A Damped Telegraph Random Process with Logistic Stationary Distribution

Abstract

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

Generalized Telegraph Process with Random Delays

“…). A similar choice has been considered by Di Crescenzo and Martinucci [6] and Di Crescenzo et al [8], who studied a damped telegraph process and a damped geometric telegraph process, respectively. Since the parameters λi are linear increasing in i, the process X(t) exhibits a damped behavior, in the sense that its sample paths are composed of line segments that become stochastically smaller and smaller.…”

“…The assumption that the parameters of U i , V i , and W i are λi implies that the random times separating consecutive velocity reversals have the same distribution of the intertimes of a simple birth process (see [17] for instance). Due to Equation (11) of [6], the density and the distribution function of the n-fold convolution U (n) for n ≥ 1 are given by Making use of (4.14) and recalling (2.1), we thus obtain Moreover, due to (2.6) and (4.12), we also have p n (t) = (1 − e −λt ) n − (1 − e −λt ) n+1 = e −λt (1 − e −λt ) n , so that, recalling (2.8), the density of Y + (t) is given, for 0 < y < t, by (4.13).…”

Analysis and Applications of the Residual Varentropy of Random Lifetimes

“…for α ∈ R. Clearly, recalling (13), one has λ 0 (t) = λ(t) for all t. Other parameterizations of λ α (t) have been treated in Bieniek and Szpak [5] as a special case of the generalized failure rate defined by Barlow and van Zwet [4]. Further forms of generalized hazard rates have been considered in the past.…”

“…Making use of Eqs. (13) and (15), one has f (x) = λ(x)e −Λ(x) , so that the function introduced in (46) can be rewritten also as follows:…”

Some Results on the Telegraph Process Confined by Two Non-Standard Boundaries

“…Since the seminal papers by Goldstein [18] and Kac [20], many generalizations of the telegraph process have been proposed in the literature, such as the asymmetric telegraph process (cf. [1], [25]), the generalized telegraph process (see, for instance, [5], [7], [8], [9], [30], [36]) or the jump-telegraph process (for example, [10], [11], [24], [31], [32]). Other recent investigations have been devoted to suitable functionals of telegraph processes (cf.…”