  We consider the boundary crossing problem for time-homogeneous diffusions and
general curvilinear boundaries. Bounds are derived for the approximation error
of the one-sided (upper) boundary crossing probability when replacing the
original boundary by a different one. In doing so we establish the existence of
the first-passage time density and provide an upper bound for this function. In
the case of processes with diffusion interval equal to whole real line this is
extended to a lower bound, as well as bounds for the first crossing time of a
lower boundary. An extension to some time-inhomogeneous diffusions is given.
These results are illustrated by numerical examples.
