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A temporal factorization at the maximum for certain positive self-similar Markov processes

Part of: Stochastic processes

Published online by Cambridge University Press:  23 November 2020

Matija Vidmar
Matija Vidmar*
Affiliation:
University of Ljubljana
*
*Postal address: Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska ulica 21, 1000 Ljubljana, Slovenia. Email: matija.vidmar@fmf.uni-lj.si
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Abstract

For a spectrally negative self-similar Markov process on $[0,\infty)$ with an a.s. finite overall supremum, we provide, in tractable detail, a kind of conditional Wiener–Hopf factorization at the maximum of the absorption time at zero, the conditioning being on the overall supremum and the jump at the overall supremum. In a companion result the Laplace transform of this absorption time (on the event that the process does not go above a given level) is identified under no other assumptions (such as the process admitting a recurrent extension and/or hitting zero continuously), generalizing some existing results in the literature.

Keywords

spectrally negative Lévy processespositive self-similar Markov processessplitting at the maximumWiener–Hopf factorizationmartingalestime-changes

MSC classification

Primary: 60G51: Processes with independent increments; L'evy processes 60G18: Self-similar processes
Secondary: 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 4 , December 2020 , pp. 1045 - 1069
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Copyright
© Applied Probability Trust 2020

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