One-Dimensional Reflected Diffusions with Two Boundaries and an Inverse First-Hitting Problem

We study an inverse first-hitting problem for a one-dimensional, time-homogeneous diffusion X(t) reflected between two boundaries a and b, which starts from a random position eta. Let a <= S <= b be a given threshold, such that P(eta epsilon [a, S]) = 1, and F an assigned distribution function. The problem consists of finding the distribution of eta such that the first-hitting time of X to S has distribution F. This is a generalization of the analogous problem for ordinary diffusions, that is, without reflecting, previously considered by the author.