Economy – Quantitative Finance – Risk Management
Scientific paper
2011-11-13
Economy
Quantitative Finance
Risk Management
26 pages, 1 figure
Scientific paper
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.
Ettinger Boris
Evans Steven N.
Hening Alexandru
No associations
LandOfFree
Killed Brownian motion with a prescribed lifetime distribution and models of default does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Killed Brownian motion with a prescribed lifetime distribution and models of default, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Killed Brownian motion with a prescribed lifetime distribution and models of default will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-340313