  We propose a new approach to the problem of the first passage time. Our
method is applicable not only to the Wiener process but also to the
non--Gaussian L$\acute{\rm e}$vy flights or to more complicated stochastic
processes whose distributions are stable. To show the usefulness of the method,
we particularly focus on the first passage time problems in the truncated
L$\acute{\rm e}$vy flights (the so-called KoBoL processes), in which the
arbitrarily large tail of the L$\acute{\rm e}$vy distribution is cut off. We
find that the asymptotic scaling law of the first passage time $t$ distribution
changes from $t^{-(\alpha +1)/\alpha}$-law (non-Gaussian L$\acute{\rm e}$vy
regime) to $t^{-3/2}$-law (Gaussian regime) at the crossover point. This result
means that an ultra-slow convergence from the non-Gaussian L$\acute{\rm e}$vy
regime to the Gaussian regime is observed not only in the distribution of the
real time step for the truncated L$\acute{\rm e}$vy flight but also in the
first passage time distribution of the flight. The nature of the crossover in
the scaling laws and the scaling relation on the crossover point with respect
to the effective cut-off length of the L$\acute{\rm e}$vy distribution are
discussed.
