Killed Brownian motion with a prescribed lifetime distribution and models of default

Boris Ettinger, Steven N. Evans and Alexandru Hening

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Abstract: 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 t\}}{dt}

Date: 2011-11, Revised 2014-01
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Published in Annals of Applied Probability 2014, Vol. 24, No. 1, 1-33

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Persistent link: https://EconPapers.repec.org/RePEc:arx:papers:1111.2976

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