An inverse first-passage problem for one-dimensional diffusions with random starting point

We consider an inverse first-passage time (FPT) problem for a homogeneous one-dimensional diffusion X(t)X(t), starting from a random position ηη. Let S(t)S(t) be an assigned boundary, such that P(η≥S(0))=1P(η≥S(0))=1, and FF an assigned distribution function. The problem consists of finding the distribution of ηη such that the FPT of X(t)X(t) below S(t)S(t) has distribution FF. We obtain some generalizations of the results of Jackson et al., 2009, which refer to the case when X(t)X(t) is Brownian motion and S(t)S(t) is a straight line across the origin.