The Inverse First Passage Time Problem for killed Brownian motion

The classical inverse first passage time problem asks whether, for a Brownian motion $(B_t)_{t\geq 0}$ and a positive random variable $\xi$, there exists a barrier $b:\mathbb{R}_+\to\mathbb{R}$ such that $\mathbb{P}\{B_s>b(s), 0\leq s \leq t\}=\mathbb{P}\{\xi>t\}$, for all $t\geq 0$. We study a variant of the inverse first passage time problem for killed Brownian motion. We show that if $\lambda>0$ is a killing rate parameter and $\mathbb{1}_{(-\infty,0]}$ is the indicator of the set $(-\infty,0]$ then, under certain compatibility assumptions, there exists a unique continuous function $b:\mathbb{R}_+\to\mathbb{R}$ such that $\mathbb{E}\left[-\lambda \int_0^t \mathbb{1}_{(-\infty,0]}(B_s-b(s))\,ds\right] = \mathbb{P}\{\zeta>t\}$ holds for all $t\geq 0$. This is a significant improvement of a result of the first two authors (Annals of Applied Probability 24(1):1--33, 2014). The main difficulty arises because $\mathbb{1}_{(-\infty,0]}$ is discontinuous. We associate a semi-linear parabolic partial differential equation (PDE) coupled with an integral constraint to this version of the inverse first passage time problem. We prove the existence and uniqueness of weak solutions to this constrained PDE system. In addition, we use the recent Feynman-Kac representation results of Glau (Finance and Stochastics 20(4):1021--1059, 2016) to prove that the weak solutions give the correct probabilistic interpretation.