Change-level detection for Levy subordinators

Let X = (Xt)t>0 be a process behaving as a general increasing Levy process (subordinator) prior to hitting a given unknown level m0, then behaving as another different subordinator once this threshold is crossed. This paper addresses the detection of this unknown threshold m0 is an element of [0, +infinity] 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 m0 is infinite (i.e. when no changes occur), its expectation E infinity(L epsilon) tends to +infinity as epsilon tends to 0, and the expected overshoot Em0([L epsilon -m0]+), while the threshold m0 is finite, is negligible compared to E infinity(L epsilon) as epsilon tends to 0. Numerical illustrations are provided when the Levy processes are gamma processes with different shape parameters. (c) 2022 Elsevier B.V. All rights reserved.