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Conditionings and path decompositions for Lévy processes

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  • Chaumont, L.

Abstract

We first give an interpretation for the conditioning to stay positive (respectively, to die at 0) for a large class of Lévy processes starting at x > 0. Next, we specify the laws of the pre-minimum and post-minimum parts of a Lévy process conditioned to stay positive. We show that, these parts are independent and have the same law as the process conditioned to die at 0 and the process conditioned to stay positive starting at 0, respectively. Finally, in some special cases, we prove the Skorohod convergence of this family of laws when x goes to 0.

Suggested Citation

  • Chaumont, L., 1996. "Conditionings and path decompositions for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 39-54, November.
    • Handle: RePEc:eee:spapps:v:64:y:1996:i:1:p:39-54
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    References listed on IDEAS

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    1. Bertoin, Jean, 1993. "Splitting at the infimum and excursions in half-lines for random walks and Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 47(1), pages 17-35, August.
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    Cited by:

    1. Kyprianou, Andreas E. & Rivero, Víctor M. & Satitkanitkul, Weerapat, 2019. "Conditioned real self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 129(3), pages 954-977.
    2. Ceren Vardar-Acar & Mine Çağlar & Florin Avram, 2021. "Maximum Drawdown and Drawdown Duration of Spectrally Negative Lévy Processes Decomposed at Extremes," Journal of Theoretical Probability, Springer, vol. 34(3), pages 1486-1505, September.
    3. Chaumont, L. & Kyprianou, A.E. & Pardo, J.C., 2009. "Some explicit identities associated with positive self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 980-1000, March.
    4. Kim, Panki & Song, Renming & Vondraček, Zoran, 2023. "Positive self-similar Markov processes obtained by resurrection," Stochastic Processes and their Applications, Elsevier, vol. 156(C), pages 379-420.
    5. Foss, Sergey G. & Puhalskii, Anatolii A., 2011. "On the limit law of a random walk conditioned to reach a high level," Stochastic Processes and their Applications, Elsevier, vol. 121(2), pages 288-313, February.
    6. Chaumont, Loïc & Rivero, Víctor, 2007. "On some transformations between positive self-similar Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1889-1909, December.
    7. Fernando Cordero, 2016. "The First Passage Time of a Stable Process Conditioned to Not Overshoot," Journal of Theoretical Probability, Springer, vol. 29(3), pages 776-796, September.
    8. Campi, Luciano & Cetin, Umut & Danilova, Albina, 2013. "Explicit construction of a dynamic Bessel bridge of dimension 3," LSE Research Online Documents on Economics 45263, London School of Economics and Political Science, LSE Library.
    9. Kyprianou, Andreas E. & Rivero, Victor & Şengül, Batı, 2017. "Conditioning subordinators embedded in Markov processes," Stochastic Processes and their Applications, Elsevier, vol. 127(4), pages 1234-1254.
    10. Leżaj, Łukasz, 2024. "Non-symmetric stable processes: Dirichlet heat kernel, Martin kernel and Yaglom limit," Stochastic Processes and their Applications, Elsevier, vol. 174(C).
    11. Shosei Takeda, 2024. "Sample Path Behaviors of Lévy Processes Conditioned to Avoid Zero," Journal of Theoretical Probability, Springer, vol. 37(4), pages 3177-3201, November.

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