On the distribution of the telegraph meander and its properties
Andrea Pedicone, Enzo Orsingher
TL;DR
The study analyzes the telegraph meander, the telegraph process conditioned to remain nonnegative on $[0,t]$, deriving its full law via a finite-velocity reflection principle that links it to the law of the underlying telegraph process with initial velocity $-c$. It provides explicit end-time densities, the characteristic function, moments, and a hyperbolic PDE governing the law, and establishes a Brownian meander as the weak limit under Kac's scaling, highlighting a deep connection between finite-velocity dynamics and Brownian barriers. The work also develops conditioned distributions on the number of Poisson events, revealing how increasing the Poisson activity concentrates the meander near the origin, and offers a potential finite-velocity alternative for meander models in applications like finance and geophysics.
Abstract
In this paper we present the distribution of the telegraph meander, a random function obtained by conditioning the telegraph process to stay above the zero level. The reflection principle for finite-velocity random motions allows the law of the telegraph meander to be expressed in terms of the spatial derivative of the law of the telegraph process with initial negative velocity. As a result, we are able to obtain the characteristic function, the moments and the hyperbolic equation that governs the law of the telegraph meander. Furthermore, we prove that Brownian meander is the weak limit of the telegraph meander.