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 L\'evy 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 $(\mathbb{T}_a^{(\mathcal{b})}, (\mathbb{X} - \mathcal{b})_{\mathbb{T}_a^{(\mathcal{b})}})$ 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 $\mathcal{b}=(\mathcal{b}_t)_{t\geq 0}$ a (possibly degenerate) subordinator independent of $X$ and $\mathcal{K}$. We proceed by providing a detailed analysis of the cases where either $\mathcal{K}$ 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 $\mathbb{T}_a^{(\mathcal{b})}$ has the same law as the first passage time of a semi-regenerative process of L\'evy type, a terminology that we introduce to mean that this process satisfies the Markov property of L\'evy processes for stopping times whose graph is included in the associated regeneration set.