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

Abstract

Consider n independent Goldstein-Kac telegraph processes X 1 (t), . . . , Xn(t), n ≥ 2, t ≥ 0, on the real line R. Each process X k (t), k = 1, . . . , n, describes a stochastic motion at constant finite speed c k > 0 of a particle that, at the initial time instant t = 0, starts from some initial point x 0 k = X k (0) ∈ R and whose evolution is controlled by a homogeneous Poisson process N k (t) of rate λ k > 0. The governing Poisson processes N k (t), k = 1, . . . , n, are supposed to be independent as well. Consider the linear combination of the processes X 1 (t), . . . , Xn(t), n ≥ 2, defined bywhere a k , k = 1, . . . , n, are arbitrary real nonzero constant coefficients.We obtain a hyperbolic system of 2 n first-order partial differential equations for the joint probability densities of the process L(t) and of the directions of motions at arbitrary time t > 0. From this system we derive a partial differential equation of order 2 n for the transition density of L(t) 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 S ± (t) = X 1 (t) ± X 2 (t) of two independent telegraph processes with arbitrary parameters, are also posed.

Some results on the telegraph process driven by gamma components

“…The final part of the proof is devoted to the investigation of the convergence of the series of Humbert functions appearing in the right-hand side of Equation (32). We start from the integral representation of the Humbert function 2 (cf.…”

“…Proof. Let α = m ∈ N in Equation (32). The discrete component can be obtained from Equations (18) and (19).…”

“…In particular, we provide an explicit expression for the distribution function of the square of X t , under the assumption of exponential distribution for the upward intertimes U and gamma distribution for the downward intertimes D. We recall that the square of the standard integrated telegraph process has been studied in [35], where its relationship with the square of the Brownian motion is also stressed. See also [33,Chapter 7] and [32] for other relevant functionals of the telegraph processes.…”

Quantitative control of Wasserstein distance between Brownian motion and the Goldstein--Kac telegraph process

“…The telegraph process has been proposed as an alternative to diffusion models and, as such, extensively studied by the probability and physics communities. Its generalisations are ubiquitous in applications, including transport phenomena in physical and biological systems, and it has produced a vast mathematics literature; standard references include [5,12,14,15,19,20,21,22,23,24,26,27,28,29,30,37,40,46] and further references may be found from therein. It is also used in the context of risk theory and to model financial markets, see [16], [29] and [38].…”

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

“…Other recent investigations have been devoted to suitable functionals of telegraph processes (cf. [21], [22] and [27]). See also Tilles and Petrovskii [34] for recent results on the reaction-telegraph process.…”