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On fluctuation-theoretic decompositions via Lindley-type recursions

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  • Boxma, Onno
  • Kella, Offer
  • Mandjes, Michel

Abstract

Consider a Lévy process Y(t) over an exponentially distributed time Tβ with mean 1/β. We study the joint distribution of the running maximum Ȳ(Tβ) and the time epoch G(Tβ) at which this maximum last occurs. Our main result is a fluctuation-theoretic distributional equality: the vector (Ȳ(Tβ),G(Tβ)) can be written as a sum of two independent vectors, the first one being (Ȳ(Tβ+ω),G(Tβ+ω)) and the second one being the running maximum and corresponding time epoch under the restriction that the Lévy process is only observed at Poisson(ω) inspection epochs (until Tβ). We first provide an analytic proof for this remarkable decomposition, and then a more elementary proof that gives insight into the occurrence of the decomposition and into the fact that ω only appears in the right hand side of the decomposition. The proof technique underlying the more elementary derivation also leads to further generalizations of the decomposition, and to some fundamental insights into a generalization of the well known Lindley recursion.

Suggested Citation

  • Boxma, Onno & Kella, Offer & Mandjes, Michel, 2023. "On fluctuation-theoretic decompositions via Lindley-type recursions," Stochastic Processes and their Applications, Elsevier, vol. 165(C), pages 316-336.
    • Handle: RePEc:eee:spapps:v:165:y:2023:i:c:p:316-336
    DOI: 10.1016/j.spa.2023.09.004
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    References listed on IDEAS

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    1. David Landriault & Jean-François Renaud & Xiaowen Zhou, 2014. "An Insurance Risk Model with Parisian Implementation Delays," Methodology and Computing in Applied Probability, Springer, vol. 16(3), pages 583-607, September.
    2. Albrecher, Hansjörg & Lautscham, Volkmar, 2013. "From Ruin To Bankruptcy For Compound Poisson Surplus Processes," ASTIN Bulletin, Cambridge University Press, vol. 43(2), pages 213-243, May.
    3. Korshunov, D., 1997. "On distribution tail of the maximum of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 72(1), pages 97-103, December.
    4. Albrecher, Hansjörg & Ivanovs, Jevgenijs, 2017. "Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations," Stochastic Processes and their Applications, Elsevier, vol. 127(2), pages 643-656.
    5. Bertoin, J. & Doney, R. A., 1994. "Cramer's estimate for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 21(5), pages 363-365, December.
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