Generalized Telegraph Process with Random Delays

Abstract

In this paper we study the distribution of the location, at time t, of a particle moving U time units upwards, V time units downwards, and W time units of no movement (idle). These are repeated cyclically, according to independent alternating renewals. The distributions of U , V , and W are absolutely continuous. The velocities are v = +1 upwards, v = −1 downwards, and v = 0 during idle periods. Let Y + (t), Y − (t), and Y 0 (t) denote the total time in (0, t) of movements upwards, downwards, and no movements, respectively. The exact distribution of Y + (t) is derived. We also obtain the probability law of X(t) = Y + (t) − Y − (t), which describes the particle's location at time t. Explicit formulae are derived for the cases of exponential distributions with equal rates, with different rates, and with linear rates (leading to damped processes).

On the Asymmetric Telegraph Processes

“…Beginning with the 1956 lecture notes by Kac [13], the telegraph processes and their numerous generalisations have been studied in great detail; see, e.g. [2], [5], [4], [7], [17], [18], [19], [20], [24], with applications in physics [23], biology [10], [11], ecology [16], and, more recently, in financial market modelling [21] (see also the bibliographies in these papers).…”

Generalized Bessel Functions in Terms of Generalized Hermite Polynomials

“…Still, there is room for further investigations this wide scenario offered by the nature of the Hermite polynomials. Furthermore, the generalized Hermite polynomials and the related special polynomials, cited above as Laguerre, Legendre and Chebyshev polynomials and different families of special functions, in particular the large class of functions recognized as belonging to the Bessel functions, can be efficiently employed to solve a large class of problems in field such as stochastic processes [21][22][23], particle physics [24,25], electromagnetisms [26][27][28], continuum mechanics [29,30], material sciences [31,32], transmission lines [33][34][35], building sciences [36][37][38] and applications in the field of special functions and orthogonal polynomials [39][40][41][42][43][44]. Further investigations will be carried out in the next future in other fields of interest.…”

Generalized Telegraph Process with Random Jumps

“…The second cycle then starts and the motion proceeds so on. In this case, similarly to (3) we define…”

“…Finally, we remark that the main technique adopted in this paper, which is based on the examination of underlying compound renewal processes, has been suggested by previous investigations (see [18] for the analysis of compound Poisson processes with jumps), and has recently been successfully exploited by Bshouty et al [3] for the study of a telegraph process with random delays. Furthermore, it should be noted that the results shown in the paper are also of interest to the analysis of growth-collapse models (see, for instance, [2]).…”