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

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

Published in at http://dx.doi.org/10.1214/12-AAP902 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• On the inverse first-passage-time problem for a Wiener process
• Existence and uniqueness of solutions to the inverse boundary crossing problem for diffusions

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