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 ζ, there exists a time-varying barrier b such that ℙ{Bs > b(s), 0 ≤ s ≥ t} = ℙ{ζ > t}. We study a "smoothed" version of this problem and ask whether there is a "barrier" b such that E[exp(-λ ∫0 ψ(Bs - b(s)) ds)] = ℙ{ζ > t}, where λ is a killing rate parameter, and ψ;: ℝ → [0, 1] is a nonincreasing function. We prove that if ψ is suitably smooth, the function t → ℙ{ζ > t} is twice continuously differentiable, and the condition 0 }/dt < λ holds for the hazard rate of ζ, 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. © Institute of Mathematical Statistics, 2014.