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The Upper Envelope of Positive Self-Similar Markov Processes

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  • J. C. Pardo

    (Université Pierre et Marie Curie)

Abstract

We establish integral tests in connection with laws of the iterated logarithm at 0 and at +∞, for the upper envelope of positive self-similar Markov processes. Our arguments are based on the Lamperti representation and on the study of the upper envelope of the future infimum due to the author (see Pardo in Stoch. Stoch. Rep. 78:123–155, [2006]). These results extend laws of the iterated logarithm for Bessel processes due to Dvoretsky and Erdős (Proceedings of the Second Berkeley Symposium, [1951]) and stable Lévy processes with no positive jumps conditioned to stay positive due to Bertoin (Stoch. Process. Appl. 55:91–100, [1995]).

Suggested Citation

  • J. C. Pardo, 2009. "The Upper Envelope of Positive Self-Similar Markov Processes," Journal of Theoretical Probability, Springer, vol. 22(2), pages 514-542, June.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:2:d:10.1007_s10959-008-0152-z
    DOI: 10.1007/s10959-008-0152-z
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    References listed on IDEAS

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    1. Bertoin, Jean, 1995. "On the local rate of growth of Lévy processes with no positive jumps," Stochastic Processes and their Applications, Elsevier, vol. 55(1), pages 91-100, January.
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    Cited by:

    1. Pei-Sen Li & Xiaowen Zhou, 2023. "Integral Functionals for Spectrally Positive Lévy Processes," Journal of Theoretical Probability, Springer, vol. 36(1), pages 297-314, March.
    2. Grégoire Véchambre, 2023. "Almost Sure Behavior for the Local Time of a Diffusion in a Spectrally Negative Lévy Environment," Journal of Theoretical Probability, Springer, vol. 36(2), pages 876-925, June.

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