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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.

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