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-lin 查看全文>>