  We propose an approach to compute the boundary crossing probabilities for a
class of diffusion processes which can be expressed as piecewise monotone (not
necessarily one-to-one) functionals of a standard Brownian motion. This class
includes many interesting processes in real applications, e.g.,
Ornstein-Uhlenbeck, growth processes and geometric Brownian motion with time
dependent drift. This method applies to both one-sided and two-sided general
nonlinear boundaries, which may be discontinuous. Using this approach explicit
formulas for boundary crossing probabilities for certain nonlinear boundaries
are obtained, which are useful in evaluation and comparison of various
omputational algorithms. Moreover, numerical computation can be easily done by
Monte Carlo integration and the approximation errors for general boundaries are
automatically calculated. Some numerical examples are presented.
