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

TL;DR

This paper addresses a smoothed inverse first passage time problem for Brownian motion, establishing conditions for the existence and uniqueness of a barrier function that models default times with applications in financial modeling.

Contribution

It introduces a novel smoothed inverse first passage time framework with existence and uniqueness results under specific smoothness and hazard rate conditions.

Findings

Existence and uniqueness of the barrier function under smoothness and hazard rate conditions.

Framework allows modeling default times with computable expected contingent claim values.

Provides a flexible approach for default modeling in financial contexts.

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.