  The inverse first passage time problem asks whether, for a Brownian motion
$B$ and a nonnegative random variable $\zeta$, there exists a time-varying
barrier $b$ such that $\mathbb{P}\{B_s>b(s),0\leq s\leq
t\}=\mathbb{P}\{\zeta>t\}$. We study a "smoothed" version of this problem and
ask whether there is a "barrier" $b$ such that $
\mathbb{E}[\exp(-\lambda\int_0^t\psi(B_s-b(s))\,ds)]=\mathbb{P}\{\zeta >t\}$,
where $\lambda$ is a killing rate parameter, and $\psi:\mathbb{R}\to[0,1]$ is a
nonincreasing function. We prove that if $\psi$ is suitably smooth, the
function $t\mapsto \mathbb{P}\{\zeta>t\}$ is twice continuously differentiable,
and the condition $0<-\frac{d\log\mathbb{P}\{\zeta>t\}}{dt}<\lambda$ holds for
the hazard rate of $\zeta$, then there exists a unique continuously
differentiable function $b$ solving the smoothed problem. We show how this
result leads to flexible models of default for which it is possible to compute
expected values of contingent claims.
