Before revision:  \mathbb R ^+\to[-\infty,+\infty] such that the survival function of the first-passage time \tau_b=\inf \{t\ge0:X(t)<b(t)\} is given by  H . In this paper  we consider a version of the IFPT problem where the barrier is fixed at zero and the problem is to find an initial distribution \mu and a time-change I such that for the time-changed process X\circ I the IFPT problem is solved by a constant barrier at the level zero.
After revision:  \mathbb R ^+\to[-\infty,+\infty] such that the survival function of the first-passage time \tau_b=\inf \{t\ge0:X(t)<b(t)\} is given by  H . In this paper , we consider a version of the IFPT problem where the barrier is fixed at zero and the problem is to find an initial distribution \mu and a time-change I such that for the time-changed process X\circ I the IFPT problem is solved by a constant barrier at the level zero.