Numerical determination of hitting time distributions from their Laplace transforms: One dimensional diffusions

In a previous paper we studied a method to determine the probability density of barrier crossing times by a Brownian motion from the knowledge of its Laplace transform. This knowledge combined with the method of maximum entropy yields quite good reconstructions. The aim of this work is to extend the previous analysis in two directions. On one hand, we consider diffusions with non constant coefficients. This forces us to determine the Laplace transform numerically or by means of simulations. On the other hand, and this is the gist of this note, as numerical problems involve errors, we consider as well two possible extensions of the maximum entropy procedure which allow us to incorporate those errors into the probability reconstruction process.