  In this paper we present the distribution of the maximum of the telegraph
process in the cases where the initial velocity is positive or negative with an
even and an odd number of velocity reversals. For the telegraph process with
positive initial velocity a reflection principle is proved to be valid while in
the case of an initial leftward displacement the conditional distributions are
perturbed by a positive probability of never visiting the half positive axis.
  Various relationships are established among the mentioned four classes of
conditional distributions of the maximum.
  The unconditional distributions of the maximum of the telegraph process are
obtained for positive and negative initial steps as well as their limiting
behaviour. Furthermore the cumulative distributions and the general moments of
the conditional maximum are presented.
