On the Asymmetric Telegraph Processes

Abstract

We study the one-dimensional random motion X = X(t), t ≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formulae for the density functions of X and for the moments of any order, as well as the distributions of the first passage times, are obtained. The limit behaviour of the moments is analysed under nonstandard Kac's scaling.

Telegraph Process with Elastic Boundary at the Origin

“…Restricting the attention to some recent contributions, we also mention Beghin et al [3] and López and Ratanov [26] for the asymmetric telegraph process, Bogachev and Ratanov [5] for the distribution of the occupation time of the positive half-line for the telegraph process, Crimaldi et al [7] for a telegraph process driven by certain random trials, De Gregorio and Macci [8] for the large deviation principle applied to the telegraph process, Di Crescenzo and Martinucci [10] for a damped telegraph process, Fontbona et al [16] for the long-time behavior of an ergodic variant of the telegraph process, Stadje and Zacks [39] for the telegraph process with random velocities, Pogorui et al [30] for estimates of the number of level-crossings for the telegraph process, Di Crescenzo and Zacks [11] for the analysis of a generalized telegraph process perturbed by Brownian motion, De Gregorio and Orsingher [9] and Garra and Orsingher [18] for certain multidimensional extension of the telegraph process. Moreover, D'Ovidio et al [14] investigate other types of multidimensional extensions of the telegraph process, whose distribution is related to space-time fractional n-dimensional telegraph equations.…”

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

“…Hence, it is not hard to see that where lim SC means that the limit is performed under the scaling conditions that allow X c (t) to converge to the Wiener process with drift η = β − α and infinitesimal variance σ 2 , i.e. (see Section 5 of López and Ratanov [25])…”

“…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.…”

Linear combinations of the telegraph random processes driven by partial differential equations

“…The classical Goldstein-Kac telegraph process, first introduced in the works [10] and [14], describes the stochastic motion of a particle that moves at constant finite speed on the real line R and alternates two possible directions of motion at random Poisson time instants. The main properties of this process and its numerous generalizations, as well as some their applications, have been studied in a series of works [1][2][3][4][5][6][7][8][9], [11][12][13], [15,16], [19,20], [22][23][24][25][26][27][28][29], [31]. An introduction to the contemporary theory of the telegraph processes and their applications in financial modelling can be found in the recently published book [22].…”