  Given a survival distribution on the positive half-axis and a Brownian
motion, a solution of the inverse first-passage problem consists of a boundary
so that the first passage time over the boundary has the given distribution. We
show that the solution of the inverse first- passage problem coincides with the
solution of a related optimal stopping problem. Consequently, methods from
optimal stopping theory may be applied in the study of the inverse
first-passage problem. We illustrate this with a study of the associated
integral equation for the boundary.
