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

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.

On the exit time from open sets of some semi-Markov processes

In this paper we characterize the distribution of the first exit time from an arbitrary open set for a class of semi-Markov processes obtained as time-changed Markov processes. We estimate the asymptotic behaviour of the survival function (for large $t$) and of the distribution function (for small $t$) and we provide some conditions for absolute continuity. We have been inspired by a problem of neurophyshiology and our results are particularly usefull in this field, precisely for the so-called Leacky Integrate-and-Fire (LIF) models: the use of semi-Markov processes in these models appear to be realistic under several aspects, e.g., it makes the intertimes between spikes a r.v. with infinite expectation, which is a desiderable property. Hence, after the theoretical part, we provide a LIF model based on semi-Markov processes.