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The Upper Envelope of Positive Self-Similar Markov Processes

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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]).

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  1. Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 4, Place Jussieu, 75252, Paris Cedex 05, France

    J. C. Pardo

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  1. J. C. Pardo
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Correspondence to J. C. Pardo.

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Research supported by a grant from CONACYT (Mexico).

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Pardo, J.C. The Upper Envelope of Positive Self-Similar Markov Processes. J Theor Probab 22, 514–542 (2009). https://doi.org/10.1007/s10959-008-0152-z

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  • DOI: https://doi.org/10.1007/s10959-008-0152-z

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