  This paper considers the class of L\'evy processes that can be written as a
Brownian motion time changed by an independent L\'evy subordinator. Examples in
this class include the variance gamma model, the normal inverse Gaussian model,
and other processes popular in financial modeling. The question addressed is
the precise relation between the standard first passage time and an alternative
notion, which we call first passage of the second kind, as suggested by Hurd
(2007) and others. We are able to prove that standard first passage time is the
almost sure limit of iterations of first passage of the second kind. Many
different problems arising in financial mathematics are posed as first passage
problems, and motivated by this fact, we are lead to consider the implications
of the approximation scheme for fast numerical methods for computing first
passage. We find that the generic form of the iteration can be competitive with
other numerical techniques. In the particular case of the VG model, the scheme
can be further refined to give very fast algorithms.
