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Splitting at the infimum and excursions in half-lines for random walks and Lévy processes

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  • Bertoin, Jean

Abstract

The central result of this paper is that, for a process X with independent and stationary increments, splitting at the infimum on a compact time interval amounts (in law) to the juxtaposition of the excursions of X in half-lines according to their signs. This identity yields a pathwise construction of X conditioned (in the sense of harmonic transform) to stay positive or negative, from which we recover the extension of Pitman's theorem for downwards-skip-free processes. We also extend for Lévy processes an identity that Karatzas and Shreve obtained for the Brownian motion. In the special case of stable processes, the sample path is studied near a local infimum.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:47:y:1993:i:1:p:17-35
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    Cited by:

    1. Ivanovs, Jevgenijs & Thøstesen, Jakob D., 2021. "Discretization of the Lamperti representation of a positive self-similar Markov process," Stochastic Processes and their Applications, Elsevier, vol. 137(C), pages 200-221.
    2. Pierre Andreoletti & Roland Diel, 2011. "Limit Law of the Local Time for Brox’s Diffusion," Journal of Theoretical Probability, Springer, vol. 24(3), pages 634-656, September.
    3. 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.
    4. Fitzsimmons, P. J. & Getoor, R. K., 1995. "Occupation time distributions for Lévy bridges and excursions," Stochastic Processes and their Applications, Elsevier, vol. 58(1), pages 73-89, July.
    5. 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.
    6. Chaumont, L., 1996. "Conditionings and path decompositions for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 39-54, November.
    7. Shi, Zhan, 1998. "A local time curiosity in random environment," Stochastic Processes and their Applications, Elsevier, vol. 76(2), pages 231-250, August.
    8. Ivanovs, Jevgenijs, 2017. "Splitting and time reversal for Markov additive processes," Stochastic Processes and their Applications, Elsevier, vol. 127(8), pages 2699-2724.
    9. Hambly, B. M. & Kersting, G. & Kyprianou, A. E., 2003. "Law of the iterated logarithm for oscillating random walks conditioned to stay non-negative," Stochastic Processes and their Applications, Elsevier, vol. 108(2), pages 327-343, December.

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