An inverse problem for the first-passage place of some diffusion processes with random starting point

We study an inverse problem for the first-passage place of a one-dimensional diffusion process X(t) (also with jumps), starting from a random position eta is an element of [a,b]. Let be tau(a,b) the first time at which X(t) exits the interval (a,b), and pi(a) = P(X(tau(a,b)) <= a) the probability of exit from the left of (a,b). Given a probability q is an element of (0, 1), the problem consists in finding the density g of eta (if it exists) such that pi(a) = q. Some explicit examples are reported.