On the Exact Distributions of the Maximum of the Asymmetric Telegraph Process

TL;DR

This paper derives the exact distribution of the maximum of the asymmetric telegraph process over any time interval, revealing cyclic behaviors and series representations involving Bessel functions, with implications for understanding oscillations in stochastic models.

Contribution

It provides explicit formulas for the maximum distribution of the asymmetric telegraph process, including singular and unconditional cases, and links these to generalized Euler-Poisson-Darboux equations.

Findings

Distribution of maximum displays cyclic behavior for certain initial velocities

Unconditional maximum distribution expressed as series of Bessel functions

All distributions are special cases of symmetric telegraph process results

Abstract

In this paper we present the distribution of the maximum of the asymmetric telegraph process in an arbitrary time interval $[0,t]$ under the conditions that the initial velocity $V(0)$ is either $c_1$ or $-c_2$ and the number of changes of direction is odd or even. For the case $V(0) = -c_2$ the singular component of the distribution of the maximum displays an unexpected cyclic behavior and depends only on $c_1$ and $c_2$, but not on the current time $t$. We obtain also the unconditional distribution of the maximum for either $V(0) = c_1$ or $V(0) = -c_2$ and its expression has the form of series of Bessel functions. We also show that all the conditional distributions emerging in this analysis are governed by generalized Euler-Poisson-Darboux equations. We recover all the distributions of the maximum of the symmetric telegraph process as particular cases of the present paper. We underline that it rarely happens to obtain explicitly the distribution of the maximum of a process. For this reason the results on the range of oscillations of a natural process like the telegraph model make it useful for many applications.