Asymptotic results for the absorption time of telegraph processes with elastic boundary at the origin

Abstract

We consider a telegraph process with elastic boundary at the origin studied recently in the literature. It is a particular random motion with finite velocity which starts at $x\ge 0$, and its dynamics is determined by upward and downward switching rates $\lambda$ and $\mu$, with $\lambda >\mu$, and an absorption probability (at the origin) $\alpha \in (0,1]$. Our aim is to study the asymptotic behavior of the absorption time at the origin with respect to two different scalings: $x\to \infty$ in the first case; $\mu \to \infty$, with $\lambda =\beta \mu$ for some $\beta >1$ and $x>0$, in the second case. We prove several large and moderate deviation results. We also present numerical estimates of $\beta$ based on an asymptotic Normality result for the case of the second scaling.