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

Consider [Formula: see text] independent Goldstein–Kac telegraph processes [Formula: see text] on the real line [Formula: see text]. Each process [Formula: see text] describes a stochastic motion at constant finite speed [Formula: see text] of a particle that, at the initial time instant [Formula: see text], starts from some initial point [Formula: see text] and whose evolution is controlled by a homogeneous Poisson process [Formula: see text] of rate [Formula: see text]. The governing Poisson processes [Formula: see text] are supposed to be independent as well. Consider the linear combination of the processes [Formula: see text] defined by [Formula: see text] where [Formula: see text] are arbitrary real nonzero constant coefficients.

We obtain a hyperbolic system of [Formula: see text] first-order partial differential equations for the joint probability densities of the process [Formula: see text] and of the directions of motions at arbitrary time [Formula: see text]. From this system we derive a partial differential equation of order [Formula: see text] for the transition density of [Formula: see text] in the form of a determinant of a block matrix whose elements are the differential operators with constant coefficients. Initial-value problems for the transition densities of the sum and difference [Formula: see text] of two independent telegraph processes with arbitrary parameters, are also posed.

On a jump-telegraph process driven by an alternating fractional Poisson process

The basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

Generalized Telegraph Process with Random Jumps

We consider a generalized telegraph process which follows an alternating renewal process and is subject to random jumps. More specifically, consider a particle at the origin of the real line at timet=0. Then it goes along two alternating velocities with opposite directions, and performs a random jump toward the alternating direction at each velocity reversal. We develop the distribution of the location of the particle at an arbitrary fixed timet, and study this distribution under the assumption of exponentially distributed alternating random times. The cases of jumps having exponential distributions with constant rates and with linearly increasing rates are treated in detail.

Probability Distribution Function for the Euclidean Distance Between Two Telegraph Processes

Consider two independent Goldstein-Kac telegraph processesX1(t) andX2(t) on the real line ℝ. The processesXk(t),k= 1, 2, describe stochastic motions at finite constant velocitiesc1> 0 andc2> 0 that start at the initial time instantt= 0 from the origin of ℝ and are controlled by two independent homogeneous Poisson processes of rates λ1> 0 and λ2> 0, respectively. We obtain a closed-form expression for the probability distribution function of the Euclidean distance ρ(t) = |X1(t) -X2(t)|,t> 0, between these processes at an arbitrary time instantt> 0. Some numerical results are also presented.

Generalized Telegraph Process with Random Delays

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 Y0(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).

A Generalized Telegraph Process with Velocity Driven by Random Trials

We consider a random trial-based telegraph process, which describes a motion on the real line with two constant velocities along opposite directions. At each epoch of the underlying counting process the new velocity is determined by the outcome of a random trial. Two schemes are taken into account: Bernoulli trials and classical Pólya urn trials. We investigate the probability law of the process and the mean of the velocity of the moving particle. We finally discuss two cases of interest: (i) the case of Bernoulli trials and intertimes having exponential distributions with linear rates (in which, interestingly, the process exhibits a logistic stationary density with nonzero mean), and (ii) the case of Pólya trials and intertimes having first gamma and then exponential distributions with constant rates.

The explicit probability distribution of the sum of two telegraph processes

We consider two independent Goldstein–Kac telegraph processes X1(t) and X2(t) on the real line ℝ, both developing with constant speed c > 0, that, at the initial time instant t = 0, simultaneously start from the origin 0 ∈ ℝ and whose evolutions are controlled by two independent homogeneous Poisson processes of the same rate λ > 0. Closed-form expressions for the transition density φ(x, t) and the probability distribution function Φ(x, t) = Pr {S(t) < x}, x ∈ ℝ, t > 0, of the sum S(t) = X1(t) + X2(t) of these processes at arbitrary time instant t > 0, are obtained. It is also proved that the shifted time derivative g(x, t) = (∂/∂t + 2λ)φ(x, t) satisfies the Goldstein–Kac telegraph equation with doubled parameters 2c and 2λ. From this fact it follows that φ(x, t) solves a third-order hyperbolic partial differential equation, but is not its fundamental solution. The general case is also discussed.

Generalized Telegraph Process with Random Delays

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).