### Probability distribution function for the Euclidean distance between two telegraph processes

Alexander D. Kolesnik

#### Abstract

Consider two independent Goldstein-Kac telegraph processes X1(t) and X2(t) on the real line R. The processes Xk(t), k = 1, 2, describe stochastic motions at finite constant velocities c1 > 0 and c2 > 0 that start at the initial time instant t = 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 instant t > 0. Some numerical results are also presented.

#### Article information

Source
Adv. in Appl. Probab., Volume 46, Number 4 (2014), 1172-1193.

Dates
First available in Project Euclid: 12 December 2014

https://projecteuclid.org/euclid.aap/1418396248

Digital Object Identifier
doi:10.1239/aap/1418396248

Mathematical Reviews number (MathSciNet)
MR3290434

Zentralblatt MATH identifier
1248.32008

#### Citation

Kolesnik, Alexander D. Probability distribution function for the Euclidean distance between two telegraph processes. Adv. in Appl. Probab. 46 (2014), no. 4, 1172--1193. doi:10.1239/aap/1418396248. https://projecteuclid.org/euclid.aap/1418396248

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