  Numerous applications all the way from biology and physics to economics
depend on the density of first crossings over a boundary. Motivated by the lack
of analytical tools for computing first-passage time densities (FPTDs) for
complex problems, we propose a new simple method based on the Independent
Interval Approximation (IIA). We generalise previous formulations of the IIA to
handle non-smooth processes, and derive a closed form expression for the FPTD
in Laplace and $z$-transform space for arbitrary boundary and starting points
in one dimension. We focus on Markov processes for which the IIA is exact. To
apply our equations, we calculate the FPTD in two cases: the Ornstein-Uhlenbeck
process and the discrete time Brownian walk. Our results are in good agreement
with Langevin dynamics simulations.
