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

## Author

Listed:
• Boris Ettinger
• Steven N. Evans
• Alexandru Hening

## 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}

## Suggested Citation

• Boris Ettinger & Steven N. Evans & Alexandru Hening, 2011. "Killed Brownian motion with a prescribed lifetime distribution and models of default," Papers 1111.2976, arXiv.org, revised Jan 2014.
• Handle: RePEc:arx:papers:1111.2976
as

File URL: http://arxiv.org/pdf/1111.2976

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