Higher-Order Regularity of the Free Boundary in the Inverse First-Passage Problem

Consider the inverse first-passage problem: Given a diffusion process \(\{\frak{X}_{t}\}_{t\geqslant 0}\) on a probability space \((\Omega,\mathcal{F},\mathbb{P})\) and a survival probability function \(p\) on \([0,\infty)\), find a boundary, \(x=b(t)\), such that \(p\) is the survival probability that \(\frak{X}\) does not fall below \(b\), i.e., for each \(t\geqslant 0\), \(p(t)= \mathbb{P}(\{\omega\in\Omega\;|\; {\frak{X}}_s(\omega) \geqslant b(s),\ \forall\, s\in(0,t)\})\). In earlier work, we analyzed viscosity solutions of a related variational inequality, and showed that they provided the only upper semi-continuous (usc) solutions of the inverse problem. We furthermore proved weak regularity (continuity) of the boundary \(b\) under additional assumptions on \(p\). The purpose of this paper is to study higher-order regularity properties of the solution of the inverse first-passage problem. In particular, we show that when \(p\) is smooth and has negative slope, the viscosity solution, and therefore also the unique usc solution of the inverse problem, is smooth. Consequently, the viscosity solution furnishes a unique classical solution to the free boundary problem associated with the inverse first-passage problem.