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

TL;DR

This paper analyzes the asymptotic behavior of the absorption time in a telegraph process with elastic boundary at the origin, providing large and moderate deviation results under different scalings.

Contribution

It introduces new asymptotic results for the absorption time of telegraph processes with elastic boundaries, including large and moderate deviation principles.

Findings

Large deviation results for absorption time as initial position grows

Moderate deviation results under parameter scaling

Numerical estimates of parameter  based on asymptotic normality

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 $x\geq 0$, 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.