## The Annals of Applied Probability

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

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

Boris Ettinger, Steven N. Evans, and 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 $0<-\frac{d\log\mathbb{P}\{\zeta>t\}}{dt}<\lambda$ 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