Existence and uniqueness of solutions to the inverse boundary crossing   problem for diffusions
  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. 
Xinfu Chen, Lan Cheng, John Chadam, David Saunders
math.PR