The hitting time of zero for a stable process

Abstract

For any two-sided jumping $\alpha$-stable process, where $1<\alpha <2$, we find an explicit identity for the law of the first hitting time of the origin. This complements existing work in the symmetric case and the spectrally one-sided case; cf. Yano-Yano-Yor (2009) and Cordero (2010), and Peskir (2008) respectively. We appeal to the Lamperti-Kiu representation of Chaumont-Pantí-Rivero (2011) for real-valued self-similar Markov processes. Our main result follows by considering a vector-valued functional equation for the Mellin transform of the integrated exponential Markov additive process in the Lamperti-Kiu representation. We conclude our presentation with some applications.

References in corpus (6)

• Recurrent extensions of self-similar Markov processes and Cramér's condition II
• Weak convergence of positive self-similar Markov processes and overshoots of Lévy processes
• The hitting time of zero for a stable process
• Hitting distributions of alpha-stable processes via path censoring and self-similarity
• On extrema of stable processes
• A convergent series representation for the density of the supremum of a stable process

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• The strong law of large numbers and a functional central limit theorem for general Markov additive processes
• Bivariate Markov chains converging to Lamperti transform Markov Additive Processes
• Stable windings at the origin
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