The double-barrier inverse first-passage problem for Wiener process with random starting point

We consider the double-barrier inverse first-passage time (IFPT) problem for Wiener process X(t), starting from a random position η. Let a<b such that P(a<η<b)=1, and F an assigned distribution function. The problem consists of finding the distribution of η such that the first-exit time of X(t) from the interval (a,b) has distribution F. Besides results for the Brownian motion with drift, we obtain some extensions to more general one-dimensional diffusions and we show how to find an approximate solution to the IFPT problem in the case of time varying barriers.