  The aim of this paper is to investigate how the correlation properties of a
stationary Markovian stochastic processes affect the First Passage Time
distribution. First Passage Time issues are a classical topic in stochastic
processes research. They also have relevant applications, for example, in many
fields of finance such as the assessment of the default risk for firms' assets.
By using some explicit examples, in this paper we will show that the tail of
the First Passage Time distribution crucially depends on the correlation
properties of the process and it is independent from its stationary
distribution. When the process includes an infinite set of time-scales bounded
from above, the FPTD shows tails modulated by some exponential decay. In the
case when the process is power-law correlated the FPTD shows power-law tails
1/t^(alfa+1)/2 and therefore the moments <t^n> of the FPTD are finite only when
n< (alfa-1)/2. We will also show that such power-law behaviour is not merely
due to the fact that the process includes an infinite and unbounded set of
time-scales. Rather, the time-scale must enter the FPTD with weights that must
be distributed according to a power-law for large time-scales values. Finally,
we will give a general result connecting the FPTD of an additive stochastic
processes x(t) to the FPTD of a generic process y(t) related by a coordinate
transformation y=f(x) to the first one.
