with random length. We show that the gamma bridge with random length is a pure jump process and that its jumping times are countable and dense in the random interval bounded by 0 and the random length. Moreover, we prove that this process is a Markov process with respect to its completed natural filtration as well as with respect to the usual augmentation of this filtration, which leads us to conclude that its completed natural filtration is right continuous. Finally, we give its canonical decomposition with respect to the usual augmentation of its natural filtration.

Suggested Citation

  • Mohamed Erraoui & Astrid Hilbert & Mohammed Louriki, 2020. "Bridges with Random Length: Gamma Case," Journal of Theoretical Probability, Springer, vol. 33(2), pages 931-953, June.
    • Handle: RePEc:spr:jotpro:v:33:y:2020:i:2:d:10.1007_s10959-019-00955-4
    DOI: 10.1007/s10959-019-00955-4
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    References listed on IDEAS

    as
    1. Hoyle, Edward & Mengütürk, Levent Ali, 2013. "Archimedean survival processes," Journal of Multivariate Analysis, Elsevier, vol. 115(C), pages 1-15.
    2. Dufresne, François & Gerber, Hans U. & Shiu, Elias S. W., 1991. "Risk Theory with the Gamma Process," ASTIN Bulletin, Cambridge University Press, vol. 21(2), pages 177-192, November.
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