  We study the inverse boundary crossing problem for diffusions. Given a
diffusion process $X_t$, and a survival distribution $p$ on $[0,\infty)$, we
demonstrate that there exists a boundary $b(t)$ such that $p(t)=\mathbb{P}[\tau
>t]$, where $\tau$ is the first hitting time of $X_t$ to the boundary $b(t)$.
The approach taken is analytic, based on solving a parabolic variational
inequality to find $b$. Existence and uniqueness of the solution to this
variational inequality were proven in earlier work. In this paper, we
demonstrate that the resulting boundary $b$ does indeed have $p$ as its
boundary crossing distribution. Since little is known regarding the regularity
of $b$ arising from the variational inequality, this requires a detailed study
of the problem of computing the boundary crossing distribution of $X_t$ to a
rough boundary. Results regarding the formulation of this problem in terms of
weak solutions to the corresponding Kolmogorov forward equation are presented.

Probability