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

TL;DR

This paper investigates the higher-order regularity of the free boundary in the inverse first-passage problem, demonstrating that smooth survival probability functions lead to smooth and classical solutions for the boundary.

Contribution

It establishes that under smoothness and negativity conditions on the survival probability, the inverse problem's viscosity solution is actually smooth, providing a classical solution to the free boundary problem.

Findings

Viscosity solutions are smooth when p is smooth with negative slope.

The smoothness of p implies the boundary b is also smooth.

The viscosity solution becomes a classical solution under these conditions.

Abstract

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.