Change-level detection for Lévy subordinators

Let $\boldsymbol{X}=(X_t)_{t\ge 0}$ be a process behaving as a general increasing Lévy process (subordinator) prior to hitting a given unknown level $m_0$, then behaving as another different subordinator once this threshold is crossed. This paper addresses the detection of this unknown threshold $m_0\in [0,+\infty]$ from an observed trajectory of the process. These kind of model and issue are encountered in many areas such as reliability and quality control in degradation problems. More precisely, we construct, from a sample path and for each $\epsilon >0$, a so-called detection level $L_\epsilon$ by considering a CUSUM inspired procedure. Under mild assumptions, this level is such that, while $m_0$ is infinite (i.e. when no changes occur), its expectation $ \mathbb{E}_{\infty}(L_{\epsilon})$ tends to $+\infty$ as $\epsilon$ tends to $0$, and the expected overshoot $ \mathbb{E}_{m_0}([L_{\epsilon} - m_0]^+)$, while the threshold $m_0$ is finite, is negligible compared to $ \mathbb{E}_{\infty}(L_{\epsilon})$ as $\epsilon$ tends to $0$. Numerical illustrations are provided when the Lévy processes are gamma processes with different shape parameters.