The Annals of Applied Probability

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 $0<-\frac{d\log\mathbb{P}\{\zeta>t\}}{dt}<\lambda$ 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.

Article information

Source
Ann. Appl. Probab., Volume 24, Number 1 (2014), 1-33.

Dates
First available in Project Euclid: 9 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1389278717

Digital Object Identifier
doi:10.1214/12-AAP902

Mathematical Reviews number (MathSciNet)
MR3161639

Zentralblatt MATH identifier
1328.60188

Subjects
Primary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 91G40: Credit risk 91G80: Financial applications of other theories (stochastic control, calculus of variations, PDE, SPDE, dynamical systems)

Keywords
Credit risk inverse first passage time problem killed Brownian motion Cox process stochastic intensity Feynman–Kac formula

Citation

Ettinger, Boris; Evans, Steven N.; Hening, Alexandru. Killed Brownian motion with a prescribed lifetime distribution and models of default. Ann. Appl. Probab. 24 (2014), no. 1, 1--33. doi:10.1214/12-AAP902. https://projecteuclid.org/euclid.aoap/1389278717


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