Constantinescu, C, Loeffen, R 
ORCID: 0000-0002-8461-6288 and Patie, P
(2022)
FIRST PASSAGE TIMES OVER STOCHASTIC BOUNDARIES FOR SUBDIFFUSIVE PROCESSES
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, 375 (3).
pp. 1629-1652.
ISSN 0002-9947, 1088-6850
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Abstract
Let X = (X<inf>t</inf>)<inf>t</inf>≥<inf>0</inf> be the subdiffusive process defined, for any t ≥ 0, by X<inf>t</inf> = X<inf>t</inf> where X = (X<inf>t</inf>)<inf>t</inf>≥<inf>0</inf> is a Lévy process and ι<inf>t</inf> = inf {s > 0; K<inf>s</inf> > t} with K = (K<inf>t</inf>)<inf>t</inf>≥<inf>0</inf> a subordinator independent of X. We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair (T(<inf>a</inf>b), (X − b)<inf>T</inf>(b)) where a T(<inf>a</inf>b) = inf {t > 0; X<inf>t</inf> > a + b<inf>t</inf>} with a ∈ R and b = (b<inf>t</inf>)<inf>t</inf>≥<inf>0</inf> a (possibly degenerate) subordinator independent of X and K. We proceed by providing a detailed analysis of the cases where either X is a self-similar or is spectrally negative. For the later, we show the fact that the process (T(<inf>a</inf>b) )<inf>a</inf>≥<inf>0</inf> is a subordinator. Our proofs hinge on a variety of techniques including excursion theory, change of measure, asymptotic analysis and on establishing a link between subdiffusive processes and a subclass of semi-regenerative processes. In particular, we show that the variable T(<inf>a</inf>b) has the same law as the first passage time of a semi-regenerative process of Lévy type, a terminology that we introduce to mean that this process satisfies the Markov property of Lévy processes for stopping times whose graph is included in the associated regeneration set.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | First passage time problems, subdiffusive diffusions, Wiener-Hopf factorization, Levy processes, time-changed, inverse subordinator, semi-regenerative processes, long-range dependence, ruin probability, stable processes |
| Divisions: | Faculty of Science & Engineering > School of Physical Sciences |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 18 Oct 2021 07:47 |
| Last Modified: | 16 Jun 2026 11:20 |
| DOI: | 10.1090/tran/8534 |
| Related Websites: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3140517 |
| Disclaimer: | The University of Liverpool is not responsible for content contained on other websites from links within repository metadata. Please contact us if you notice anything that appears incorrect or inappropriate. |
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