First passage times over stochastic boundaries for subdiffusive processes
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- by C. Constantinescu, R. Loeffen and P. Patie PDF
- Trans. Amer. Math. Soc. 375 (2022), 1629-1652 Request permission
Abstract:
Let $\mathbb {X}=(\mathbb {X}_t)_{t\geq 0}$ be the subdiffusive process defined, for any $t\geq 0$, by $\mathbb {X}_t = X_{\ell _t}$ where $X=(X_t)_{t\geq 0}$ is a Lévy process and $\ell _t=\inf \{s>0; \mathcal {K}_s>t \}$ with $\mathcal {K}=(\mathcal {K}_t)_{t\geq 0}$ a subordinator independent of $X$. We start by developing a composite Wiener-Hopf factorization to characterize the law of the pair $(\mathbb {T}_a^{({b})}, (\mathbb {X} - {b})_{\mathbb {T}_a^{({b})}})$ where \begin{equation*}\mathbb {T}_a^{({b})} = \inf \{t>0; \mathbb {X}_t > a+ {b}_t \} \end{equation*} with $a \in \mathbb {R}$ and ${b}=({b}_t)_{t\geq 0}$ a (possibly degenerate) subordinator independent of $X$ and $\mathcal {K}$. We proceed by providing a detailed analysis of the cases where either $\mathbb {X}$ is a self-similar or is spectrally negative. For the later, we show the fact that the process $(\mathbb {T}_a^{({b})})_{a\geq 0}$ 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 $\mathbb {T}_a^{({b})}$ 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.References
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Additional Information
- C. Constantinescu
- Affiliation: Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3 BX, United Kingdom
- MR Author ID: 894153
- Email: C.Constantinescu@liverpool.ac.uk
- R. Loeffen
- Affiliation: University of Manchester, School of Mathematics, Manchester M13 9PL, United Kingdom
- MR Author ID: 821022
- Email: ronnie.loeffen@manchester.ac.uk
- P. Patie
- Affiliation: Cornell University, School of Operations Research and Information Engineering, 220 Rhodes Hall, Ithaca, New York 14853
- MR Author ID: 702262
- ORCID: 0000-0003-4221-0439
- Email: ppatie@cornell.edu
- Received by editor(s): January 27, 2021
- Received by editor(s) in revised form: June 4, 2021
- Published electronically: January 10, 2022
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 375 (2022), 1629-1652
- MSC (2020): Primary 60K15, 60G40; Secondary 60G51, 60G52, 60G18
- DOI: https://doi.org/10.1090/tran/8534
- MathSciNet review: 4378073