  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.
