  We develop some sufficient conditions for the stochastic ordering between
hitting times, in a fixed state, for two Markov chains. In particular, we focus
attention on the so called \emph{skip-free} case. In the analysis of such a
case, we develop a special type of coupling. We also compare different types of
relations between two, non-necessarily skip-free, Markov chains on the same
state space. Such relations have a natural role in establishing the usual and
the asymptotic stochastic ordering between the probability distributions of
hitting times. Finally, we present some discussions and examples related with
words' occurrences.
