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

July, 2012
Report Number: 
Boris Ettinger and Steven N. Evans and Alexandru Hening

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 $\mathbb{P}\{B_s > b(s), \, 0 \le s \le t\} = \mathbb{P}\{\zeta > t\}$. 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 non-increasing 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.

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