On the Generalized Telegraph Process with Deterministic Jumps

Abstract

We consider a semi-Markovian generalization of the integrated telegraph process subject \ud to jumps. It describes a motion on the real line characterized by two alternating velocities \ud with opposite directions, where a jump along the alternating direction occurs at each \ud velocity reversal. We obtain the formal expressions of the forward and backward transition \ud densities of the motion. We express them as series in the case of Erlang-distributed random \ud times separating consecutive jumps. Furthermore, a closed form of the transition density \ud is given for exponentially distributed times, with constant jumps and random initial velocity. \ud In this case we also provide mean and variance of the process, and study the limiting \ud behaviour of the probability law, which leads to a mixture of three Gaussian densities

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 Telegraph Process with Random Jumps

“…If V (t) = c, which corresponds to the top diagram of Figure 2, then W (t) = w if and only if Y (w) = t − w. Hence, the subdensity ψ c (w, t) is equal to f Y (t − w, w). This gives the first identity of (12), whereas the second identity immediately follows from (4) and (8). Equation (13) is obtained by a similar reasoning and recalling that V (0) = c. The proof of (11) follows by differentiation of (10), and by rearranging the terms.…”

“…This process includes the occurrence of a deterministic jump along the alternating direction at each velocity reversal. Other results on the jump-telegraph process, which include some limit theorems, are provided in [8] and [14]. Moreover, an application to option pricing models of the telegraph process with random jumps has appeared recently in [15].…”

Telegraph Processes with Random Jumps and Complete Market Models

“…Such a model with deterministic velocities and jumps is studied in detail by [21,9]. Moreover, earlier we proposed the option pricing model based on jump-telegraph processes, [21].…”