First passage times over stochastic boundaries for subdiffusive processes

Abstract

Let $\mathbb{X}=({\mathbb{X}}_t{)}_{t\ge 0}$ be the subdiffusive process defined, for any $t\ge 0$, by ${\mathbb{X}}_t=X_{{\ell }_t}$ where $X=(X_t{)}_{t\ge 0}$ is a Lévy process and ${\ell }_t=\inf \{s>0;{\mathcal{K}}_s>t\}$ with $\mathcal{K}=({\mathcal{K}}_t{)}_{t\ge 0}$ 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^{(\mathcal{b})} = \inf \{t>0;\: \mathbb{X}_t > a+ \mathcal{b}_t \} \end{equation*} with $a\in \mathbb{R}$ and $b=(b_t{)}_{t\ge 0}$ a (possibly degenerate) subordinator independent of $X$ and $\mathcal{K}$. We proceed by providing a detailed analysis of the cases where either $\mathcal{K}$ is a stable subordinator or $X$ is spectrally negative. 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.