First passage times over stochastic boundaries for subdiffusive processes

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.