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Positive self-similar Markov processes obtained by resurrection

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  • Kim, Panki
  • Song, Renming
  • Vondraček, Zoran

Abstract

In this paper we study positive self-similar Markov processes obtained by (partially) resurrecting a strictly α-stable process at its first exit time from (0,∞). We construct those processes by using the Lamperti transform. We explain their long term behavior and give conditions for absorption at 0 in finite time. In case the process is absorbed at 0 in finite time, we give a necessary and sufficient condition for the existence of a recurrent extension. The motivation to study resurrected processes comes from the fact that their jump kernels may explode at zero. We establish sharp two-sided jump kernel estimates for a large class of resurrected stable processes.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:156:y:2023:i:c:p:379-420
    DOI: 10.1016/j.spa.2022.11.014
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    References listed on IDEAS

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    1. Chaumont, L., 1996. "Conditionings and path decompositions for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 64(1), pages 39-54, November.
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