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

Abstract

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