First passage times over stochastic boundaries for subdiffusive processes

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...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2022-03, Vol.375 (3), p.1629-1652
Main Authors: Constantinescu, C., Loeffen, R., Patie, P.
Format: Article
Language:English
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/tran/8534