  Several formulae for the probability that a Wiener process with stochastic
drift process and random variance crosses a one-sided stochastic boundary
process in a finite time interval are derived. A non-explicit formula is first
obtained by the Girsanov theorem when considering an equivalent probability
measure where the boundary is constant equal to its starting value and the
drift is null. A more explicit formula is then obtained by using a
decomposition of the Radon-Nikodym derivative inverse as the product of a
random variable known with information from the final value of the Wiener
process and an independent random variable. We also give an explicit formula
based on one theoretical approximation for applications. To apply the Girsanov
theorem, we assume that the drift minus the deviation of the boundary from its
starting value divided by the standard deviation is absolutely continuous and
that its derivative satisfies Novikov's condition. We also derive a formula in
the two-sided boundary case when the difference between the deviation of each
boundary from their starting value is linear.
