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 $xgeq 0$, and its dynamics is determined by upward and downward switching rates $lambda$ and $mu$, with $lambda>mu$, and an absorption probability (at the origin) $alphain(0,1]$. Our aim is to study the asymptotic behavior of the absorption time at the origin with respect to two different scalings: $xtoinfty$ in the first case; $mutoinfty$, with $lambda=betamu$ 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.
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