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Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon

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  • Bladt, Mogens
  • Ivanovs, Jevgenijs

Abstract

There is an abundance of useful fluctuation identities for one-sided Lévy processes observed up to an independent exponentially distributed time horizon. We show that all the fundamental formulas generalize to time horizons having matrix exponential distributions, and the structure is preserved. Essentially, the positive killing rate is replaced by a matrix with eigenvalues in the right half of the complex plane which, in particular, applies to the positive root of the Laplace exponent and the scale function. Various fundamental properties of thus obtained matrices and functions are established, resulting in an easy to use toolkit. An important application concerns deterministic time horizons which can be well approximated by concentrated matrix exponential distributions. Numerical illustrations are also provided.

Suggested Citation

  • Bladt, Mogens & Ivanovs, Jevgenijs, 2021. "Fluctuation theory for one-sided Lévy processes with a matrix-exponential time horizon," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 105-123.
    • Handle: RePEc:eee:spapps:v:142:y:2021:i:c:p:105-123
    DOI: 10.1016/j.spa.2021.08.002
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    Cited by:

    1. Søren Asmussen & Mogens Bladt, 2022. "From PH/MAP to ME/RAP," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 173-175, April.

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