Analysis of an Inverse First Passage Problem from Risk Management

We study the following "inverse first passage time" problem. Given a diffusion process $X_{t}$ and a probability distribution $q$ on $[0,\infty)$, does there exist a boundary $b(t)$ such that $q(t)=\mathbb{P}[\tau\leq t]$, where τ is the first hitting time of $X_{t}$ to the time-dependent level $b(t)$? A free boundary problem for a parabolic partial differential operator is associated with the inverse first passage time problem. We prove the existence and uniqueness of a viscosity solution to this problem. We also investigate the small time behavior of the boundary $b(t)$, presenting both upper and lower bounds. Finally, we derive some integral equations characterizing the boundary.